tag:blogger.com,1999:blog-59697752041434895762019-11-17T09:15:36.735-08:00Drill for Exam Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-5969775204143489576.post-19172405179883706952019-01-12T22:23:00.002-08:002019-01-17T03:27:09.813-08:00$\def\D{\displaystyle}$<br />1 (Edexcel, Further Pure Math, 2013 jan, Paper 2, No 2)<br />Using the identities $\D<br />\sin (A + B) = \sin A \cos B + \cos A \sin B \\<br />\cos (A + B) = \cos A \cos B – \sin A \sin B \\<br />\tan A=\D\frac{\sin A}{\cos A}$<br />(a) show that $\D \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$ (3)<br />(b) Hence show that<br />(i) $\D \tan 105^{\circ}=\frac{1+\sqrt{3}}{1-\sqrt{3}}$<br />(ii) $\D \tan 15^{\circ}=\frac{\sqrt{3}-1}{1+\sqrt{3}}$ (4)<br /><br />Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-75869759614190842972019-01-04T23:05:00.000-08:002019-01-04T23:05:28.196-08:00Probability (Edexcel Mathematic B)$\def\D{\displaystyle}$<br />1.[Edexcel 2012,winter,mathematic B, Paper 01, no 21]<br />&nbsp;A jar contains 3 red sweets, 4 blue sweets and 7 yellow sweets. One sweet is taken, at random, from the jar and not replaced. Another sweet is then taken, at random, from the jar. A tree diagram representing these two events is shown below.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-KNVXK5lsla8/XDAp_uqSfdI/AAAAAAAACSI/Ssc3fpKLTmgNFvFl8xiM7mh1426TeNmCQCEwYBhgL/s1600/edprobability01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="894" data-original-width="679" height="400" src="https://3.bp.blogspot.com/-KNVXK5lsla8/XDAp_uqSfdI/AAAAAAAACSI/Ssc3fpKLTmgNFvFl8xiM7mh1426TeNmCQCEwYBhgL/s400/edprobability01.png" width="303" /></a></div>(a) Complete the tree diagram representing these two events. (2)<br />(b) Find the probability that both sweets are red. Give your answer as a simplified fraction.<br />..............................................................(2)<br /><br />2[Edexcel 2012,winter, mathematic B, Paper 02, no 5]<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-hss8XIUNV3k/XDAp_1rf5FI/AAAAAAAACSM/34Q7gLsAu18KQe56d9Bu5XYITbYfUS0WACEwYBhgL/s1600/edprobability02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="475" data-original-width="616" height="307" src="https://4.bp.blogspot.com/-hss8XIUNV3k/XDAp_1rf5FI/AAAAAAAACSM/34Q7gLsAu18KQe56d9Bu5XYITbYfUS0WACEwYBhgL/s400/edprobability02.png" width="400" /></a></div>Figure 2 shows a diagram of routes to a factory. There are four road junctions labelled $\D A, B, C$ and $\D D$ and four towns labelled $\D W, X, Y$ and $\D Z.$ Mr Driver is approaching junction $\D A$ from Town $\D W,$ as shown, when he realises that he does not know how to get to the factory. He decides that at each road junction he will choose a road to take at random, but he will not turn around and go back along the road he has just travelled.<br />(a) Write down the probability that Mr Driver will choose the direct road to the factory at road junction $\D A.$ (1)<br />(b) Show that the probability that Mr Driver will pass through exactly two road junctions and reach the factory is $\D \frac{5}{18}.$ (3)<br />During the journey, if Mr Driver takes the road towards Town $\D X,$ the road towards Town $\D Y$ or the road towards Town $\D Z$ he will not arrive at the factory.<br />(c) Find the probability that Mr Driver will not arrive. (3)<br /><br />3. [Edexcel 2012, Summer, Mathematics, Paper 01, No. 26]<br />A bag contains 3 red balls and 5 black balls. Two balls are to be taken at random, without replacement, from the bag.<br />(a) Complete the probability tree diagram.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-hyInf_QK340/XDAqAmeLk9I/AAAAAAAACSs/S8jGz3IV3SQpVBZRHB7PqKb4rAXtBLNyQCEwYBhgL/s1600/edprobability03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="637" data-original-width="737" height="345" src="https://1.bp.blogspot.com/-hyInf_QK340/XDAqAmeLk9I/AAAAAAAACSs/S8jGz3IV3SQpVBZRHB7PqKb4rAXtBLNyQCEwYBhgL/s400/edprobability03.png" width="400" /></a></div>(b) Find the probability that the two balls taken are of the same colour.<br />. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2)<br /><br />4. [Edexcel 2012, Summer, mathematic B, Paper 02, no 5]<br />Iftekhar travels to work each day either by bus or by train. The probability that he takes the bus is 4/5. If he takes the bus, the probability that he buys a newspaper is 3/4. If he takes the train, the probability that he buys a newspaper is 2/3.<br />(a) Draw a tree diagram to represent this information. (4)<br />(b) Calculate the probability that one particular day, Iftekhar will not buy a newspaper. (3)<br /><br />5. [Edexcel 2013, winter, Mathematics, Paper 01, No. 25]<br />There are 100 coloured discs in a bag. Of these, 25 are brown, 40 are green and the others are neither brown nor green. A disc is to be chosen at random from the bag.<br />(a) Calculate the probability that the disc is either brown or green.<br />..................................................(2)<br />This disc is then returned to the bag. Two discs are now to be chosen at random from the bag without replacement.<br />(b) Calculate the probability that one disc will be brown and one disc will be green.<br />..................................................(3)<br /><br />6. [Edexcel 2013, Winter, mathematic B, Paper 02, no 6]<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FV5mnKa7WoM/XDAqBOQ7kDI/AAAAAAAACS0/-DVjSryMoNQVRscPp1Y6muw36j5Q0DuUQCEwYBhgL/s1600/edprobability06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="641" data-original-width="877" height="291" src="https://2.bp.blogspot.com/-FV5mnKa7WoM/XDAqBOQ7kDI/AAAAAAAACS0/-DVjSryMoNQVRscPp1Y6muw36j5Q0DuUQCEwYBhgL/s400/edprobability06.png" width="400" /></a></div>A survey was carried out into the time it took students to travel to school on Monday. Information about the results of this survey is shown in the histogram in Figure 2. No student took more than 70 minutes to travel to school. 35 students took between 30 minutes and 40 minutes to travel to school.<br />(a) Calculate how many students took part in the survey. (4)<br />One of these students is to be chosen at random.<br />(b) Calculate the probability that this student took more than 30 minutes to travel to school. (2)<br />A similar survey was carried out on Tuesday and the results were compared with those of Monday’s survey.<br />On Tuesday, 8 fewer students took less than 10 minutes to travel to school. The number of students that took between 10 minutes and 30 minutes to travel to school<br />was the same on both Monday and Tuesday. 3 more students took between 30 minutes and 40 minutes to travel to school, 5 fewer students took more than 40 minutes to travel to school. No student took more than 70 minutes to travel to school. One of the students from Tuesday’s survey is to be chosen at random.<br />(c) Calculate the probability that this student took more than 30 minutes to travel to school. (3)<br /><br /><br />7. [Edexcel 2013, Summer, Mathematics B, Paper 01, No. 22]<br />There are only red and blue counters in a bag. When a counter is taken at random from the bag, the probability that the counter is blue is $\D \frac{2}{5}$.&nbsp; Given that there are 60 counters in the bag,<br />(a) find the number of blue counters in the bag.<br />.............................................................. (2)<br />Some more blue counters are added to the 60 counters already in the bag. The number of extra blue counters added is $\D x.$ When a counter is now taken at random from the bag, the probability that the counter is blue is $\D \frac{1}{2}.$<br />(b) Find the value of $\D x.$<br />$\D x =$ ..............................................................(2)<br /><br />8. [Edexcel 2013, Summer, Mathematics B, Paper 02, No. 8]<br />At Trafalgar High school 120 students took examinations in Mathematics $\D (M),$ English $\D (E)$ and Science $\D (S).$ Every student passed at least one of these subjects and $\D x$ pupils passed all three subjects. 25 students passed both Mathematics and English.<br />(a) Write down an expression in terms of $\D x$ for the number of students who passed both Mathematics and English but not Science. (1)<br />Given that<br />18 students passed both Mathematics and Science<br />17 students passed both English and Science<br />21 students passed Mathematics only<br />22 students passed English only<br />37 students passed Science only<br />(b) show all this information on Figure 4. (3)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-MbI6JaH8b_8/XDAqBZ07nwI/AAAAAAAACSw/4GGm8-H1MW4oH5C74BRfSjA16aGV6rF7wCEwYBhgL/s1600/edprobability08.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="438" data-original-width="497" height="352" src="https://2.bp.blogspot.com/-MbI6JaH8b_8/XDAqBZ07nwI/AAAAAAAACSw/4GGm8-H1MW4oH5C74BRfSjA16aGV6rF7wCEwYBhgL/s400/edprobability08.png" width="400" /></a></div>(c) Find the value of $\D x.$ (2)<br />(d) Find the value of<br />(i) n$\D (M \cup S)$<br />(ii) n$\D (M \cap E \cap S' )$ (2)<br />A student is to be chosen at random from the 120 who took examinations in Mathematics, English and Science.<br />(e) Given that this student passed the Science examination, find the probability that the student also passed the English examination. (3)<br /><br /><br /><br />9. [Edexcel 2013, Summer, Mathematics B, Paper 01R, No. 11]<br />A bag contains 15 red balls and 20 black balls. Balls are to be taken out of the bag at random, one at a time and not replaced. Find the probability that<br />(a) the first ball taken out of the bag is red,<br />..............................................................(1)<br />(b) the first two balls taken out of the bag are both red.<br />..............................................................(2)<br /><br />10. [Edexcel 2014, winter, Mathematics B, Paper 01, No. 15]<br />A box contains balls of different colours. The box is opened and a ball is selected at random. The probability that the ball is white is 0.9 and the probability that the ball is black is 0.04<br />(a) Write down the probability that the ball is either white or black.<br />..............................................................(1)<br />(b) Find the probability that the ball is neither white nor black.<br />..............................................................(2)<br /><br />11. [Edexcel 2014, winter, Mathematics B, Paper 02, No. 6]<br />On school days, Fatima goes to school by bus. The probability that it will rain on a school day is $\D \frac{2}{7}.$ When it rains, the probability that the bus will be late is $\D \frac{1}{5}.$ When it does not rain, the probability that the bus will not be late is $\D \frac{5}{6}.$<br />(a) Complete the probability tree diagram.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-qCZem5jA5gQ/XDAqBulz3SI/AAAAAAAACS4/ItbmJNZvjmQvSa8nE3aY4vFHY1NvRj_iACEwYBhgL/s1600/edprobability11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="973" data-original-width="810" height="400" src="https://4.bp.blogspot.com/-qCZem5jA5gQ/XDAqBulz3SI/AAAAAAAACS4/ItbmJNZvjmQvSa8nE3aY4vFHY1NvRj_iACEwYBhgL/s400/edprobability11.png" width="332" /></a></div>Calculate the probability that on a school day,<br />(b) it will be raining and the bus will be late, (2)<br />(c) the bus will be late. (3)<br /><br />12. [Edexcel 2014, winter, Mathematics, Paper 02R, No. 7]<br />A sports club has 80 members. For the three activities Swimming $\D (S),$ Cycling $\D (C)$ and Running $\D (R),$<br />8 members take part in all three activities,<br />3 members do not take part in any of the three activities,<br />22 members take part in only Swimming,<br />23 members take part in Swimming and Cycling,<br />19 members take part in Swimming and Running,<br />14 members take part in Cycling and Running.<br />(a) Using this information place the number of members in the appropriate subsets of the Venn diagram. (3)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Mu6CEfXcTW8/XDAqCMPiQKI/AAAAAAAACS8/FFpScXoJc6MA0xvlonA1EzJGNfnF5z4FwCEwYBhgL/s1600/edprobability12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="487" data-original-width="768" height="252" src="https://2.bp.blogspot.com/-Mu6CEfXcTW8/XDAqCMPiQKI/AAAAAAAACS8/FFpScXoJc6MA0xvlonA1EzJGNfnF5z4FwCEwYBhgL/s400/edprobability12.png" width="400" /></a></div>The number of members who take part in only Cycling is twice the number of members who take part in only Running. Let the number of members who take part in only Running be $\D x$ and, using all the given information,<br />(b) form an equation in $\D x.$ (1)<br />(c) Solve your equation to find the value of $\D x.$ (2)<br /><br />13. [Edexcel 2014, Summer, mathematic B, Paper 01, no 27]<br />An archer shoots an arrow at a target. The probability that he will hit the target is 3/4. After the first shot, the target is moved further away from the archer. The archer shoots a second arrow at the target and the probability that he will hit the target is now 3/5.<br />(a) Complete the probability tree diagram.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-E1V9nn8hhKQ/XDAqCRC35mI/AAAAAAAACTA/yCgKqmniFsggDh37u6CX4HlYGc-Ov2KDwCEwYBhgL/s1600/edprobability13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="535" data-original-width="726" height="293" src="https://4.bp.blogspot.com/-E1V9nn8hhKQ/XDAqCRC35mI/AAAAAAAACTA/yCgKqmniFsggDh37u6CX4HlYGc-Ov2KDwCEwYBhgL/s400/edprobability13.png" width="400" /></a></div>Calculate the probability that the archer will<br />(b) hit the target with his first shot but miss the target with his second shot,<br />.......................................................(2)<br />(c) hit the target at least once if he takes both shots.<br />.......................................................(3)<br /><br /><br />14. [Edexcel 2014, Summer, mathematic B, Paper 02, no 6]<br />There are 159 people living in a street. The table below shows information about the number of people living in each of 30 houses in the street.<br />$\D\begin{array}{|c|c|}\hline<br />\mbox{Number (n) of people}&amp;\mbox{Number of houses with n}\\<br />\mbox{living in a house}&amp;\mbox{people living in the house}\\<br />\hline<br />1&amp; 2\\<br />2&amp; 3\\<br />3 &amp;1\\<br />4 &amp;4\\<br />5 &amp;3\\<br />6 &amp;6\\<br />7 &amp;8\\<br />8 &amp;2\\<br />9 &amp;1\\<br />\hline<br />\end{array}$<br />(a) Find<br />(i) the modal number of people living in a house,<br />(ii) the median number of people living in a house,<br />(iii) the mean number of people living in a house.<br />(5)<br />Two houses in the street are chosen at random.<br />(b) Calculate the probability that 4 people live in one of the houses and 2 people live in the other of the houses. (2)<br />One of the people living in the street is chosen at random.<br />(c) Find the probability that this person lives in a house in which at least 5 people live. (2)<br /><br />15. [Edexcel 2014, Summer, mathematic B, Paper 02R, no 5]<br />A bag contains 10 counters. Of these counters, 7 are black and 3 are white. Two of these counters are to be taken at random, without replacement, from the bag.<br />(a) Complete the probability tree diagram. (3)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-66WlNAJCMYw/XDAp_pgAJdI/AAAAAAAACSo/daob86TKEG0sHc5IyrRN57h0dr_rOn7gQCEwYBhgL/s1600/edprobabilit15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="764" data-original-width="759" height="400" src="https://4.bp.blogspot.com/-66WlNAJCMYw/XDAp_pgAJdI/AAAAAAAACSo/daob86TKEG0sHc5IyrRN57h0dr_rOn7gQCEwYBhgL/s400/edprobabilit15.png" width="396" /></a></div>(b) Find the probability that the two counters taken are of different colours. (3)<br /><div><br /></div><div><div>Answer</div><div>1.(a) $\D \frac{4}{13},\frac{7}{13}$</div><div>$\D \frac{3}{13},\frac{6}{13}$</div><div>(b) $\D \frac{3}{14}\times\frac{2}{13}$</div><div>2(a) $\D \frac{1}{3}$</div><div>(b) $\D \frac{1}{3}\times\frac{1}{3}+\frac{1}{3}\times\frac{1}{2}$</div><div>(c) $D \frac{1}{3}$</div><div>3(a) $\D 5/8,3/7,4/7$</div><div>(b) $\D \frac{13}{28}$</div><div>4 (a)%fig<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-hpM_J4pyK9M/XDBWI9f9zqI/AAAAAAAACTI/4eyWutVqHP0NYzc5EZe7Abjpv1f8gkjswCEwYBhgL/s1600/edprobabilit04ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="475" height="266" src="https://4.bp.blogspot.com/-hpM_J4pyK9M/XDBWI9f9zqI/AAAAAAAACTI/4eyWutVqHP0NYzc5EZe7Abjpv1f8gkjswCEwYBhgL/s320/edprobabilit04ans.png" width="320" /></a></div></div><div>(b) $\D \frac{4}{15}$</div><div>5(a) $\D \frac{13}{20}$</div><div>(b) $\D \frac{22}{99}$</div><div>6(a) 166</div><div>(b) $\D \frac{28}{83}$</div><div>(c) $\D \frac{9}{26}$</div><div>7(a) $D x=24$</div><div>(b) 12</div><div>8 (a) $\D 25-x$</div><div>(b) %fig<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-a4rlELnseSQ/XDBWI1_djFI/AAAAAAAACTQ/HTM-YyniFI4-M9nNr0DjS54bFiJ-sATrgCEwYBhgL/s1600/edprobabilit08ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="276" data-original-width="334" height="264" src="https://2.bp.blogspot.com/-a4rlELnseSQ/XDBWI1_djFI/AAAAAAAACTQ/HTM-YyniFI4-M9nNr0DjS54bFiJ-sATrgCEwYBhgL/s320/edprobabilit08ans.png" width="320" /></a></div></div><div>(c) 10</div><div>(d) (i) 98(ii) 15</div><div>(e) $\D \frac{17}{64}$</div><div>9 (a) 3/7</div><div>(b) 3/17</div><div>10 (a) 0.94</div><div>(b) 0.06</div><div>11 (a) %fig<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-CGBedUPRGG0/XDBWI69iMRI/AAAAAAAACTY/hHjI_jGCBec-XAnmqK99XyXRDWEFMSytwCEwYBhgL/s1600/edprobabilit11ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="734" data-original-width="599" height="320" src="https://1.bp.blogspot.com/-CGBedUPRGG0/XDBWI69iMRI/AAAAAAAACTY/hHjI_jGCBec-XAnmqK99XyXRDWEFMSytwCEwYBhgL/s320/edprobabilit11ans.png" width="261" /></a></div></div><div>(b) 2/35</div><div>(c) 37/210</div><div>12 (a) 3,22,15,11,6</div><div>(b) 80</div><div>(c) 5</div><div>(d) Swimming</div><div>(e)(i)39(ii) 34(iii) 8/39</div><div>13 (a) %fig<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-pQYGH2n91GA/XDBWJ_5FMLI/AAAAAAAACTg/Ycd72M3fzeEUAcMO_a9LJeqIlgNWoNn2wCEwYBhgL/s1600/edprobabilit13ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="804" data-original-width="520" height="320" src="https://3.bp.blogspot.com/-pQYGH2n91GA/XDBWJ_5FMLI/AAAAAAAACTg/Ycd72M3fzeEUAcMO_a9LJeqIlgNWoNn2wCEwYBhgL/s320/edprobabilit13ans.png" width="206" /></a></div></div><div>(b) 3/10</div><div>(c) 9/10</div><div>14 (a) (i) 7 (ii) 6 (iii) 5.3</div><div>(b) 24/870</div><div>(c) 132/159</div><div>15 (a) 3/10,</div><div>6/9,3/9,</div><div>7/9,2/9</div><div>(b) 7/15</div></div><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-77459694641814111922019-01-02T09:05:00.003-08:002019-01-02T09:06:43.796-08:00Set (CIE)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 21, question 6)<br />By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly.<br />(i) $\D A\cap&nbsp; B = (A'\cap&nbsp; B)'$&nbsp; <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-aftsJsvlzwI/XCzn-xJX26I/AAAAAAAACQQ/umT64FnFbAgkXiWV1wvk2W4dsP9Cr_KNQCEwYBhgL/s1600/set_1i.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="562" data-original-width="1414" height="158" src="https://1.bp.blogspot.com/-aftsJsvlzwI/XCzn-xJX26I/AAAAAAAACQQ/umT64FnFbAgkXiWV1wvk2W4dsP9Cr_KNQCEwYBhgL/s400/set_1i.png" width="400" /></a></div>(ii) $\D X\cap&nbsp; Y = X'\cup&nbsp; Y'$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-XqhNzf0-_mU/XCzn_ITFLSI/AAAAAAAACQU/8Xc3sRQOpBcr23eYlKM6wYl3Ds_UIMiggCEwYBhgL/s1600/set_1ii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="583" data-original-width="1406" height="165" src="https://1.bp.blogspot.com/-XqhNzf0-_mU/XCzn_ITFLSI/AAAAAAAACQU/8Xc3sRQOpBcr23eYlKM6wYl3Ds_UIMiggCEwYBhgL/s400/set_1ii.png" width="400" /></a></div>(iii) $\D (P\cap&nbsp; Q)\cup&nbsp; (Q\cap&nbsp; R) = Q\cap&nbsp; (P\cup&nbsp; R)$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-_APew6RxeDo/XCzn_BfPixI/AAAAAAAACQY/XadvOlFOuh00W_LJSi_pDJxWiHhmZ0YZQCEwYBhgL/s1600/set_1iii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="634" data-original-width="1462" height="172" src="https://4.bp.blogspot.com/-_APew6RxeDo/XCzn_BfPixI/AAAAAAAACQY/XadvOlFOuh00W_LJSi_pDJxWiHhmZ0YZQCEwYBhgL/s400/set_1iii.png" width="400" /></a></div>3 (CIE 2012, w, paper 13, question 1)<br />(a) On the Venn diagrams below, shade the region corresponding to the set given below each Venn diagram.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-2xAdwn5ekFU/XCzn_rwmx0I/AAAAAAAACRM/19UadMORCas5PotN253XePsIa39SlEr_wCEwYBhgL/s1600/set_3a.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="560" data-original-width="1253" height="178" src="https://3.bp.blogspot.com/-2xAdwn5ekFU/XCzn_rwmx0I/AAAAAAAACRM/19UadMORCas5PotN253XePsIa39SlEr_wCEwYBhgL/s400/set_3a.png" width="400" /></a></div><br />(b) It is given that sets $\D&nbsp; E, B, S$ and $\D F$ are such that<br />$\D E$ = {students in a school},<br />$\D B =$ {students who are boys},<br />$\D S =$ {students in the swimming team},<br />$\D F =$ {students in the football team}.<br />Express each of the following statements in set notation.<br />(i) All students in the football team are boys. <br />(ii) There are no students who are in both the swimming team and the football team. <br /><br />4 (CIE 2012, w, paper 21, question 2)<br />(a) It is given that $\D E$ is the set of integers, $\D P$ is the set of prime numbers between 10 and 50, $\D F$ is the set of multiples of 5, and $\D T$ is the set of multiples of 10. Write the following statements using set notation.<br />(i) There are 11 prime numbers between 10 and 50. <br />(ii) 18 is not a multiple of 5. <br />(iii) All multiples of 10 are multiples of 5. <br />(b) (i) In the Venn diagram below shade the region that represents $\D (A'\cap&nbsp; B) \cup (A\cap&nbsp; B').$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ZGJOedRgAug/XCzoAMdneAI/AAAAAAAACRM/Mca7I4pBGGkt9-OImSPMDggfHdBJDblgwCEwYBhgL/s1600/set_4bi.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="385" data-original-width="689" height="178" src="https://1.bp.blogspot.com/-ZGJOedRgAug/XCzoAMdneAI/AAAAAAAACRM/Mca7I4pBGGkt9-OImSPMDggfHdBJDblgwCEwYBhgL/s320/set_4bi.png" width="320" /></a></div>(ii) In the Venn diagram below shade the region that represents $\D Q\cap&nbsp; (R\cup&nbsp; S' ).$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-e0XqmOsqVzw/XCzoAcVL4uI/AAAAAAAACRI/o_qN9gGGjScE1Q6Al6C0MDV0pKdbvG7OwCEwYBhgL/s1600/set_4bii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="483" data-original-width="786" height="196" src="https://2.bp.blogspot.com/-e0XqmOsqVzw/XCzoAcVL4uI/AAAAAAAACRI/o_qN9gGGjScE1Q6Al6C0MDV0pKdbvG7OwCEwYBhgL/s320/set_4bii.png" width="320" /></a></div><br />5 (CIE 2013, s, paper 12, question 1)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-URu45HQpMy0/XCzoA1ncqxI/AAAAAAAACRY/arDozYBcQdQ7HnpyqcXpTSrIyHzOjvHBACEwYBhgL/s1600/set_5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="582" data-original-width="1069" height="174" src="https://4.bp.blogspot.com/-URu45HQpMy0/XCzoA1ncqxI/AAAAAAAACRY/arDozYBcQdQ7HnpyqcXpTSrIyHzOjvHBACEwYBhgL/s320/set_5.png" width="320" /></a></div>The Venn diagram shows the universal set $\D E$, the set $\D A$ and the set $\D B.$ Given that $\D n(B ) = 5, n(A') = 10$<br />and $\D n(E) = 26,$ find<br />(i) $\D n(A \cap B),$ <br />(ii) $\D n(A),$ <br />(iii) $\D n(B' \cap A).$ <br /><br />6 (CIE 2013, s, paper 21, question 9)<br />It is given that $\D x \in R$ and that <br />$\D E = {x : − 5 &lt; x &lt; 12},$<br />$\D S = {x : 5x + 24 &gt; x^2},$<br />$\D T = {x : 2x + 7 &gt; 15}.$<br />Find the values of $\D x$ such that<br />(i) $\D x \in&nbsp; S ,$ <br />(ii) $\D x \in S\cup T ,$ <br />(iii) $\D x \in (S\cap T )'.$ <br /><br />7 (CIE 2013, w, paper 11, question 4)<br />The sets $\D A$ and $\D B$ are such that<br />$\D A=\{ x:\cos x=\frac{1}{2} , 0^{\circ}\le x\le 620^{\circ},\}$<br />$\D B=\{ x:\tan x=\sqrt{3} , 0^{\circ}\le x\le 620^{\circ},\}$<br />(i) Find $\D n(A).$ <br />(ii) Find $\D n(B).$ <br />(iii) Find the elements of $\D A \cup B.$ <br />(iv) Find the elements of $\D A \cap B.$ <br /><br />8 (CIE 2013, w, paper 23, question 5)<br />(a) (i) In the Venn diagram below shade the region that represents $\D (A\cup B)'$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UIX-00ZcyNY/XCzoBEpW1BI/AAAAAAAACRM/Mrh2WFK9x0MHA-8pqrcCETZsTKGx60hMACEwYBhgL/s1600/set_8ai.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="409" data-original-width="716" height="182" src="https://3.bp.blogspot.com/-UIX-00ZcyNY/XCzoBEpW1BI/AAAAAAAACRM/Mrh2WFK9x0MHA-8pqrcCETZsTKGx60hMACEwYBhgL/s320/set_8ai.png" width="320" /></a></div>(ii) In the Venn diagram below shade the region that represents $\D P\cap Q\cap R'.$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-fYolM05ZTek/XCzoBLenXGI/AAAAAAAACRQ/f54_DihZ_cM-ruO-rn9VZfyfOMl1l4lWwCEwYBhgL/s1600/set_8aii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="528" data-original-width="758" height="222" src="https://2.bp.blogspot.com/-fYolM05ZTek/XCzoBLenXGI/AAAAAAAACRQ/f54_DihZ_cM-ruO-rn9VZfyfOMl1l4lWwCEwYBhgL/s320/set_8aii.png" width="320" /></a></div>(b) Express, in set notation, the set represented by the shaded region. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-kpV7__xnH9U/XCzoB2gT9nI/AAAAAAAACRQ/Ba08-xEclJAx5zQPEVpKTnogqRYTZwGbgCEwYBhgL/s1600/set_8b.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="405" data-original-width="718" height="180" src="https://2.bp.blogspot.com/-kpV7__xnH9U/XCzoB2gT9nI/AAAAAAAACRQ/Ba08-xEclJAx5zQPEVpKTnogqRYTZwGbgCEwYBhgL/s320/set_8b.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br />(c) The universal set $\D E$ and the sets $\D V$ and $\D W$ are such that $\D n(E) = 40, n(V ) = 18$ and $\D n(W) = 14.$ Given that $\D n(V\cap W) = x$ and $\D n((V\cup W)') = 3x$ find the value of $\D x.$<br />You may use the Venn diagram below to help you. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-iqhgaN-EJAM/XCzoB_GNYwI/AAAAAAAACRU/Eyr3wW6xWfwqcrVT7d9a4ZkFvnQnNfHNwCEwYBhgL/s1600/set_8c.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="493" data-original-width="900" height="175" src="https://4.bp.blogspot.com/-iqhgaN-EJAM/XCzoB_GNYwI/AAAAAAAACRU/Eyr3wW6xWfwqcrVT7d9a4ZkFvnQnNfHNwCEwYBhgL/s320/set_8c.png" width="320" /></a></div><br />9 (CIE 2014, s, paper 11, question 3)<br />(a) On the Venn diagrams below, shade the regions indicated.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-IgWEzGNo-yU/XCzoCaFwNQI/AAAAAAAACRY/uGMjSRnVw2UpMrPwBDSY-YhfB8hsf_i8QCEwYBhgL/s1600/set_9a.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="497" data-original-width="1600" height="122" src="https://3.bp.blogspot.com/-IgWEzGNo-yU/XCzoCaFwNQI/AAAAAAAACRY/uGMjSRnVw2UpMrPwBDSY-YhfB8hsf_i8QCEwYBhgL/s400/set_9a.png" width="400" /></a></div><br />(b) Sets $\D P$ and $\D Q$ are such that $\D P=\{x:x^2+2x=0\},$ and $\D Q=\{x:x^2+2x+7=0\},$ where $\D x\in R.$<br />(i) Find $\D n(P).$ <br />(ii) Find $\D n(Q).$ <br /><br />10 (CIE 2014, s, paper 12, question 2)<br />(a) On the Venn diagrams below, draw sets $\D A$ and $\D B$ as indicated.<br />(i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-rxPnABt9knE/XCzn9R0EH6I/AAAAAAAACRQ/fF4DEw1wfPQHPImgjSzkXdVzeT8g5aZiwCEwYBhgL/s1600/set_10ai.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="503" data-original-width="674" height="238" src="https://4.bp.blogspot.com/-rxPnABt9knE/XCzn9R0EH6I/AAAAAAAACRQ/fF4DEw1wfPQHPImgjSzkXdVzeT8g5aZiwCEwYBhgL/s320/set_10ai.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-TqUoLtIj8tE/XCzn9-slIJI/AAAAAAAACRY/ex8x8P2lWY4iWVY7hHiuevG3oTwZ3y4OACEwYBhgL/s1600/set_10aii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="515" data-original-width="649" height="253" src="https://2.bp.blogspot.com/-TqUoLtIj8tE/XCzn9-slIJI/AAAAAAAACRY/ex8x8P2lWY4iWVY7hHiuevG3oTwZ3y4OACEwYBhgL/s320/set_10aii.png" width="320" /></a></div>(ii)<br /><br />(b) The universal set $\D E$ and sets $\D P$ and $\D Q$ are such that $\D n(E) = 20, n(P \cup&nbsp; Q) = 15, n(P) = 13$&nbsp; and $\D n(P \cap Q) = 4.$ Find<br />(i) $\D n(Q),$ <br />(ii) $\D n(P \cup Q)',$ <br />(iii) $\D n(P \cap Q').$ <br /><br />11 (CIE 2014, s, paper 23, question 4)<br />(a) Illustrate the following statements using the Venn diagrams below.<br />(i) $\D A \cup B = A$ (ii) $\D A \cap&nbsp; B \cap C = \emptyset.$ <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ShNaqr0cWGs/XCzn9g45b4I/AAAAAAAACRU/RmMAtWQilacV-B_EJcPhXBP-FmjpT_VQwCEwYBhgL/s1600/set_11ai.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="493" data-original-width="718" height="219" src="https://4.bp.blogspot.com/-ShNaqr0cWGs/XCzn9g45b4I/AAAAAAAACRU/RmMAtWQilacV-B_EJcPhXBP-FmjpT_VQwCEwYBhgL/s320/set_11ai.png" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-TqUoLtIj8tE/XCzn9-slIJI/AAAAAAAACRY/ex8x8P2lWY4iWVY7hHiuevG3oTwZ3y4OACEwYBhgL/s1600/set_10aii.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="515" data-original-width="649" height="253" src="https://2.bp.blogspot.com/-TqUoLtIj8tE/XCzn9-slIJI/AAAAAAAACRY/ex8x8P2lWY4iWVY7hHiuevG3oTwZ3y4OACEwYBhgL/s320/set_10aii.png" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div>(b) It is given that $\D E$ is the set of integers between 1 and 100 inclusive. $\D S$ and $\D C$ are subsets of $\D E$, where $\D S$ is the set of square numbers and $\D C$ is the set of cube numbers. Write the following statements using set notation.<br />(i) 50 is not a cube number. <br />(ii) 64 is both a square number and a cube number. <br />(iii) There are 90 integers between 1 and 100 inclusive which are not square numbers. <br /><br />12 (CIE 2014, w, paper 13, question 3)<br />The universal set $\D E$ is the set of real numbers. Sets $\D A, B$ and $\D C$ are such that<br />$\D A = \{x:x^2+5x+6=0\},$<br />$\D B = \{x:(x-3)(x+2)(x+1)=0\},$<br />$\D C = \{x:x^2+x+3=0\}.$<br />(i) State the value of each of $\D n(A), n(B)$ and $\D n(C).$ <br />(ii) List the elements in the set $\D A \cup B.$ <br />(iii) List the elements in the set $\D A \cap B.$ <br />(iv) Describe the set $\D C'.$ <br /><br />13 (CIE 2014, w, paper 21, question 1)<br />(a) On each of the Venn diagrams below shade the region which represents the given set.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-p-PCe2qMjvw/XCzn-fHu9vI/AAAAAAAACRU/yohKQbcO6oA1wnuXzYxb0QU7C2zOS0MsgCEwYBhgL/s1600/set_13a.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="674" data-original-width="1492" height="144" src="https://4.bp.blogspot.com/-p-PCe2qMjvw/XCzn-fHu9vI/AAAAAAAACRU/yohKQbcO6oA1wnuXzYxb0QU7C2zOS0MsgCEwYBhgL/s320/set_13a.png" width="320" /></a></div>(b) In a year group of 98 pupils, $\D F$ is the set of pupils who play football and $\D H$ is the set of pupils who play hockey. There are 60 pupils who play football and 50 pupils who play hockey. The number that play both sports is $\D x$ and the number that play neither is $\D 30 - 2x.$ Find the value of $\D x.$ <br /><div><br /></div><h3>Answers</h3><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-bPqr0g1VFHg/XCzrh8wl8gI/AAAAAAAACRo/boDizHDZqDE8pX749GN3NJ6evwcsxJc5QCEwYBhgL/s1600/setans5_9.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="492" data-original-width="200" height="320" src="https://4.bp.blogspot.com/-bPqr0g1VFHg/XCzrh8wl8gI/AAAAAAAACRo/boDizHDZqDE8pX749GN3NJ6evwcsxJc5QCEwYBhgL/s320/setans5_9.png" width="130" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-z6PalwMWVO8/XCzrhm92bMI/AAAAAAAACRg/kASogREL_SkkfcsxbpkaeL89tpKa4gwCgCEwYBhgL/s1600/setans10_14.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="719" data-original-width="221" height="320" src="https://1.bp.blogspot.com/-z6PalwMWVO8/XCzrhm92bMI/AAAAAAAACRg/kASogREL_SkkfcsxbpkaeL89tpKa4gwCgCEwYBhgL/s320/setans10_14.png" width="98" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-EutB7k7U-K0/XCzrhnOUAFI/AAAAAAAACRk/atnExgPZqU03EiqXRGn_ixj-tn3tAuzIwCLcBGAs/s1600/setans1_4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="492" data-original-width="225" height="320" src="https://3.bp.blogspot.com/-EutB7k7U-K0/XCzrhnOUAFI/AAAAAAAACRk/atnExgPZqU03EiqXRGn_ixj-tn3tAuzIwCLcBGAs/s320/setans1_4.png" width="146" /></a></div><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-50687317605975098692019-01-02T04:39:00.001-08:002019-01-23T20:36:22.335-08:00Algebra (CIE)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 12, question 4)<br />Solve the simultaneous equations<br />$\D 5x + 3y = 2$ and $\D \frac{2}{x}-\frac{3}{y}=1.$<br /><br /><br />2 (CIE 2012, w, paper 22, question 1)<br />Solve the equation $\D |7x + 5| = |3x – 13|.$ <br /><br />3 (CIE 2012, w, paper 23, question 1)<br />Solve the equation $\D |5x + 7| = 13.$ <br /><br />4 (CIE 2015, s, paper 22, question 5)<br />Solve the simultaneous equations<br />$\D \begin{array}{rcl}<br />2x^2+3y^2&amp;=&amp;7y,\\x+y&amp;=&amp;4.<br />\end{array}$<br /><br /><br />5 (CIE 2016, w, paper 21, question 1)<br />Solve the equation $\D |4x - 3 |= x.$ <br />&nbsp; &nbsp; &nbsp; <br />6 (CIE 2017, march, paper 22, question 1)<br />Solve the equation $\D |5 - 3x |= 10.$ <br /><br />7 (CIE 2017, s, paper 22, question 1)<br />Solve $\D |5x + 3 |= |1 - 3x |.$ <br /><br />8 (CIE 2017, w, paper 21, question 4)<br />Solve the following simultaneous equations for $\D x$ and $\D y,$ giving each answer in its simplest surd form.<br />$\D \begin{array}{rcl}<br />\sqrt{3}x + y&amp; =&amp; 4\\<br />x - 2y &amp;=&amp; 5 \sqrt{3}<br />\end{array}$<br /><br /><br />9 (CIE 2017, w, paper 22, question 1)<br />If $\D z = 2 + \sqrt{3}$ find the integers $\D a$ and $\D b$ such that $\D az^2 + bz = 1 + \sqrt{3}.$&nbsp; <br /><br />10 (CIE 2017, w, paper 22, question 3)<br />Solve the inequality $\D |3x - 1|&gt;&nbsp; 3 + x.$ <br /><br />11 (CIE 2017, w, paper 23, question 2)<br />Solve the equation $\D |3x - 1| = |5 + x| .$&nbsp; <br /><br />12 (CIE 2018, s, paper 11, question 1)<br />Solve the equations<br />$\D \begin{array}{rcl}<br />y - x &amp;=&amp; 4,\\<br />x^2 + y^2 - 8x - 4y - 16 &amp;=&amp; 0.<br />\end{array}$<br /><br /><br /><h3>Answers</h3>1.&nbsp; $\D x = \frac{1}{5},y=\frac{1}{3}$:<br />$\D x = 4; y = -6$<br />2. $\D x = 0.8;-4.5$<br />3. $\D 1.2,-4$<br />4. $\D x = \frac{4}{3},<br />\frac{8}{3}$<br />$\D x = 3; y = 1$<br />5. $\D x = 1; 0.6$<br />6. $\D -5/3; 5$<br />7. $\D x = -2; x = -0.25$<br />8. $\D x = 2 +\sqrt{3},y=1-2\sqrt{3}$<br />9. $\D a = 1; b = -3$<br />10. $\D x &gt; 2; x &lt; -.5$<br />11. $\D x = -1$<br />12.&nbsp; $\D x = 4; y = 8$<br />$\D x = -2; y = 2$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-27840410012729539832019-01-02T04:00:00.000-08:002019-01-02T04:00:45.722-08:00Function (CIE)$\newcommand{\D}{\displaystyle}$<br />1 (CIE 2012, s, paper 12, question 10)<br />(a) It is given that $\D f(x) =\frac{1}{2+x}$&nbsp; for $\D x \not= -2, x\in R.$<br />(i) Find $\D f ″(x).$ <br />(ii) Find $\D f^{-1}&nbsp; (x).$ <br />(iii) Solve $\D f^2(x) = -1.$ <br />(b) The functions g, h and k are defined, for $\D x\in R,$ by<br />\begin{eqnarray*}<br />g(x)&amp;=&amp;\frac{1}{x+5},x\not=-5\\<br />h(x)&amp;=&amp;x^2-1,\\<br />k(x)&amp;=&amp;2x+1.<br />\end{eqnarray*}<br />Express the following in terms of g, h and/or k.<br />(i) $\D \frac{1}{(x^2-1)+5}$ <br />(ii) $\D \frac{2}{x+5}+1$ <br /><br /><br />2 (CIE 2012, s, paper 21, question 12or)<br />A function g is defined by $\D g : x \mapsto 5x^2 + px + 72,$ where $\D p$ is a constant. The function can also be written as $\D g : x \mapsto 5(x - 4)^2 + q.$<br />(i) Find the value of $\D p$ and of $\D q.$ <br />(ii) Find the range of the function g. <br />(iii) Sketch the graph of the function on the axes provided. <br />(iv) Given that the function $\D h$ is defined by $\D h : x \mapsto \ln x,$ where $\D x &gt; 0,$ solve the equation $\D gh(x) = 12.$ <br /><br /><br />3 (CIE 2012, w, paper 11, question 9)<br />A function g is such that $\D g(x) = \frac{1}{2x-1}$&nbsp; for $\D 1 \le x \le&nbsp; 3.$<br />(i) Find the range of $\D g.$ <br />(ii) Find $\D g^{-1}(x).$ <br />(iii) Write down the domain of $\D g^{-1}(x).$ <br />(iv) Solve $\D g^2(x) = 3.$ <br /><br />4 (CIE 2012, w, paper 23, question 12either)<br />(i) Express $\D 4x^2 + 32x + 55$ in the form $\D (ax + b)^2 + c,$ where a, b and c are constants and a is positive. <br />The functions f and g are defined by<br />\begin{eqnarray*}<br />f:x&amp;\mapsto&amp; 4x^2+32x+55 \mbox{ for } x&gt;-4\\<br />g:x&amp;\mapsto&amp;\frac{1}{x}\mbox{ for }x&gt;0.<br />\end{eqnarray*}<br />(ii) Find $\D f^{-1}(x).$ <br />(iii) Solve the equation $\D fg(x) = 135.$ <br /><br />5 (CIE 2012, w, paper 23, question 12or)<br />The functions h and k are defined by<br />\begin{eqnarray*}<br />h:x&amp;\mapsto&amp; \sqrt{2x-7} \mbox{ for } x&gt;c\\<br />k:x&amp;\mapsto&amp;\frac{3x-4}{x-2}\mbox{ for }x&gt;2.<br />\end{eqnarray*}<br /><br />(i) State the least possible value of c. <br />(ii) Find $\D h^{-1}(x).$ <br />(iii) Solve the equation $\D k(x) = x.$ <br />(iv) Find an expression for the function $\D k^2,$ in the form $\D k^2 : x \mapsto a + \frac{b}{x}$ where a and b are constants. <br /><br />6 (CIE 2013, s, paper 21, question 11)<br />A one-one function f is defined by $\D f(x)= (x- 1)^2- 5$ for $\D x \ge&nbsp; k .$<br />(i) State the least value that k can take. <br />For this least value of k<br />(ii) write down the range of f, <br />(iii) find $\D f^{-1}(x),$ <br />(iv) sketch and label, on the axes below, the graph of $\D y = f(x)$ and of $\D y= f^{-1}(x),$ <br />(v) find the value of x for which $\D f(x)= f^{-1}(x).$&nbsp; <br /><br />7 (CIE 2013, w, paper 11, question 12)<br />(a) A function f is such that $\D f (x)= 3x^2- 1$ for $\D - 10 \le x \le 8.$<br />(i) Find the range of f. <br />(ii) Write down a suitable domain for f for which $f^{-1}$ exists. <br />(b) Functions g and h are defined by $\D g(x)= 4e^x- 2$ for $\D x \in R,$ $h(x) = \ln 5x$ for $\D x &gt; 0.$<br />(i) Find $\D g^{-1} (x).$ <br />(ii) Solve $\D gh(x) = 18.$ <br /><br />8 (CIE 2013, w, paper 13, question 5)<br />For $\D x\in&nbsp; R,$ the functions f and g are defined by<br />\begin{eqnarray*}<br />f(x)&amp;=&amp;2x^3,\\<br />g(x)&amp;=&amp;4x-5x^2.<br />\end{eqnarray*}<br />(i) Express $\D f^2\left(\frac{1}{2}\right)$ as a power of 2. <br />(ii) Find the values of x for which f and g are increasing at the same rate with respect to x. <br /><br />9 (CIE 2014, s, paper 21, question 12)<br />The functions f and g are defined by<br />\begin{eqnarray*}<br />f(x)&amp;=&amp;\frac{2x}{x+1}\mbox{ for } x&gt;0,\\<br />g(x)&amp;=&amp;\sqrt{x+1}\mbox{ for } x&gt;-1.<br />\end{eqnarray*}<br />(i) Find $\D fg(8)$. <br />(ii) Find an expression for $\D f^2(x),$&nbsp; giving your answer in the form $\D \frac{ax}{bx+c},$&nbsp; where a, b and c are integers to be found. <br />(iii) Find an expression for $\D g^{-1}(x),$ stating its domain and range. <br />(iv) On the same axes, sketch the graphs of $\D y=g(x)$ and $\D y=g^{-1}(x),$ indicating the geometrical relationship between the graphs. <br /><br /><br />10 (CIE 2014, s, paper 22, question 11)<br />The functions f and g are defined, for real values of x greater than 2, by<br />\begin{eqnarray*}<br />f(x)&amp;=&amp;2^x-1,\\<br />g(x)&amp;=&amp;x(x+1).<br />\end{eqnarray*}<br />(i) State the range of f. <br />(ii) Find an expression for $\D f^{-1} (x),$ stating its domain and range. <br />(iii) Find an expression for $\D gf (x)$ and explain why the equation $\D gf (x) = 0$ has no solutions. <br /><br />11 (CIE 2014, s, paper 23, question 12)<br />The function f is such that $\D f(x) = \sqrt{x-3}$ for $\D 4\le x\le 28.$<br />(i) Find the range of f. <br />(ii) Find $\D f^2 (12).$ <br />(iii) Find an expression for $\D f^{-1} (x).$ <br />The function g is defined by&nbsp; $\D g(x)=\frac{120}{x}$&nbsp; for $\D x\ge 0.$<br />(iv) Find the value of x for which $\D gf (x) = 20.$ <br /><br />12 (CIE 2014, w, paper 21, question 4)<br />The functions f and g are defined for real values of x by<br />\begin{eqnarray*}<br />f(x)&amp;=&amp;\sqrt{x-1}-3 \mbox{ for } x&gt;1,\\<br />g(x)&amp;=&amp; \frac{x-2}{2x-3} \mbox{ for }x&gt;2.<br />\end{eqnarray*}<br />(i) Find $\D gf(37).$ <br />(ii) Find an expression for $\D f^{-1} (x).$ <br />(iii) Find an expression for $\D g^{-1} (x) .$&nbsp; <br /><br />13 (CIE 2014, w, paper 23, question 7)<br />The functions f and g are defined for real values of x by<br />\begin{eqnarray*}<br />f(x)&amp;=&amp; \frac{2}{x}+1 \mbox{ for }x&gt;1,\\<br />g(x)&amp;=&amp;x^2+2.<br />\end{eqnarray*}<br /><br />Find an expression for<br />(i) $\D f^{-1}(x),$ <br />(ii) $\D gf(x),$ <br />(iii) $\D fg(x).$ <br />(iv) Show that $\D ff(x)=\frac{3x+2}{x+2}$ and solve $\D ff(x)=x.$ <br /><br />14 (CIE 2015, s, paper 11, question 8)<br />It is given that<br />\begin{eqnarray*}<br />f(x)&amp;=&amp;3e^{2x} \mbox{ for }x\ge 0,\\<br />g(x)&amp;=&amp;(x+2)^2+5 \mbox{ for } x\ge 0.<br />\end{eqnarray*}<br />(i) Write down the range of f and of g. <br />(ii) Find $\D g^{-1},$ stating its domain. <br />(iii) Find the exact solution of $\D gf(x) = 41.$ <br />(iv) Evaluate $\D f'(\ln 4).$ <br /><br />Answers<br /><br />1.(a)(i) $\D 2(2+x)^{-3}$<br />(ii) $\D \frac{1-2x}{x}$<br />(iii) $\D x=-\frac{7}{3}$<br />(b) $\D gh,kg$<br />2. (i) $p=-40,q=-8$<br />(ii) $g(x)&gt;-8$<br />(iii)<br />(iv) $\D x=e^2,x=e^6$<br />3.(i) $\D 0.2\le x\le 1$<br />(ii) $\D g^{-1}(x)=\frac{1+x}{2x}$<br />(iii) $\D 0.2\le x\le 1$<br />(iv) $x+1.25$<br />4(i) $\D (2x+8)^2-9$<br />(ii) $\D f^{-1}=\frac{\sqrt{x+9}-8}{2}$<br />(iii) $\D x=0.5$<br />5(i) 3.5 (ii) $\D h^{-1}(x)=\frac{x^2+7}{2}$<br />(iii) $\D x=4$ (iv) 5-4/x<br />6(i)1 (ii) $\D f\ge -5$<br />(iii) $\D 1+\sqrt{x+5}$ (v)4<br />7(a)(i) $\D -1\le y\le 299$<br />(ii) $\D x\ge 0$<br />(b)(i) $\D \ln\left(\frac{x+2}{4}\right)$<br />(ii) $\D x=1$<br />8(i) $\D 2^{-5}$ (ii) $\D x=1/3,-2$<br />9(i) $\D 3/2$<br />(ii) $\D 4x/(3x+1)$<br />(iii) $\D g^{-1}(x)=x^2-1$<br />D: $x&gt;0$ R:$\D g^{-1}(x)&gt;-1$<br />(iv)<br />10(i) $\D f(x)&gt;3$<br />(ii) $\D f^{-1}(x)=\log_2(x+1)$<br />$x&gt;3,y&gt;2$<br />(iii) no solution<br />11(i) $\D 3&lt;f&lt;7$<br />(ii) $\D 2+\sqrt{2}$<br />(iii) $\D f^{-1}(x)=(x-2)^2+3$<br />(iv) $\D x=19$<br />12(i) $\D 1/3$<br />(ii) $\D (x+3)^2+1$<br />(iii)$\D \frac{3x-2}{2x-1}$<br />13(i) $\D 2/(x-1)$<br />(ii) $\D gf(x)=(2/x+1)^2+2$<br />(iii)$\D fg(x)=2/(x^2+2)+1$<br />(iv) $\D x=2$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-52817479912656708362018-12-27T22:53:00.000-08:002018-12-27T22:53:37.441-08:00Trigonometry (Selected Problems)$\def\D{\displaystyle}$<br />Question (1): In $\Delta ABC$, if $\cot A+\cot B+\cot C=\sqrt 3$, then $\Delta ABC$ is equilateral.<br /><br />$\D \begin{array}{|rl|}\hline<br />\cot(A+B)&amp;=\D\frac{\cot A\cot B-1}{\cot A+\cot B}\\ \hline<br />\end{array}$<br /><br />Proof:<br />$\cot C=\cot(180-(B+C))=-\cot(A+B)=\frac{1-\cot A\cot B}{\cot A+\cot B}.$<br />Let $\cot A=x,\cot B=y.$ Hence,<br />\begin{eqnarray*}<br />x+y+\frac{1-xy}{x+y}&amp;=&amp;\sqrt 3\\<br />(x^2+2xy+y^2) +(1-xy)&amp;=&amp;\sqrt 3x+\sqrt 3y\\<br />y^2+(x-\sqrt 3)y+(x^2-\sqrt 3x+1)&amp;=&amp;0\qquad \qquad (1)<br />\end{eqnarray*}<br />For real&nbsp; solutions, $b^2-4ac\ge 0.$ Thus<br />\begin{eqnarray*}<br />(x-\sqrt 3)^2-4(1)(x^2-\sqrt 3x+1)&amp;\ge 0\\<br />3x^2-2\sqrt 3x+1&amp;\le&amp;0\\<br />(\sqrt 3x-1)^2&amp;\le&amp;0\\<br />\sqrt 3x-1&amp;=&amp;0.<br />\end{eqnarray*}<br />Thus $x=1/\sqrt 3$. By (1), $y=1/\sqrt 3$. Therefore<br />$\cot A=\cot B=\frac{1}{\sqrt 3}\Longrightarrow A=B=60^{\circ}.$<br /><br />Question (2): In $\Delta ABC$, if $\sin^2 A+\sin^2 B+\sin^2 C= 2$, then $\Delta ABC$ is a right triangle.<br />$\D\begin{array}{|rl|}\hline<br />\sin^2A&amp;=\D \frac{1-\cos2A}{2}\\<br />\cos A+\cos B&amp;=\D2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\\<br />\sin^2A&amp;=1-\cos^2A\\<br />\sin A&amp;=\sin(180^{\circ}-A)\\<br />\hline \end{array}$<br /><br />Proof:<br />\begin{eqnarray*}<br />\sin^2A+\sin^2B&amp;=&amp;\frac{1-\cos2A}{2}+\frac{1-\cos2B}{2} \\<br />&amp;=&amp;1-\frac{1}{2}(\cos2A+\cos2B)\\<br />&amp;=&amp;1-\frac{1}{2}\left(2\cos \frac{2A+2B}{2}\cos \frac{2A-2B}{2}\right)\\<br />\sin^2A+\sin^2B &amp;=&amp;1-\cos(A+B)\cos(A-B)\cdots (1)\\<br />\sin^2C&amp;=&amp;\sin^2(180^{\circ}-C)=\sin^2(A+B)\\<br />\sin^2C&amp;=&amp;1-\cos^2(A+B)\cdots (2)<br />\end{eqnarray*}<br />(1)+(2):<br />$\D\sin^2A+\sin^2B+\sin^2C$\begin{eqnarray*}<br />&amp;=&amp;2-\cos(A+B)(\cos(A+B)+\cos(A-B))\\<br />2&amp;=&amp;2-\cos(A+B)\left(\cos A\cos B-\sin A\sin B \right.\\<br />&amp;&amp;\left.\qquad \qquad +\cos A\cos B+\sin A\sin B\right)\\<br />0&amp;=&amp;-2\cos(A+B)\cos A\cos B<br />\end{eqnarray*}<br />Hence, $\D \cos(A+B)=0$ or $$\D \cos A=0$$ or $$\D\cos B=0.$$<br />Thus&nbsp; $$\D A+B=90^{\circ}$$, ie $$\D C=90^{\circ}$$ or $$\D A=90^{\circ}$$ or $$\D B=90^{\circ}.$$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-77443678943940307662018-12-26T07:00:00.002-08:002018-12-26T07:06:25.571-08:00AP GP Series (IB Standard Level)$\def\D{\displaystyle}$<br />1.) In an arithmetic sequence, $\D u_1 = 2$ and $\D u_3 = 8.$<br />(a) Find $\D d.$<br />(b) Find $\D u_{20}.$<br />(c) Find $\D S_{20}.$ (Total 6 marks)<br /><br />2.) In an arithmetic sequence $\D u_1 = 7, u_{20} = 64$ and $\D u_n = 3709.$<br />(a) Find the value of the common difference.<br />(b) Find the value of $\D n.$ (Total 5 marks)<br /><br />3.) Consider the arithmetic sequence 3, 9, 15, ..., 1353.<br />(a) Write down the common difference.<br />(b) Find the number of terms in the sequence.<br />(c) Find the sum of the sequence. (Total 6 marks)<br /><br />4.) An arithmetic sequence, $\D u_1, u_2, u_3, \ldots ,$ has $\D d = 11$ and $\D u_{27} = 263.$<br />(a) Find $u_1.$<br />(b) (i) Given that $\D u_n = 516,$ find the value of $\D n.$<br />(ii) For this value of $\D n,$ find $\D S_n.$ (Total 6 marks)<br /><br />5.) The first three terms of an infinite geometric sequence are 32, 16 and 8.<br />(a) Write down the value of $\D r.$<br />(b) Find $\D u_6.$<br />(c) Find the sum to infinity of this sequence. (Total 5 marks)<br /><br />6.) The $\D n^{th}$ term of an arithmetic sequence is given by $\D u_n = 5 + 2n.$<br />(a) Write down the common difference.<br />(b) (i) Given that the $\D n^{th}$ term of this sequence is 115, find the value of $\D n.$<br />(ii) For this value of $\D n,$ find the sum of the sequence. (Total 6 marks)<br /><br />7.) In an arithmetic series, the first term is $\D -7$ and the sum of the first 20 terms is 620.<br />(a) Find the common difference.<br />(b) Find the value of the $\D 78 ^{th}$ term. (Total 5 marks)<br /><br />8.) In a geometric series, $\D u_1 =<br />\frac{1}{81}$ and $\D u_4 =\frac{1}{3}.$<br />(a) Find the value of $\D r.$<br />(b) Find the smallest value of $\D n$ for which $\D S_n &gt; 40.$ (Total 7 marks)<br /><br />9.) (a) Expand $\D \sum_{r=4}^{7} 2^r$ as the sum of four terms.<br />(b) (i) Find the value of $\D \sum_{r=4}^{30} 2^r.$<br />(ii) Explain why $\D \sum_{r=4}^{\infty} 2^r$ cannot be evaluated. (Total 7 marks)<br /><br />10.) In an arithmetic sequence, $\D S_{40} = 1900$ and $\D u_{40} = 106.$ Find the value of $\D u_1$ and of $\D d.$ (Total 6 marks)<br /><br />11.) Consider the arithmetic sequence 2, 5, 8, 11, ....<br />(a) Find $\D u_{101}.$<br />(b) Find the value of $\D n$ so that $\D u_n = 152.$ (Total 6 marks)<br /><br />12.) Consider the infinite geometric sequence $\D 3000, - 1800, 1080, -648, … .$<br />(a) Find the common ratio.<br />(b) Find the 10 th term.<br />(c) Find the exact sum of the infinite sequence. (Total 6 marks)<br /><br />13.) Consider the infinite geometric sequence $\D 3, 3(0.9), 3(0.9)^2, 3(0.9)^3, … .$<br />(a) Write down the 10 th term of the sequence. Do not simplify your answer.<br />(b) Find the sum of the infinite sequence. (Total 5 marks)<br /><br />14.) In an arithmetic sequence $\D u_{21} = -37$ and $\D u_4 = -3.$<br />(a) Find<br />(i) the common difference;<br />(ii) the first term.<br />(b) Find $\D S_{10}.$ (Total 7 marks)<br /><br />15.) Let $\D u_n = 3 - 2n.$<br />(a) Write down the value of $\D u_1, u_2,$ and $\D u_3.$<br />(b) Find $\D \sum_{n=1}^{20} (3-2n)$ (Total 6 marks)<br /><br />16.) A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row.<br />(a) Calculate the number of seats in the 20th row.<br />(b) Calculate the total number of seats. (Total 6 marks)<br /><br />17.) A sum of \$5000 is invested at a compound interest rate of 6.3 \% per annum.<br />(a) Write down an expression for the value of the investment after$\D n$full years.<br />(b) What will be the value of the investment at the end of five years?<br />(c) The value of the investment will exceed \$ 10 000 after $\D n$ full years.<br />(i) Write down an inequality to represent this information.<br />(ii) Calculate the minimum value of $\D n.$ (Total 6 marks)<br /><br />18.) Consider the infinite geometric sequence 25, 5, 1, 0.2, … .<br />(a) Find the common ratio.<br />(b) Find<br />(i) the 10th term;<br />(ii) an expression for the n th term.<br />(c) Find the sum of the infinite sequence. (Total 6 marks)<br /><br />19.) The first four terms of a sequence are 18, 54, 162, 486.<br />(a) Use all four terms to show that this is a geometric sequence.<br />(b) (i) Find an expression for the $\D n$ th term of this geometric sequence.<br />(ii) If the $\D n$ th term of the sequence is 1062 882, find the value of $\D n.$ (Total 6 marks)<br /><br />20.) (a) Write down the first three terms of the sequence $\D u_n = 3n,$ for $\D n\ge 1.$<br />(b) Find<br />(i) $\D \sum_{n=1}^{20} 3n$<br />(ii) $\D \sum_{n=21}^{100} 3n$ (Total 6 marks)<br /><br />21.) Consider the infinite geometric series 405 + 270 + 180 +....<br />(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.<br />(b) Find the fifteenth term of this series.<br />(c) Find the exact value of the sum of the infinite series. (Total 6 marks)<br /><br />22.) (a) Consider the geometric sequence $\D -3, 6, -12, 24, ….$<br />(i) Write down the common ratio.<br />(ii) Find the 15th term.<br />Consider the sequence $\D x - 3, x +1, 2x + 8,\ldots.$<br />(b) When $\D x = 5,$ the sequence is geometric.<br />(i) Write down the first three terms.<br />(ii) Find the common ratio.<br />(c) Find the other value of $\D x$ for which the sequence is geometric.<br />(d) For this value of $\D x,$ find<br />(i) the common ratio;<br />(ii) the sum of the infinite sequence. (Total 12 marks)<br /><br /><h3>Answers&nbsp;</h3>1 (a) $\D =3$<br />(b) $\D u_{20}=59$<br />(c) $\D S_{20}=610$<br />2 (a) $\D d=3$<br />(b) $\D n=1235$<br />3(a) $\D d=6$<br />(b)$\D n=226$<br />(c) $\D S_{226}=153228$<br />4 (a) $\D -23$<br />(b)(i) $\D 50$<br />(ii) 12325<br />5 (a) $\D r=\frac{1}{2}$<br />(b) $\D u_6=-1$<br />(c) $\D S=64$<br />6 (a) $\D d=2$<br />(b)(i) $\D n=55$<br />(ii) $\D S_{55}=3355$<br />7 (a) $\D d=4$<br />(b) $\D u_{78}=301$<br />8 (a) $\D r=3$<br />(b) $\D n=8$<br />9 (a) $\D 2^4+2^5+2^6+2^7$<br />(b)(i) 2147483632<br />(ii) $\D r\ge 1$<br />10 $\D u_1=-11,d=3$<br />11 $\D u_{101}=302,n=51$<br />12 $\D r=-0.6$<br />(b) $\D u_{10}=-30.2$<br />(c) 1875<br />13 (a) $\D u_{10}=3(0.9)^9$<br />(b) $\D S=30$<br />14 (a)(i) $\D d=-2,$<br />(ii) $\D u_1=3$<br />(b) $\D S_{10}=-60$<br />15 (a) $\D 1,-1,-3$<br />(b) $\D S_{20}=-360$<br />16 $\D u_{20}=53$<br />(b) $\D S_{20}=680$<br />17 (a) $\D 5000(1.063)^n$<br />(b) (a) $\D 6786$<br />(c) (i) $\D 5000(1.063)^n&gt;10000$<br />(ii) 12 years<br />18 (a) $\D r=1/5$<br />(b) (i) 0.0000128<br />(ii) $\D u_n=25(.2)^n-1$<br />(c) $\D S=31.25$<br />19 (a)<br />(b) (i)$\D u_n=18\times 3^{n-1}$<br />(ii) $\D n=11$<br />20 (a) 3,6,9<br />(b) (i) 630<br />(ii) 14520<br />21 (a) $\D r=2/3$<br />(b) $\D u_{15}=1.39$<br />(c) $\D S=1215$<br />22 (a) (i) $\D r=-2$<br />(ii) $\D u_{15}=-49152$<br />(b) (i) 2,6,18<br />(ii) $\D r=3$<br />(c)&nbsp; $\D x=-5$<br />(d) (i) $\D r=0.5$<br />(ii) $\D S=-16$<br /><br />Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-702748784464325372018-12-25T09:14:00.002-08:002018-12-29T00:52:55.846-08:00Graph (CIE)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 11, question 1)<br />(i) Sketch the graph of $\D y = |2x - 5|,$ showing the coordinates of the points where the graph meets the coordinate axes. <br />(ii) Solve $\D |2x - 5| = 3 .$ <br /><br />2 (CIE 2012, s, paper 12, question 7)<br />(i) Sketch the graph of $\D y = |x^2 - x - 6|,$ showing the coordinates of the points where the curve meets the coordinate axes. <br />(ii) Solve $|x^2 - x - 6| = 6.$ <br /><br />3 (CIE 2012, s, paper 21, question 3)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MMcH2Gj7qzQ/XCJd8kttcbI/AAAAAAAACNE/lIvk1NAV9X0WyoAbnfjJ6TiBUrO9cI1pgCEwYBhgL/s1600/graph03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="847" data-original-width="1502" height="180" src="https://1.bp.blogspot.com/-MMcH2Gj7qzQ/XCJd8kttcbI/AAAAAAAACNE/lIvk1NAV9X0WyoAbnfjJ6TiBUrO9cI1pgCEwYBhgL/s320/graph03.png" width="320" /></a></div>The diagram shows a sketch of the curve $\D y = a\sin(bx) + c$ for $\D 0^{\circ}\le&nbsp; x \le&nbsp; 180^{\circ}.$ Find the<br />values of $\D a, b$ and $\D c.$ <br />(b) Given that $\D f(x) = 5\cos3x + 1,$ for all $\D x,$ state<br />(i) the period of $\D f,$ <br />(ii) the amplitude of $\D f.$ <br /><br />4 (CIE 2012, w, paper 11, question 1)<br />(i) Sketch the graph of $\D y = |3 + 5x|,$ showing the coordinates of the points where your graph meets the coordinate axes. <br />(ii) Solve the equation $\D |3 + 5x| = 2.$ <br /><br />5 (CIE 2012, w, paper 12, question 9)<br />(a) (i) Using the axes below, sketch for $\D 0\le&nbsp; &nbsp;x \le&nbsp; \pi,$ the graphs of $\D y = \sin 2x$ and $\D y = 1 + \cos 2x.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-B1Sr5nQmP7c/XCJd82Nm_pI/AAAAAAAACNM/t3xFdAyGoAItZlr48s6aUA9ASr61eHFDgCEwYBhgL/s1600/graph05.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="735" data-original-width="1570" height="149" src="https://1.bp.blogspot.com/-B1Sr5nQmP7c/XCJd82Nm_pI/AAAAAAAACNM/t3xFdAyGoAItZlr48s6aUA9ASr61eHFDgCEwYBhgL/s320/graph05.png" width="320" /></a></div>(ii) Write down the solutions of the equation $\D \sin 2x - \cos 2x = 1,$ for $\D 0 \le x \le&nbsp; \pi.$ <br />(b) (i) Write down the amplitude and period of $\D 5 \cos 4x - 3.$ <br />(ii) Write down the period of $\D 4 \tan 3x.$ <br /><br />6 (CIE 2012, w, paper 13, question 4)<br />(i) On the axes below sketch, for $\D 0\le&nbsp; &nbsp;x \le&nbsp; \pi,$ the graphs of $\D y = \tan x$ and $\D y = 1 + 3\sin 2x.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-7Nr4OkP31E0/XCJd9t5UL-I/AAAAAAAACNU/VqTEbat2hi4cUmdFKkZrFOXhatbWJToKQCEwYBhgL/s1600/graph06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1250" data-original-width="1585" height="252" src="https://1.bp.blogspot.com/-7Nr4OkP31E0/XCJd9t5UL-I/AAAAAAAACNU/VqTEbat2hi4cUmdFKkZrFOXhatbWJToKQCEwYBhgL/s320/graph06.png" width="320" /></a></div>Write down<br />(ii) the coordinates of the stationary points on the curve $\D y = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi,$ <br />(iii) the number of solutions of the equation $\D \tan x = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi.$ <br /><br />7 (CIE 2012, w, paper 21, question 3)<br />(i) On the grid below draw, for $\D 0^{\circ} \le&nbsp; x \le&nbsp; 360^{\circ},$ the graphs of $\D y = 3 \sin 2x$ and $\D y = 2 + \cos x.$ <br />(ii) State the number of values of $\D x$ for which $\D 3 \sin 2x = 2 + \cos x$ in the interval $\D 0^{\circ} \le&nbsp; x \le&nbsp; 360^{\circ}.$ <br /><br />8 (CIE 2013, s, paper 11, question 1)<br />On the axes below sketch, for $\D 0 \le&nbsp; x \le&nbsp; 2\pi,$ the graph of<br />(i) $\D y = \cos x - 1,$ <br />(ii) $\D y = \sin 2x.$ <br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-n82w7U8iaqU/XCJd-_1RJAI/AAAAAAAACO4/ZG7IMTIYlV88_zZKKjbcWVpSkcqP0qobgCEwYBhgL/s1600/graph08.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="917" data-original-width="1424" height="206" src="https://1.bp.blogspot.com/-n82w7U8iaqU/XCJd-_1RJAI/AAAAAAAACO4/ZG7IMTIYlV88_zZKKjbcWVpSkcqP0qobgCEwYBhgL/s320/graph08.png" width="320" /></a></div>(iii) State the number of solutions of the equation $\D \cos x - \sin 2x = 1,$ for $\D 0 \le x \le&nbsp; 2\pi.$ <br /><br />9 (CIE 2013, s, paper 21, question 2)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-_wjGLaDu6Io/XCJigIUZbaI/AAAAAAAACPY/NBFyETzu-qYsxUOfG4hVSwzzQHMFRVILgCLcBGAs/s1600/graph09.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="552" data-original-width="1304" height="135" src="https://3.bp.blogspot.com/-_wjGLaDu6Io/XCJigIUZbaI/AAAAAAAACPY/NBFyETzu-qYsxUOfG4hVSwzzQHMFRVILgCLcBGAs/s320/graph09.png" width="320" /></a></div>The velocity-time graph represents the motion of a particle moving in a straight line.<br />(i) Find the acceleration during the first 5 seconds. <br />(ii) Find the length of time for which the particle is travelling with constant velocity. <br />(iii) Find the total distance travelled by the particle. <br /><br />10 (CIE 2013, s, paper 21, question 4)<br />(i) Sketch the graph of $\D y = |4x - 2|$, showing the coordinates of the points where the graph meets the axes. <br />(ii) Solve the equation $\D |4x - 2| = x.$ <br /><br />11 (CIE 2013, s, paper 22, question 3)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-qfW-vrdm_ps/XCJd_4BWyuI/AAAAAAAACO4/uoIaARojvAMZBMe2hQIeeKWjcUViLpNigCEwYBhgL/s1600/graph11a.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1014" data-original-width="1581" height="205" src="https://4.bp.blogspot.com/-qfW-vrdm_ps/XCJd_4BWyuI/AAAAAAAACO4/uoIaARojvAMZBMe2hQIeeKWjcUViLpNigCEwYBhgL/s320/graph11a.png" width="320" /></a></div>(i) Write down the letter of each graph which does not represent a function. <br />(ii) Write down the letter of each graph which represents a function that does not have an inverse. <br />(b)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-4a_gk-GNJzI/XCJeA4Bkf9I/AAAAAAAACPA/3fos8TkutW8O9XJq5bd2z5O6WbdxL2okACEwYBhgL/s1600/graph11b.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="982" data-original-width="1035" height="303" src="https://3.bp.blogspot.com/-4a_gk-GNJzI/XCJeA4Bkf9I/AAAAAAAACPA/3fos8TkutW8O9XJq5bd2z5O6WbdxL2okACEwYBhgL/s320/graph11b.png" width="320" /></a></div>The diagram shows the graph of a function $\D y = f(x).$ On the same axes sketch the graph of $\D y = f^{-1}(x).$<br /><br />12 (CIE 2013, s, paper 22, question 10)<br />(a) The function $\D f$ is defined, for $\D 0^{\circ} \le x\le&nbsp; 360^{\circ},$ by $\D f(x) = 1 + 3 \cos 2x.$<br />(i) Sketch the graph of $\D y = f(x)$ on the axes below. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-_s8eMGAMe0A/XCJeBC6kiAI/AAAAAAAACPA/QGMkw856fpUHLcVSGIMbAm0DbWjVbcYBwCEwYBhgL/s1600/graph12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="729" data-original-width="1172" height="199" src="https://1.bp.blogspot.com/-_s8eMGAMe0A/XCJeBC6kiAI/AAAAAAAACPA/QGMkw856fpUHLcVSGIMbAm0DbWjVbcYBwCEwYBhgL/s320/graph12.png" width="320" /></a></div>(ii) State the amplitude of $\D f.$ <br />(iii) State the period of $\D f.$ <br />(b) Given that $\D \cos x = p ,$ where $\D 270^{\circ} &lt; x &lt; 360^{\circ},$ find&nbsp; cosec $\D x$ in terms of $\D p.$ <br /><br />13 (CIE 2013, w, paper 11, question 1)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-PJNCwObtils/XCJeB1aTsAI/AAAAAAAACPI/CzE35v_X46IVbG5O7ccmsMT3QOSCT1FhgCEwYBhgL/s1600/graph13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="776" data-original-width="1600" height="155" src="https://4.bp.blogspot.com/-PJNCwObtils/XCJeB1aTsAI/AAAAAAAACPI/CzE35v_X46IVbG5O7ccmsMT3QOSCT1FhgCEwYBhgL/s320/graph13.png" width="320" /></a></div>The diagram shows the graph of $\D y = a \sin(bx) + c$ for $\D 0 \le&nbsp; x \le&nbsp; 2\pi,$ where $\D a, b$ and $\D c$ are positive integers. State the value of $\D a,$ of $\D b$ and of $\D c.$ <br /><br />14 (CIE 2013, w, paper 11, question 8)<br />(i) On the grid below, sketch the graph of $\D y = |(x - 2) (x + 3)|$ for $\D - 5 \le&nbsp; x \le&nbsp; 4,$ and state the coordinates of the points where the curve meets the coordinate axes. <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-GJll7EAf5aQ/XCJeCKa5joI/AAAAAAAACPE/uk-2fwEF_6A1dR7YbE26rKs0UbosDvw0ACEwYBhgL/s1600/graph14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="916" data-original-width="1600" height="183" src="https://2.bp.blogspot.com/-GJll7EAf5aQ/XCJeCKa5joI/AAAAAAAACPE/uk-2fwEF_6A1dR7YbE26rKs0UbosDvw0ACEwYBhgL/s320/graph14.png" width="320" /></a></div>(ii) Find the coordinates of the stationary point on the curve $\D y = |(x - 2) (x + 3)| .$ <br />(iii) Given that $\D k$ is a positive constant, state the set of values of $\D k$ for which $\D |(x - 2) (x + 3)| = k$ has 2 solutions only. <br /><br />15 (CIE 2013, w, paper 23, question 4)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Y-Ynb4BbFrI/XCJeCwGjZLI/AAAAAAAACO4/uR9evHKk1kYswLJv8q-NB-7foGmzKNb2wCEwYBhgL/s1600/graph15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1115" data-original-width="1364" height="261" src="https://1.bp.blogspot.com/-Y-Ynb4BbFrI/XCJeCwGjZLI/AAAAAAAACO4/uR9evHKk1kYswLJv8q-NB-7foGmzKNb2wCEwYBhgL/s320/graph15.png" width="320" /></a></div>(a) (i) The diagram shows the graph of $\D y = A + C \tan(Bx)$ passing through the points (0, 3) and $\D \left(\frac{\pi}{2},3\right).$ Find the value of $\D A$ and of $\D B.$ <br />(ii) Given that the point $\D \left(\frac{\pi}{8},7\right)$&nbsp; also lies on the graph, find the value of $\D C.$ <br />(b) Given that $\D f (x) = 8 - 5 \cos 3x,$ state the period and the amplitude of $\D f.$ <br /><br />16 (CIE 2014, s, paper 11, question 9a)<br />(a) The diagram shows the velocity-time graph of a particle $\D P$ moving in a straight line with velocity $D v$ ms$\D^{-1}$ at time $\D t$ s after leaving a fixed point.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-8cRC7LNLR00/XCJeDMCmskI/AAAAAAAACPA/wob8n7D9KWQ1K19jtcuySrcShe1futF3QCEwYBhgL/s1600/graph16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="742" data-original-width="1600" height="148" src="https://4.bp.blogspot.com/-8cRC7LNLR00/XCJeDMCmskI/AAAAAAAACPA/wob8n7D9KWQ1K19jtcuySrcShe1futF3QCEwYBhgL/s320/graph16.png" width="320" /></a></div>Find the distance travelled by the particle $\D P.$ <br />(b) The diagram shows the displacement-time graph of a particle $\D Q$ moving in a straight line with displacement $\D s$ m from a fixed point at time $\D t$ s.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-8jomURWroIU/XCJeEN-GIHI/AAAAAAAACO8/1nLXTWzUvogsBeqY8jczkCALx1-BpQUPACEwYBhgL/s1600/graph17b.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1079" data-original-width="1600" height="215" src="https://2.bp.blogspot.com/-8jomURWroIU/XCJeEN-GIHI/AAAAAAAACO8/1nLXTWzUvogsBeqY8jczkCALx1-BpQUPACEwYBhgL/s320/graph17b.png" width="320" /></a></div>On the axes below, plot the corresponding velocity-time graph for the particle $\D Q.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-PZNuytZRGIs/XCJeEF8faGI/AAAAAAAACPM/Dyc5uJQWVbE22wKOiUF6Fcj1DQ3x_dGjwCEwYBhgL/s1600/graph17c.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="814" data-original-width="1600" height="162" src="https://3.bp.blogspot.com/-PZNuytZRGIs/XCJeEF8faGI/AAAAAAAACPM/Dyc5uJQWVbE22wKOiUF6Fcj1DQ3x_dGjwCEwYBhgL/s320/graph17c.png" width="320" /></a></div><br />(c) The displacement $\D s$ m of a particle $\D R,$ which is moving in a straight line, from a fixed point at time $\D t$ s is given by $\D s = 4t - 16 \ln(t+1)+ 13.$<br />(i) Find the value of $\D t$ for which the particle $\D R$ is instantaneously at rest. <br />(ii) Find the value of $\D t$ for which the acceleration of the particle $\D R$ is 0.25ms$\D ^{-1}.$ <br /><br />17 (CIE 2014, s, paper 11, question 9b)<br />18 (CIE 2014, s, paper 12, question 3)<br />(i) Sketch the graph of $\D y = |(2x+1)(x-2)|$ for $\D -2\le&nbsp; x\le&nbsp; &nbsp;3,$ showing the coordinates of the points where the curve meets the x- and y-axes. <br />(ii) Find the non-zero values of $\D k$ for which the equation $\D |(2x+1)(x-2)| = k$ has two solutions only.<br /><br /><br />19 (CIE 2014, s, paper 21, question 3)<br />(i) On the axes below, sketch the graph of $\D y = |(x-4)(x+2)|$ showing the coordinates of the points where the curve meets the x-axis. <br />(ii) Find the set of values of k for which ^x - 4h^x + 2h = k has four solutions. <br /><br />20 (CIE 2014, w, paper 11, question 2)<br />(a) On the axes below, sketch the curve $\D y = 3 \cos 2x - 1$ for&nbsp; $\D 0^{\circ}\le&nbsp; x \le&nbsp; 180^{\circ}.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-o1Iud5cHCwE/XCJeFi_msEI/AAAAAAAACPI/M6nvlaaH-XswPU0eF-dSLYAYAV86lPuhACEwYBhgL/s1600/graph20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1007" data-original-width="1600" height="201" src="https://3.bp.blogspot.com/-o1Iud5cHCwE/XCJeFi_msEI/AAAAAAAACPI/M6nvlaaH-XswPU0eF-dSLYAYAV86lPuhACEwYBhgL/s320/graph20.png" width="320" /></a></div>(b) (i) State the amplitude of $\D 1 - 4 \sin 2x.$ <br />(ii) State the period of $\D 5 \tan 3x + 1.$ <br /><br />21 (CIE 2014, w, paper 13, question 1)<br />The diagram shows the graph of $\D y = a \cos bx + c$ for $\D 0^{\circ} \le x \le&nbsp; 360^{\circ},$ where $\D a, b$ and $\D c$ are<br />positive integers. State the value of each of $\D a, b$ and $\D c.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-MU0HlWF-6zQ/XCJeGiZuiMI/AAAAAAAACPQ/B8M7Vun-LJsTIH3PJwCglokZrSEZdwO2QCEwYBhgL/s1600/graph21.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="732" data-original-width="1600" height="146" src="https://3.bp.blogspot.com/-MU0HlWF-6zQ/XCJeGiZuiMI/AAAAAAAACPQ/B8M7Vun-LJsTIH3PJwCglokZrSEZdwO2QCEwYBhgL/s320/graph21.png" width="320" /></a></div>22 (CIE 2014, w, paper 13, question 2)<br />The line $\D 4y = x + 8$ cuts the curve $\D xy = 4 + 2x$ at the points $\D A$ and $\D B.$ Find the exact length of $\D AB.$ <br /><br /><h3>Answers</h3><br />1. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-emZk-eregW0/XCJd7ScNTZI/AAAAAAAACPM/FEypuZA0hTYM5lVHJTx5uM0H1cJgW4jmACEwYBhgL/s1600/graph01ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="263" data-original-width="464" height="181" src="https://4.bp.blogspot.com/-emZk-eregW0/XCJd7ScNTZI/AAAAAAAACPM/FEypuZA0hTYM5lVHJTx5uM0H1cJgW4jmACEwYBhgL/s320/graph01ans.png" width="320" /></a></div>(ii) $\D x = 1, 4$<br />2. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-w5DbrdgcniQ/XCJd7anVeOI/AAAAAAAACPA/tHWV25vgf-ADTH-2WmgtKi-VCSVaXCIdQCEwYBhgL/s1600/graph02ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="291" data-original-width="420" height="221" src="https://2.bp.blogspot.com/-w5DbrdgcniQ/XCJd7anVeOI/AAAAAAAACPA/tHWV25vgf-ADTH-2WmgtKi-VCSVaXCIdQCEwYBhgL/s320/graph02ans.png" width="320" /></a></div>(ii) $\D x = 0, 1$<br />3. (a) $\D a = 3, b = 8, c = 7$<br />(b) $\D 2\pi/3, 5$<br />4. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-9RU08F80PRY/XCJd88XMJAI/AAAAAAAACPE/d7YJXg5dV54wDjSFMRNlC5A1mhjUXoERQCEwYBhgL/s1600/graph04ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="164" data-original-width="283" src="https://2.bp.blogspot.com/-9RU08F80PRY/XCJd88XMJAI/AAAAAAAACPE/d7YJXg5dV54wDjSFMRNlC5A1mhjUXoERQCEwYBhgL/s1600/graph04ans.png" /></a></div>(ii) $\D x = -1=5,-1$<br />5. (ai)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Lt8zMHVrTnI/XCJd9coJqvI/AAAAAAAACPA/6Hd0sYbitxQ-MtZd6lgbcwvUIbqTeLY3ACEwYBhgL/s1600/graph05ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="211" data-original-width="439" height="153" src="https://3.bp.blogspot.com/-Lt8zMHVrTnI/XCJd9coJqvI/AAAAAAAACPA/6Hd0sYbitxQ-MtZd6lgbcwvUIbqTeLY3ACEwYBhgL/s320/graph05ans.png" width="320" /></a></div>(ii) $\D x = \pi/4, \pi/2$<br />(b)(i)Amp=5,Period= $\D \pi/2$<br />(ii)Period= $\D \pi/3$<br />6. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-mInntO-pV98/XCJd95r1BMI/AAAAAAAACPQ/ShrF8wmUrnggFlykyGsP-LmQ4tes7GMdQCEwYBhgL/s1600/graph06ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="192" data-original-width="497" height="122" src="https://3.bp.blogspot.com/-mInntO-pV98/XCJd95r1BMI/AAAAAAAACPQ/ShrF8wmUrnggFlykyGsP-LmQ4tes7GMdQCEwYBhgL/s320/graph06ans.png" width="320" /></a></div>(ii) $\D (\pi/4, 4), (3\pi/4,-2)$<br />(iii) $\D 3$<br />7. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-cAlaCuSVEho/XCJd-cGNyYI/AAAAAAAACPA/p6xg40yQmBcYpKqm5-3CbtSDUmOWIRaqwCEwYBhgL/s1600/graph07ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="335" data-original-width="724" height="148" src="https://2.bp.blogspot.com/-cAlaCuSVEho/XCJd-cGNyYI/AAAAAAAACPA/p6xg40yQmBcYpKqm5-3CbtSDUmOWIRaqwCEwYBhgL/s320/graph07ans.png" width="320" /></a></div>(ii) 4<br />8. (ii)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-YMfb2kEMKwQ/XCJd_any39I/AAAAAAAACO8/JEiIOVjhpYoOeljJ4-XP4rOztJVQRwO3ACEwYBhgL/s1600/graph08ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="257" data-original-width="628" height="130" src="https://2.bp.blogspot.com/-YMfb2kEMKwQ/XCJd_any39I/AAAAAAAACO8/JEiIOVjhpYoOeljJ4-XP4rOztJVQRwO3ACEwYBhgL/s320/graph08ans.png" width="320" /></a></div>(iii) 3<br />9. 3.2,15,312<br />10. (ii) 2/5<br />11. (ai) A,E<br />(ii) C,D<br />(b)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-DbbYdFZdD1o/XCJeAn3aJSI/AAAAAAAACPA/5mo7k7xOBhAwiO6flI0TtBtrikNoQFj_gCEwYBhgL/s1600/graph11ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="293" data-original-width="337" height="278" src="https://4.bp.blogspot.com/-DbbYdFZdD1o/XCJeAn3aJSI/AAAAAAAACPA/5mo7k7xOBhAwiO6flI0TtBtrikNoQFj_gCEwYBhgL/s320/graph11ans.png" width="320" /></a></div>12. (a)(i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MPS9dzMXSVQ/XCJeBpRTeYI/AAAAAAAACPE/F3KdWAGFW6cXs8hb1ZJsKtKT5N-_DxejQCEwYBhgL/s1600/graph12ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="260" data-original-width="386" height="215" src="https://1.bp.blogspot.com/-MPS9dzMXSVQ/XCJeBpRTeYI/AAAAAAAACPE/F3KdWAGFW6cXs8hb1ZJsKtKT5N-_DxejQCEwYBhgL/s320/graph12ans.png" width="320" /></a></div>(ii)3<br />(iii)180<br />(b) $\D \frac{-1}{\sqrt{1-p^2}}$<br />13. $\D a = 3, b = 2, c = 1$<br />14. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-NdcueTkp9vM/XCJeCxgqlmI/AAAAAAAACPM/G4xWNCBjqM0KPTDWdipLJ1rfb5JoRiqdQCEwYBhgL/s1600/graph14ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="394" data-original-width="618" height="204" src="https://2.bp.blogspot.com/-NdcueTkp9vM/XCJeCxgqlmI/AAAAAAAACPM/G4xWNCBjqM0KPTDWdipLJ1rfb5JoRiqdQCEwYBhgL/s320/graph14ans.png" width="320" /></a></div>(ii) $\D (-0.5, 25/4)$<br />(iii) $\D k &gt; 25/4$<br />15. (a)(i) $\D A = 3,B = 2$<br />(ii) $\D C = 4$<br />(b) 120,5<br />16. (a) 480<br />17. (b)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-4wuHYj0AORs/XCJeD4P8vmI/AAAAAAAACPI/gobOTXNpKrQT6sfeHo46cDk9C64tvgD6ACEwYBhgL/s1600/graph17ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="239" data-original-width="404" height="189" src="https://4.bp.blogspot.com/-4wuHYj0AORs/XCJeD4P8vmI/AAAAAAAACPI/gobOTXNpKrQT6sfeHo46cDk9C64tvgD6ACEwYBhgL/s320/graph17ans.png" width="320" /></a></div>(c)&nbsp; 3,7<br />18. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-BkhWD28wPfo/XCJeE2_1VkI/AAAAAAAACPI/pwJlhdI0sBQ-x0elg3NA4aMzwfqQyFHXQCEwYBhgL/s1600/graph18ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="338" data-original-width="393" height="275" src="https://4.bp.blogspot.com/-BkhWD28wPfo/XCJeE2_1VkI/AAAAAAAACPI/pwJlhdI0sBQ-x0elg3NA4aMzwfqQyFHXQCEwYBhgL/s320/graph18ans.png" width="320" /></a></div>(ii) $\D k &gt; 25/8$<br />19. (i)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-hX7JnFVdIdo/XCJeFaaY4zI/AAAAAAAACPE/L8QV9H2mfIg20dxiOFs5nw9gUT7hardIQCEwYBhgL/s1600/graph19ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="654" height="211" src="https://2.bp.blogspot.com/-hX7JnFVdIdo/XCJeFaaY4zI/AAAAAAAACPE/L8QV9H2mfIg20dxiOFs5nw9gUT7hardIQCEwYBhgL/s320/graph19ans.png" width="320" /></a></div>(ii) $\D 0 &lt; k &lt; 9$<br />20. (a)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-EsxWwYu_VBc/XCJeGLaCTDI/AAAAAAAACPQ/T_5khFvYntMaUZgLSGvARx_29M9Y9xdHwCEwYBhgL/s1600/graph20ans.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="392" data-original-width="691" height="181" src="https://2.bp.blogspot.com/-EsxWwYu_VBc/XCJeGLaCTDI/AAAAAAAACPQ/T_5khFvYntMaUZgLSGvARx_29M9Y9xdHwCEwYBhgL/s320/graph20ans.png" width="320" /></a></div>(b) $\D 4,\pi/3$<br />21. $\D a = 3, b = 2, c = 4$<br />22. $\D 2\sqrt{17}$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com1tag:blogger.com,1999:blog-5969775204143489576.post-43773526014385906952018-12-23T05:51:00.000-08:002018-12-29T00:53:30.964-08:00Straight Line Equations (CIE)$\def\D{\displaystyle}\newcommand{\vcol}{\begin{array}{|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}\newcommand{\vicol}{\begin{array}{|c|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}$1 (CIE 2012, s, paper 11, question 7)<br />The table shows values of variables $\D x$ and $\D y.$<br />$\D \vicol{x&amp; 1&amp; 3&amp; 6&amp; 10&amp; 14}{y&amp; 2.5&amp; 4.5&amp; 0 &amp;–20 &amp;–56}$<br /><br />(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = Ax + Bx^2,$ where $\D A$ and $\D B$ are constants. <br />(ii) Use your graph to find the value of $\D A$ and of $\D B.$ <br /><br />2 (CIE 2012, s, paper 22, question 7)<br />The table shows experimental values of variables $\D x$ and $\D y.$<br />$\D \vcol{x&amp; 5&amp; 30&amp; 150&amp; 400}{y&amp; 8.9&amp; 21.9&amp; 48.9&amp; 80.6}$<br />(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. <br />(ii) Use your graph to estimate the value of $\D a$ and of $\D b.$ <br /><br />3 (CIE 2012, w, paper 11, question 10)<br />The table shows values of the variables $\D x$ and $\D y.$<br />$\D\vicol{ x^{\circ}&amp; 10&amp; 30 &amp;45 &amp;60 &amp;80}{ y &amp;11.2&amp; 16 &amp;19.5&amp; 22.4&amp; 24.7}$<br />(i) Using the graph paper below, plot a suitable straight line graph to show that, for 10° $\D \le x\le$ 80°, $\D \sqrt{y} = A \sin x + B,$ where $\D A$ and $\D B$ are positive constants. <br />(ii) Use your graph to find the value of $\D A$ and of $\D B.$ <br />(iii) Estimate the value of $\D y$ when $\D x = 50.$ <br />(iv) Estimate the value of $\D x$ when $\D y = 12.$ <br /><br />4 (CIE 2012, w, paper 22, question 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UU89Dl6oUBQ/XB-RtTOCNBI/AAAAAAAACMY/OKBHwNutUDgEJcNxidpD3zRYfvlqAhrlgCEwYBhgL/s1600/straightline04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="446" data-original-width="756" height="188" src="https://3.bp.blogspot.com/-UU89Dl6oUBQ/XB-RtTOCNBI/AAAAAAAACMY/OKBHwNutUDgEJcNxidpD3zRYfvlqAhrlgCEwYBhgL/s320/straightline04.png" width="320" /></a></div><br />The variables $\D x$ and $\D y$ are related in such a way that when $\D \lg y$ is plotted against $\D \lg x$ a straight line graph is obtained as shown in the diagram. The line passes through the points (2, 4) and (8, 7).<br />(i) Express $\D y$ in terms of $\D x,$ giving your answer in the form $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. <br />Another method of drawing a straight line graph for the relationship $\D y = ax^b,$ found in part (i), involves plotting $\D \lg x$ on the horizontal axis and $\D \lg(y^2)$ on the vertical axis. For this straight line graph what is<br />(ii) the gradient, <br />(iii) the intercept on the vertical axis? <br /><br />5 (CIE 2012, w, paper 23, question 9)<br />The table shows experimental values of two variables $\D x$ and $\D y.$<br />$\D \vcol{ x&amp; 1&amp; 2&amp; 3&amp; 4}{y&amp; 9.41 &amp;1.29&amp; – 0.69&amp; – 1.77}$<br />It is known that $\D x$ and $\D y$ are related by the equation $\D y = \frac{a}{x^2}+bx,$&nbsp; where $\D a$ and $\D b$ are constants.<br />(i) A straight line graph is to be drawn to represent this information. Given that $\D x^2y$ is plotted on the vertical axis, state the variable to be plotted on the horizontal axis. <br />(ii) On the grid opposite, draw this straight line graph. <br />(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ <br />(iv) Estimate the value of $\D y$ when $\D x$ is 3.7. <br /><br />6 (CIE 2013, s, paper 11, question 2)<br />Variables $\D x$ and $\D y$ are such that $\D y= Ab^x,$&nbsp; where $\D A$ and $\D b$ are constants. The diagram shows the graph of $\D \ln y$ against $\D x,$ passing through the points (2, 4) and (8, 10). Find the value of $\D A$ and of $\D b.$ <br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-0Sn7QSJC_Wc/XB-RtUFoI4I/AAAAAAAACMo/PTj_Re3dPyA59VxSWS_5-8a5OXc1f8lNwCEwYBhgL/s1600/straightline06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="928" data-original-width="1423" height="208" src="https://1.bp.blogspot.com/-0Sn7QSJC_Wc/XB-RtUFoI4I/AAAAAAAACMo/PTj_Re3dPyA59VxSWS_5-8a5OXc1f8lNwCEwYBhgL/s320/straightline06.png" width="320" /></a></div><br />7 (CIE 2013, s, paper 22, question 1)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-wHvNnDDXgZs/XB-RtRv1_ZI/AAAAAAAACMk/fhRpe8BF8S4JmqiHU5DZtq4tEzN0kJNNQCEwYBhgL/s1600/straightline07.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="608" data-original-width="972" height="200" src="https://4.bp.blogspot.com/-wHvNnDDXgZs/XB-RtRv1_ZI/AAAAAAAACMk/fhRpe8BF8S4JmqiHU5DZtq4tEzN0kJNNQCEwYBhgL/s320/straightline07.png" width="320" /></a></div><br />Variables $\D x$ and $\D y$ are such that when $\D \sqrt{y}$ is plotted against $\D x^2$ a straight line graph passing through the points (1, 3) and (4, 18) is obtained. Express $\D y$ in terms of $\D x.$ <br /><br />8 (CIE 2013, w, paper 13, question 10)<br />The variables $\D s$ and $\D t$ are related by the equation $\D t= ks^n,$ where $\D k$ and $\D n$ are constants. The table below shows values of variables $\D s$ and $\D t.$<br />$\D \vcol{s&amp; 2&amp; 4&amp; 6&amp; 8}{t&amp; 25.00&amp; 6.25&amp; 2.78&amp; 1.56}$<br />(i) A straight line graph is to be drawn for this information with $\D \lg t$ plotted on the vertical axis. State the variable which must be plotted on the horizontal axis. <br />(ii) Draw this straight line graph on the grid below. <br />(iii) Use your graph to find the value of $\D k$ and of $\D n.$ <br />(iv) Estimate the value of $\D s$ when $\D t = 4.$ <br /><br />9 (CIE 2013, w, paper 21, question 8)<br />The table shows experimental values of two variables $\D x$ and $\D y.$<br />$\D \vcol{x&amp; 2 &amp;4&amp; 6&amp; 8}{y&amp; 9.6&amp; 38.4&amp; 105&amp; 232}$<br />It is known that $\D x$ and $\D y$ are related by the equation $\D y= ax^3+ bx,$ where $\D a$ and $\D b$ are constants.<br />(i) A straight line graph is to be drawn for this information with $\D \frac{y}{x}$ on the vertical axis. State the variable which must be plotted on the horizontal axis. <br />(ii) Draw this straight line graph on the grid below. <br />(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ <br />(iv) Estimate the value of $\D x$ for which $\D 2y = 25x.$ <br /><br />10 (CIE 2014, s, paper 11, question 8)<br />The table shows values of variables $\D V$ and $\D p.$<br />$\D \vcol{ V &amp;10&amp; 50&amp; 100&amp; 200}{p&amp; 95.0&amp; 8.5&amp; 3.0&amp; 1.1}$<br />(i) By plotting a suitable straight line graph, show that $\D V$ and $\D p$ are related by the equation $\D p = kV^n ,$<br />where $\D k$ and $\D n$ are constants. <br />Use your graph to find<br />(ii) the value of $\D n,$ <br />(iii) the value of $\D p$ when $\D V = 35.$ <br /><br />11 (CIE 2014, s, paper 13, question 10)<br />The table shows experimental values of $\D x$ and $\D y.$<br />$\D \vcol{x&amp; 1.50 &amp;1.75&amp; 2.00&amp; 2.25}{y&amp; 3.9&amp; 8.3 &amp;19.5&amp; 51.7}$<br />(i) Complete the following table.<br />$\D \vcol{x^2&amp;\qquad &amp;\qquad &amp;\qquad &amp;\qquad}{\lg y&amp;&amp;&amp;&amp;}$<br /><br />(ii) By plotting a suitable straight line graph on the graph paper, show that $\D x$ and $\D y$ are related by the equation $\D y= Ab^{x^2},$&nbsp; where $\D A$ and $\D b$ are constants. <br />(iii) Use your graph to find the value of $\D A$ and of $\D b.$ <br />(iv) Estimate the value of $\D y$ when $\D x = 1.25.$ <br /><br />12 (CIE 2014, s, paper 22, question 10)<br />Two variables $\D x$ and $\D y$ are connected by the relationship $\D y = Ab^x ,$ where $\D A$ and $\D b$ are constants.<br />(i) Transform the relationship $\D y = Ab^x$ into a straight line form. <br />An experiment was carried out measuring values of $\D y$ for certain values of $\D x.$ The values of $\D \ln y$ and $\D x$ were plotted and a line of best fit was drawn. The graph is shown on the grid below.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-i4xWbLGgLcs/XB-RuplPvEI/AAAAAAAACMw/1oya_WFM3JgO0XMODGI7rpmjOcbxtq5PQCEwYBhgL/s1600/straightline12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1062" data-original-width="1006" height="320" src="https://1.bp.blogspot.com/-i4xWbLGgLcs/XB-RuplPvEI/AAAAAAAACMw/1oya_WFM3JgO0XMODGI7rpmjOcbxtq5PQCEwYBhgL/s320/straightline12.png" width="303" /></a></div><br />(ii) Use the graph to determine the value of $\D A$ and the value of $\D b,$ giving each to 1 significant figure. <br />(iii) Find $\D x$ when $\D y = 220.$ <br /><br />13 (CIE 2014, w, paper 11, question 9)<br />The table shows experimental values of variables $\D x$ and $\D y.$<br />$\D \vicol{x&amp; 2&amp; 2.5&amp; 3 &amp;3.5&amp; 4}{y&amp; 18.8&amp; 29.6&amp; 46.9&amp; 74.1 &amp;117.2}$<br />(i) By plotting a suitable straight line graph on the grid below, show that $\D x$ and $\D y$ are related by the equation $\D y = ab^x ,$ where $\D a$ and $\D b$ are constants. <br />(ii) Use your graph to find the value of $\D a$ and of $\D b.$ <br /><br />14 (CIE 2014, w, paper 23, question 6)<br />Variables $\D x$ and $\D y$ are such that, when $\D \ln y$ is plotted against $\D 3^x ,$ a straight line graph passing through (4, 19) and (9, 39) is obtained.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-k6yk6VTIkRE/XB-RuxIQGmI/AAAAAAAACMs/_cmCHRPkQ4oNMRucJbB35Jh64cGSFhVtQCEwYBhgL/s1600/straightline14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="698" data-original-width="1600" height="139" src="https://1.bp.blogspot.com/-k6yk6VTIkRE/XB-RuxIQGmI/AAAAAAAACMs/_cmCHRPkQ4oNMRucJbB35Jh64cGSFhVtQCEwYBhgL/s320/straightline14.png" width="320" /></a></div><br />(i) Find the equation of this line in the form $\D \ln y= m3^x+ c,$&nbsp; where $\D m$ and $\D c$ are constants to be found. <br />(ii) Find $\D y$ when $\D x = 0.5.$ <br />(iii) Find $\D x$ when $\D y = 2000.$ <br /><br /><h3>Answers</h3><br />1. (i) $\D y/x = A + Bx$<br />$\D \vicol{x&amp; 1&amp; 3&amp; 6&amp; 10&amp; 14}{y/x&amp; 2.5&amp; 1.5&amp; 0&amp; -2&amp; -4}$<br />(ii) $\D B = -0.5;A = 3$<br />2. (i) $\D \ln y = ln a + b ln x$<br />(ii) $\D b = 0.5; a = 4$<br />(iii) 32 to 49<br />3. (i) $\D \vicol{\sin x&amp; 0.17&amp; 0.5&amp; 0.71&amp; 0.87&amp; 0.98}{\sqrt{y}&amp; 3.35&amp; 4 &amp;4.42&amp; 4.73&amp; 4.97}$<br />(ii) $\D A = 2;B = 3$<br />(iii) $\D y = 20.5$<br />(iv) $\D x = 14.5$<br />4. (i) $\D y = 1000\sqrt{x}$<br />(ii) $\D m = 1$<br />(iii) $\D c = 6$<br />5. (i) $\D x^3$<br />(ii) $\D \vcol{x^3&amp; 1&amp; 8&amp; 27&amp; 64}{x^2y&amp; 9.41 &amp;5.16&amp; -6.21&amp; -28.32}$<br />(iii) $\D a = 10; b = -0.6$<br />(iv) $\D -1.48$<br />6. $\D b = e;A = e^2$<br />7. $\D y = (5x^2 - 2)^2$<br />8. (i) $\D \lg s$<br />(ii) $\D \vcol{\lg s&amp; 0.3 &amp;0.6&amp; 0.78&amp; 0.9}{lg t&amp; 1.4&amp; 0.8&amp; 0.44&amp; 0.19}$<br />(iii) $\D n = -2; k = 100$<br />(iv) $\D s = 4.9$<br />9. (i) $\D x^2$<br />(ii) $\D \vcol{x^2&amp; 4&amp; 16&amp; 36&amp; 64}{\frac{y}{x}&amp; 4.8&amp; 9.6&amp; 17.5&amp; 29}$<br />(iv) $\D 4.8$<br />10. (i)<br />(ii) $\D n = 1.5$<br />(iii) $\D 15$<br />11. (i) $\D \vcol{x^2&amp; 2.25&amp; 3.06&amp; 4&amp; 5.06}{\lg y&amp; 0.59&amp; 0.92 &amp;1.29&amp; 1.71}$<br />(ii)<br />(iii) $\D b = 2.5;A = 0.5$<br />(iv) $\D 2.1$<br />12. (i) $\D \log y = \log A + x \log b$<br />(ii) $\D 0.5$ (iii) $\D 4.4$<br />13. (i)<br />(ii) $\D b = 2.5; a = 3$<br />14. (i) $\D \ln y = 4(3^x) + 3$<br />(ii) $\D y = 20500$<br />(iii) $\D x = 0.127$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-35838001292532010802018-12-22T06:54:00.000-08:002018-12-29T00:54:05.298-08:00Area of Sector (CIE, IGCSE Edexcel)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 12, question 8)<br /><a href="https://4.bp.blogspot.com/-GbhAFpCHW3o/XB5NpEmozvI/AAAAAAAACLM/DyOpN8wSHuIwIHhAXKMdTpSHr59Z95QdACEwYBhgL/s1600/sector01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="361" data-original-width="569" height="253" src="https://4.bp.blogspot.com/-GbhAFpCHW3o/XB5NpEmozvI/AAAAAAAACLM/DyOpN8wSHuIwIHhAXKMdTpSHr59Z95QdACEwYBhgL/s400/sector01.png" width="400" /></a><br />The figure shows a circle, centre $\D&nbsp; O,$ with radius 10 cm. The lines $\D XA$ and $\D XB$ are tangents to the circle at $\D A$ and $\D B$ respectively, and angle $\D AOB$ is $\D \frac{2\pi}{3}$ radians.<br />(i) Find the perimeter of the shaded region. <br />(ii) Find the area of the shaded region. <br /><br />2 (CIE 2012, s, paper 21, question 11)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-aKHN98dvcE8/XB5NpLGY4QI/AAAAAAAACLI/1cMyN3jpB24Rm5vo4JdKJwZ5SrC_INudQCEwYBhgL/s1600/sector02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="293" data-original-width="441" height="211" src="https://3.bp.blogspot.com/-aKHN98dvcE8/XB5NpLGY4QI/AAAAAAAACLI/1cMyN3jpB24Rm5vo4JdKJwZ5SrC_INudQCEwYBhgL/s320/sector02.png" width="320" /></a></div>The diagram shows a right-angled triangle $\D ABC$ and a sector $\D CBDC$ of a circle with centre $\D C$ and radius 12 cm. Angle $\D ACB = 1$ radian and $\D ACD$ is a straight line.<br />(i) Show that the length of $\D AB$ is approximately 10.1 cm. <br />(ii) Find the perimeter of the shaded region. <br />(iii) Find the area of the shaded region. <br /><br />3 (CIE 2012, w, paper 12, question 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-umaFazlKYlg/XB5No-yiChI/AAAAAAAACL8/HAuM7r4P5SAFT95pvdBDIrCsxXEDehxNACEwYBhgL/s1600/sector03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="417" data-original-width="730" height="227" src="https://3.bp.blogspot.com/-umaFazlKYlg/XB5No-yiChI/AAAAAAAACL8/HAuM7r4P5SAFT95pvdBDIrCsxXEDehxNACEwYBhgL/s400/sector03.png" width="400" /></a></div>The diagram shows an isosceles triangle $\D OBD$ in which $\D OB = OD = 18$ cm and angle $\D BOD = 1.5$ radians. An arc of the circle, centre $\D O$ and radius 10 cm, meets $\D OB$ at $\D A$ and $\D OD$ at $\D C.$<br />(i) Find the area of the shaded region. <br />(ii) Find the perimeter of the shaded region. <br /><br />4 (CIE 2012, w, paper 13, question 9)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-0Np7TL5WMe4/XB5NqK1He0I/AAAAAAAACMI/0lI1gx5e_hYah1dnYtOrJo5Wkclf3_YNgCEwYBhgL/s1600/sector04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="396" data-original-width="959" height="165" src="https://2.bp.blogspot.com/-0Np7TL5WMe4/XB5NqK1He0I/AAAAAAAACMI/0lI1gx5e_hYah1dnYtOrJo5Wkclf3_YNgCEwYBhgL/s400/sector04.png" width="400" /></a></div>The diagram shows four straight lines, $\D AD, BC, AC$ and $\D BD.$ Lines $\D AC$ and $\D BD$ intersect at $\D O$ such that angle $\D AOB$ is $\D \frac{\pi}{6}$ radians. $\D AB$ is an arc of the circle, centre $\D O$ and radius 10 cm, and $\D CD$ is an arc of the circle, centre $\D O$ and radius 20 cm.<br />(i) Find the perimeter of $\D ABCD.$ <br />(ii) Find the area of $\D ABCD.$ <br /><br />5 (CIE 2012, w, paper 21, question 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-dGfqXsSfr7w/XB5NqL4AKiI/AAAAAAAACMI/9e-4FBxF330cgOsTdVDKpg2O7LGeDpp3wCEwYBhgL/s1600/sector05.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="451" height="281" src="https://3.bp.blogspot.com/-dGfqXsSfr7w/XB5NqL4AKiI/AAAAAAAACMI/9e-4FBxF330cgOsTdVDKpg2O7LGeDpp3wCEwYBhgL/s320/sector05.png" width="320" /></a></div><br />In the diagram $\D PQ$ and $\D RS$ are arcs of concentric circles with centre $\D O$ and angle $\D POQ = 1$ radian. The radius of the larger circle is $\D x$ cm and the radius of the smaller circle is $\D y$ cm.<br />(i) Given that the perimeter of the shaded region is 20 cm, express $\D y$ in terms of $\D x.$ <br />(ii) Given that the area of the shaded region is 16cm$\D^2,$ express $\D y^2$ in terms of $\D x^2.$ <br />(iii) Find the value of $\D x$ and of $\D y.$ <br /><br />6 (CIE 2013, s, paper 11, question 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-WVCGgRZZ8us/XB5NqY-xBeI/AAAAAAAACMA/S1KpjWqZ2TUobrqFa0v8LCbE2VUt45OuQCEwYBhgL/s1600/sector06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="489" data-original-width="565" height="275" src="https://3.bp.blogspot.com/-WVCGgRZZ8us/XB5NqY-xBeI/AAAAAAAACMA/S1KpjWqZ2TUobrqFa0v8LCbE2VUt45OuQCEwYBhgL/s320/sector06.png" width="320" /></a></div><br />The diagram shows a square $\D ABCD$ of side 16 cm. $\D M$ is the mid-point of $\D AB.$ The points $\D E$ and $\D F$ are on $\D AD$ and $\D BC$ respectively such that $\D AE = BF = 6$ cm. $\D EF$ is an arc of the circle centre $\D M,$ such that angle $\D EMF$ is $\D \theta$ radians.<br />(i) Show that $\D \theta = 1.855$ radians, correct to 3 decimal places. <br />(ii) Calculate the perimeter of the shaded region. <br />(iii) Calculate the area of the shaded region. <br /><br />7 (CIE 2013, s, paper 22, question 6)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-IxyUvpy1A3k/XB5NrE7YriI/AAAAAAAACL8/vi1nl8KGImQ0ZaTnuDSeBdkUW-FA-A_AgCEwYBhgL/s1600/sector07.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="333" data-original-width="312" height="200" src="https://1.bp.blogspot.com/-IxyUvpy1A3k/XB5NrE7YriI/AAAAAAAACL8/vi1nl8KGImQ0ZaTnuDSeBdkUW-FA-A_AgCEwYBhgL/s200/sector07.png" width="186" /></a></div>The shaded region in the diagram is a segment of a circle with centre $\D O$ and radius $\D r$ cm. Angle $\D AOB = \frac{\pi}{3}$ radians.<br />(i) Show that the perimeter of the segment is $\D r\left(\frac{3+\pi}{3}\right).$ <br />(ii) Given that the perimeter of the segment is 26 cm, find the value of $\D r$ and the area of the<br />segment. <br /><br />8 (CIE 2013, w, paper 13, question 8)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-euZonldSY0E/XB5NrZPfWBI/AAAAAAAACMA/Fi7zzhvfhgYLudx7LVH2j8iq9YpjtZDaQCEwYBhgL/s1600/sector08.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="374" data-original-width="394" height="303" src="https://4.bp.blogspot.com/-euZonldSY0E/XB5NrZPfWBI/AAAAAAAACMA/Fi7zzhvfhgYLudx7LVH2j8iq9YpjtZDaQCEwYBhgL/s320/sector08.png" width="320" /></a></div>The diagram shows two concentric circles, centre $\D O,$ radii 4 cm and 6 cm. The points $\D A$ and $\D B$ lie on the larger circle and the points $\D C$ and $\D D$ lie on the smaller circle such that $\D ODA$ and $\D OCB$ are straight lines.<br />(i) Given that the area of triangle $\D OCD$ is 7.5 cm$\D ^2,$ show that $\D \theta = 1.215$ radians, to 3 decimal places. <br />(ii) Find the perimeter of the shaded region. <br />(iii) Find the area of the shaded region. <br /><br />9 (CIE 2013, w, paper 21, question 10)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-H6IIcisyuZ0/XB5Nrusk5eI/AAAAAAAACME/u3G4-IrVLOY_ZyVlyuBRbrq3cGalBDabQCEwYBhgL/s1600/sector09.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="364" data-original-width="383" height="304" src="https://4.bp.blogspot.com/-H6IIcisyuZ0/XB5Nrusk5eI/AAAAAAAACME/u3G4-IrVLOY_ZyVlyuBRbrq3cGalBDabQCEwYBhgL/s320/sector09.png" width="320" /></a></div>The diagram shows a circle with centre $\D O$ and a chord $\D AB.$ The radius of the circle is 12 cm andangle AOB is 1.4 radians.<br />(i) Find the perimeter of the shaded region. <br />(ii) Find the area of the shaded region. <br /><br />10 (CIE 2014, s, paper 12, question 7)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-nmddyoMe_KM/XB5NsN1xeAI/AAAAAAAACMI/fN1q5VjED_8ohrrB-EkbckD9EfHi7o8GwCEwYBhgL/s1600/sector10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="269" data-original-width="273" src="https://4.bp.blogspot.com/-nmddyoMe_KM/XB5NsN1xeAI/AAAAAAAACMI/fN1q5VjED_8ohrrB-EkbckD9EfHi7o8GwCEwYBhgL/s1600/sector10.png" /></a></div>The diagram shows a circle, centre $\D O,$ radius 8 cm. Points $\D P$ and $\D Q$ lie on the circle such that the chord $\D PQ = 12$ cm and angle $\D POQ = \theta$ radians.<br />(i) Show that $\D \theta = 1.696,$ correct to 3 decimal places. <br />(ii) Find the perimeter of the shaded region. <br />(iii) Find the area of the shaded region. <br /><br />11 (CIE 2014, s, paper 23, question 1)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-ExnyQAsa8Fk/XB5Nspehy0I/AAAAAAAACME/ofjaZNzDgiABUwdn6HRGuncjlxo_TJtUQCEwYBhgL/s1600/sector11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="298" data-original-width="303" height="196" src="https://3.bp.blogspot.com/-ExnyQAsa8Fk/XB5Nspehy0I/AAAAAAAACME/ofjaZNzDgiABUwdn6HRGuncjlxo_TJtUQCEwYBhgL/s200/sector11.png" width="200" /></a></div><br />The diagram shows a sector of a circle of radius $\D r$ cm. The angle of the sector is 1.6 radians and the area of the sector is 500 cm$\D ^2 .$<br />(i) Find the value of $\D r.$ <br />(ii) Hence find the perimeter of the sector. <br /><br />12 (CIE 2014, w, paper 13, question 6)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-cX7zE7f8Vpk/XB5Ns04OXQI/AAAAAAAACMI/Q_VNWsmByJsuuRz8HnW7bNirmypYzzeDQCEwYBhgL/s1600/sector12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="292" data-original-width="348" height="268" src="https://1.bp.blogspot.com/-cX7zE7f8Vpk/XB5Ns04OXQI/AAAAAAAACMI/Q_VNWsmByJsuuRz8HnW7bNirmypYzzeDQCEwYBhgL/s320/sector12.png" width="320" /></a></div>The diagram shows a sector, $\D AOB,$ of a circle centre $\D O,$ radius 12 cm. Angle $\D AOB = 0.9$ radians. The point $\D C$ lies on $\D OA$ such that $\D OC = CB.$<br />(i) Show that $\D OC = 9.65$ cm correct to 3 significant figures. <br />(ii) Find the perimeter of the shaded region. <br />(iii) Find the area of the shaded region. <br /><br />13 (CIE 2014, w, paper 21, question 11)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-0hpnh6X361E/XB5NtYnTg9I/AAAAAAAACME/rykRbtThgkY6gtSXsPA7HbUMUPtxdvMBQCEwYBhgL/s1600/sector13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="390" data-original-width="607" height="205" src="https://2.bp.blogspot.com/-0hpnh6X361E/XB5NtYnTg9I/AAAAAAAACME/rykRbtThgkY6gtSXsPA7HbUMUPtxdvMBQCEwYBhgL/s320/sector13.png" width="320" /></a></div>The diagram shows a sector $\D OPQ$ of a circle with centre $\D O$ and radius $\D x$ cm. Angle $\D POQ$ is 0.8 radians. The point $\D S$ lies on $\D OQ$ such that $\D OS = 5$ cm. The point $\D R$ lies on $\D OP$ such that angle $\D ORS$ is a right angle. Given that the area of triangle $\D ORS$ is one-fifth of the area of sector $\D OPQ,$ find<br />(i) the area of sector $\D OPQ$ in terms of $\D x$ and hence show that the value of $\D x$ is 8.837 correct to 4 significant figures, <br />(ii) the perimeter of $\D PQSR,$ <br />(iii) the area of $\D PQSR.$ <br /><br />Answers<br />1. (i) $\D 55.6$<br />(ii) $\D 68.5$<br />2. (ii) $\D 54.3$<br />(iii) $\D 187$<br />3. (i) $\D 86.6$<br />(ii) $\D 55.5$<br />4. (i) $\D 73.9,$<br />(ii) $\D 231$<br />5. (i) $\D y = 3x - 20$<br />(ii) $\D y^2 = x^2 -32$<br />(iii) $\D x = 9; y = 7$<br />6. (ii) $\D P = 54.6$<br />(iii) $\D A = 115.25$<br />7. (ii) $\D r = 12.7;A = 14.6$<br />8. (ii) $\D 15.9$<br />(iii) $\D 14.4$<br />9. (i) $\D 74.1$<br />(ii) $\D 422$<br />10.&nbsp; $\D P=48.7,A=178.5$<br />11.&nbsp; $\D 25; 90$<br />12.&nbsp; $\D P = 22.8;A = 19.4$<br />13.&nbsp; &nbsp;(ii) $\D P = 19.8;A = 25$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-58911362526608654582018-12-22T03:08:00.000-08:002019-01-17T05:58:31.119-08:00Logarithms (CIE, Myanmar Grade 9)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 11, question 8)<br />(a) Find the value of $\D x$ for which $\D 2\lg x - \lg(5x + 60) = 1 .$ <br />(b) Solve $\D \log_5 y = 4\log_y 5 .$ <br /><br />2 (CIE 2012, w, paper 11, question 3)<br />Given that $\D p = \log_q 32,$ express, in terms of $\D p,$<br />(i) $\D \log_q 4,$ <br />(ii) $\D \log_q 16q.$ <br /><br />3 (CIE 2012, w, paper 12, question 4)<br />Given that $\D \log_a pq = 9$ and $\D \log_a p^2q = 15,$ find the value of<br />(i) $\D \log_a p$ and of $\D \log_a q,$ <br />(ii) $\D \log_p a + \log_q a.$ <br /><br />4 (CIE 2013, s, paper 11, question 4)<br />(i) Given that $\D \log_4 x=\frac{1}{2},$&nbsp; find the value of $\D x.$ <br />(ii) Solve $\D 2\log_4y- \log_4 (5y-12)=\frac{1}{2}.$ <br /><br />5 (CIE 2013, w, paper 13, question 2)<br />Solve $\D 2 \lg y - \lg(5y+60)=1.$ <br /><br />6 (CIE 2013, w, paper 21, question 4)<br />Given that $\D \log_p X= 5$ and $\D \log_p Y= 2,$ find<br />(i) $\D \log_p X^2,$ <br />(ii) $\D \log_p\frac{1}{X},$&nbsp; <br />(iii) $\D \log_{XY} p .$ <br /><br />7 (CIE 2014, s, paper 22, question 6)<br />(a) (i) State the value of $\D u$ for which $\D \lg u = 0.$ <br />(ii) Hence solve $\lg |2x + 3| = 0.$ <br />(b) Express $\D 2 \log_315- (\log_a5) (\log_3a),$ where $\D a &gt; 1,$ as a single logarithm to base 3. <br /><br />8 (CIE 2014, s, paper 23, question 2)<br />Using the substitution $\D u= \log_3 x,$ solve, for $\D x,$ the equation $\D \log_3x -12 \log_x3= 4 .$ <br /><br />9 (CIE 2014, w, paper 13, question 7)<br />Solve the equation $\D 1+ 2 \log_5 x= \log_5(18x-9).$ <br /><br />10 (CIE 2014, w, paper 21, question 3)<br />Solve the following simultaneous equations.<br />$\D \begin{array}{rcl}<br />\log_2(x+3)&amp;=&amp;2+\log_2y\\<br />\log_2(x+y)&amp;=&amp;3<br />\end{array}$ <br /><br />Answers<br />1. (a) $\D x = 60$<br />(b) $\D y = 25,\frac{1}{25}$<br />2. (i) $\D 2p/5$<br />(ii) $\D 1 + 4p/5$<br />3. (i) $\D \log_a p = 6,\log_a q = 3$<br />(ii) $\D 0.5$<br />4. (i) $\D 2$<br />(ii) $\D y = 4, 6$<br />5. $\D 60$<br />6. $\D 10;-5;1/7$<br />7. (a) $\D 1;x = -1,-2$<br />(b) $\log_3 45$<br />8. $\D 729; 1/9$<br />9. $\D x = 3/5; 3$<br />10. $\D x = 5.8; y = 2.2$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-35838089065869022912018-12-22T01:27:00.000-08:002018-12-23T14:39:30.279-08:00Selected Trigonometric Examples<h2>$\def\D{\displaystyle}$</h2><h3>Example 1&nbsp;</h3>Prove that $\D \sin^4x+\cos^4x=\frac{1}{4}(3+\cos 4x).$<br /><br />$\D \begin{array}{|rl|}\hline (x+y)^2&amp;=x^2+y^2+2xy \\ \sin^2 x&amp;=\frac{1-\cos 2x}{2} \\ \sin^2 x+\cos^2 x&amp;=1\\ 2\sin x\cos x&amp;=\sin 2x\\ \hline \end{array}$<br /><br /><h3>Proof:&nbsp;</h3>\begin{eqnarray*}<br />\left(\sin^2x+\cos^2x\right)^2&amp;=&amp;\sin^4x+\cos^4x+2\sin^2x\cos^2x\\<br />1^2&amp;=&amp;\sin^4x+\cos^4x+\frac{1}{2}(2\sin x\cos x)^2\\<br />1&amp;=&amp;\sin^4x+\cos^4x+\frac{1}{2}\sin^22x\\<br />&amp;=&amp;\sin^4x+\cos^4x+\frac{1}{2}\times \frac{1-\cos4x}{2}\\<br />\sin^4x+\cos^4x&amp;=&amp;1-\left( \frac{1}{4}-\frac{\cos4x}{4}\right) \\<br />&amp;=&amp;\frac{3}{4}+\frac{\cos4x}{4}<br />\end{eqnarray*}<br />Hence $\D \sin^4x+\cos^4x=\frac{1}{4}\left(3+\cos 4x\right).$<br /><br /><h3>Example 2&nbsp;</h3>If $\D \sin x+\cos x=a,$ then show that $\D \sin^6x+\cos^6x=\frac{1}{4}\left(4-3\left(a^2-1\right)^2\right).$<br /><br />$\D\begin{array}{|rcl|}\hline<br />(x+y)^3&amp;=&amp;x^3+y^3+3xy(x+y)\\<br />(x+y)^2&amp;=&amp;x^2+y^2+2xy\\<br />1&amp;=&amp;\sin^2x+\cos^2x\\ \hline<br />\end{array}$<br /><br /><h3>Proof:&nbsp;</h3>\begin{eqnarray*}<br />a^2&amp;=&amp;\left(\sin x+\cos x\right)^2\\<br />&amp;=&amp;\sin^2x+\cos^2x+2\sin x\cos x\\<br />&amp;=&amp;1+2\sin x\cos x\\<br />\sin x\cos x&amp;=&amp;\frac{a^2-1}{2}<br />\end{eqnarray*}<br />\begin{eqnarray*}<br />\left(\sin^2x+\cos^2x\right)^3<br />&amp;=&amp;\left(\sin^2x\right)^3+\left(\cos^2x\right)^3\\<br />&amp;&amp;+3\sin^2x\cos^2x\left(\sin^2x+\cos^2x\right)\\<br />1^3&amp;=&amp;\sin^6x+\cos^6x+3(\sin x\cos x)^2\times 1\\<br />1&amp;=&amp;\sin^6x+\cos^6x+3\left(\frac{a^2-1}{2}\right)^2\\<br />\sin^6x+\cos^6x&amp;=&amp;1-\frac{3}{4}(a^2-1)^2\\<br />&amp;=&amp;\frac{1}{4}[4-3(a^2-1)^2]<br />\end{eqnarray*}<br /><br /><h3>Example 3&nbsp;</h3>If $\D \cos x-\sin x=\sqrt{2}\sin x$, show that $\D cos x+\sin x=\sqrt{2}\cos x.$<br /><br /><h3>Proof:&nbsp;</h3>$\D \begin{array}{lrll}<br />&amp;\cos x&amp;=\sin x+\sqrt{2}\sin x&amp;\cdots (1)\\<br />(1)\times \sqrt{2}:&amp;&nbsp; \sqrt{2}\cos x&amp;=\sqrt{2}\sin x+2\sin x&amp;\cdots (2)\\<br />(2)-(1):&amp; \sqrt{2}\cos x-\cos x&amp;=\sin x &amp;<br />\end{array}$<br />Hence $\D \sqrt{2}\cos x=\sin x+\cos x.$<br /><br /><h3>Example 4</h3>Prove that $\frac{\cos 3x+\sin 3x}{\cos x-\sin x}=1+2\sin 2x.$<br />$\D \begin{array}{|rl|}\hline<br />\cos x-\cos y&amp;\D =-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\\<br />\sin x+\sin y&amp;=\D \quad 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\\ \hline<br />\end{array}$<br /><div><br /></div><h3>Proof:</h3>$\D \begin{array}{rll}<br />\cos 3x-\cos x&amp;=-2\sin \frac{3x+x}{2}\sin \frac{3x-x}{2}\\<br />&amp;=-2\sin 2x\sin x&amp;\cdots (1)\\<br />\sin 3x+\sin x&amp;=2\sin \frac{3x+x}{2}\cos\frac{3x-x}{2}\\<br />&amp;=2\sin2x\cos x&amp;\cdots (2)<br />\end{array}$<br /><br />(1)+(2): $\D \cos 3x+\sin 3x-(\cos x-\sin x) =2\sin 2x(\cos x-\sin x).$<br />$\div (\cos x-\sin x): \frac{\cos 3x+\sin 3x}{\cos x-\sin x}-1=2\sin x.$<br />Therefore $\frac{\cos 3x+\sin 3x}{\cos x-\sin x}=1+2\sin x.$<br /><br />Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-31444744010731875132018-12-21T00:52:00.000-08:002018-12-29T00:57:10.368-08:00Indices (CIE, Myanmar Grade 9)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 12, question 2)<br />Using the substitution $\D u = 2^x,$ find the values of $\D x$ such that $\D 2^{2x+2} = 5 (2^x) - 1 .$ <br /><br />2 (CIE 2012, s, paper 21, question 5)<br />(a) Solve the equation $\D 3^{2x} = 1000,$ giving your answer to 2 decimal places. <br />(b) Solve the equation $\D \frac{36^{2y-5}}{6^{3y}}=\frac{6^{2y-1}}{216^{y+6}}.$ <br /><br />3 (CIE 2012, w, paper 11, question 4)<br />Using the substitution $\D u = 5^x,$ or otherwise, solve<br />$\D 5^{2x+1} = 7(5^x) - 2.$ <br /><br />4 (CIE 2012, w, paper 22, question 6)<br />(i) Given that $\D \frac{2^{x-3}}{8^{2y-3}}=16^{x-y},$&nbsp; show that $\D 3x + 2y = 6.$ <br />(ii) Given also that $\D \frac{5^y}{125^{x-2}}=25,$ find the value of $\D x$ and of $\D y.$ <br /><br />5 (CIE 2012, w, paper 23, question 11)<br />(a) Solve $\D \left( 2^{x-2}\right)^\frac{1}{2}=100,$&nbsp; giving your answer to 1 decimal place. <br />(b) Solve $\D \log_y 2 = 3 - \log_y 256.$ <br />(c) Solve $\D \frac{6^{5z-2}}{36^z}= \frac{216^{z-1}}{36^{3-z}}.$<br /><br />6 (CIE 2013, s, paper 22, question 2)<br />(a) Solve the equation $\D 3^{p+1} = 0.7 ,$ giving your answer to 2 decimal places. <br />(b) Express $\D \frac{y\times (4x^3)^2}{\sqrt{8y^3}}$&nbsp; in the form $\D 2^a \times&nbsp; x^b \times&nbsp; y^c,$ where $\D a, b$ and $\D c$ are constants. <br /><br />7 (CIE 2013, w, paper 21, question 5)<br />Solve the simultaneous equations<br />$\D \begin{array}{rcl}<br />\frac{4^x}{256^y}&amp;=&amp;1024,\\<br />3^{2x}\times 9^y&amp;=&amp;243.<br />\end{array}$ <br /><br />8 (CIE 2014, s, paper 12, question 5)<br />(i) Given that $\D 2^{5x}\times 4^y= \frac{1}{8},$&nbsp; show that $\D 5x + 2y = -3.$ <br />(ii) Solve the simultaneous equations $\D 2^{5x}\times 4^y= \frac{1}{8}$ and $\D 7^x\times 49^{2y}=1.$ <br /><br />9 (CIE 2014, s, paper 13, question 2)<br />Given that $\D 2^{4x}\times 4^y\times 8^{x-y}=1$ and $\D 3^{x+y}= \frac{1}{3},$&nbsp; find the value of $\D x$ and of $\D y.$ <br /><br />10 (CIE 2014, s, paper 21, question 11)<br />(a) Solve $\D 2^{x^2-5x}=\frac{1}{64}.$ <br />(b) By changing the base of $\D \log_{2a} 4&nbsp; ,$ express $\D (\log_{2a} 4)(1+\log_a 2)$ as a single logarithm to base $\D a.$ <br /><br />11 (CIE 2014, w, paper 11, question 4)<br />(i) Using the substitution $\D y = 5^x ,$ show that the equation $\D 5^{2x+1}-5^{x+1}+2=2(5^x)$ can be written in the form $\D ay^2+by+2=0,$ where $\D a$ and $\D b$ are constants to be found. <br />(ii) Hence solve the equation $\D 5^{2x+1}-5^{x+1}+2=2(5^x).$ <br /><br />12 (CIE 2014, w, paper 13, question 10)<br />Solve the following simultaneous equations.<br />$\D \begin{array}{rcl}<br />\frac{5^x}{25^{3y-2}}&amp;=&amp;1\\<br />\frac{3^x}{27^{y-1}}&amp;=&amp;81<br />\end{array}$ <br /><br /><h3>Answers</h3>1. $\D x = 0,-2$<br />2. (a) $\D 3.14$<br />(b) $\D y = -4.5$<br />3. $\D x = 0;-0.569$<br />4. (ii) $\D x = 14/9; y = 2/3$<br />5. (a) $\D x = 15.3$<br />(b) $\D y = 8$<br />(c) $\D z = 3.5$<br />6. (a) $\D p = -1.32$<br />(b) $\D a = 5/2; b = 6; c = -1/2$<br />7. $\D x = 3; y = -0.5$<br />8. $\D x = -2/3; y = 1/6$<br />9. $\D x = -1/8; y = -7/8$<br />10. (a) $\D x = 2; 3$<br />(b) $\D \log_a 4$<br />11. (i) $\D 5y^2 - 7y + 2 = 0$<br />(ii) $\D x = 0; y = 1; x = -0.569; y =2/5$<br />12. (a) $\D x = 6; y = 5/3$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-77656790040875349072018-12-18T06:18:00.000-08:002018-12-29T00:57:48.241-08:00Application of exponent$\def\D{\displaystyle}$<br /><br />1 (CIE 2012, s, paper 12, question 9)<br />Variables $\D N$ and $\D x$ are such that $\D N = 200 + 50e^{\frac{x}{100}}.$<br />(i) Find the value of $\D N$ when $\D x = 0.$ <br />(ii) Find the value of $\D x$ when $\D N = 600.$ <br />(iii) Find the value of $\D N$ when $\D \frac{dN}{dx}=45.$ <br /><br />2 (CIE 2014, w, paper 21, question 5)<br />The number of bacteria $\D B$ in a culture, $\D t$ days after the first observation, is given by<br />$B= 500 +400e^{0.2t}.$<br />(i) Find the initial number present. <br />(ii) Find the number present after 10 days. <br />(iii) Find the rate at which the bacteria are increasing after 10 days. <br />(iv) Find the value of $\D t$ when $\D B = 10000.$ <br /><br />3 (CIE 2014, w, paper 23, question 4)<br />The profit \\D P$made by a company in its nth year is modelled by $P=1000e^{an+b}.$<br />In the first year the company made \$2000 profit.<br />(i) Show that $\D a + b = \ln 2.$ <br />In the second year the company made \$3297 profit.<br />(ii) Find another linear equation connecting$\D a$and$\D b.$<br />(iii) Solve the two equations from parts (i) and (ii) to find the value of$\D a$and of$\D b.$<br />(iv) Using your values for$\D a$and$\D b,$find the profit in the 10th year. <br /><br />4 (CIE 2016, w, paper 21, question 4)<br />The number of bacteria,$\D N,$present in a culture can be modelled by the equation$\D N= 7000+ 2000e^{-0.05t},$where$\D t$is measured in days. Find<br />(i) the number of bacteria when$\D t = 10,$<br />(ii) the value of$\D t$when the number of bacteria reaches 7500, <br />(iii) the rate at which the number of bacteria is decreasing after 8 days. <br /><br />5 (CIE 2017, march, paper 12, question 11)<br />It is given that$\D y = Ae^{bx} ,$where$\D A$and$\D b$are constants. When$\D \ln y$is plotted against$\D x$a straight line graph is obtained which passes through the points (1.0, 0.7) and (2.5, 3.7).<br />(i) Find the value of$\D A$and of$\D b.$<br />(ii) Find the value of$\D y$when$\D x = 2.$<br /><br />6 (CIE 2017, march, paper 22, question 2)<br />The value,$\D V$dollars, of a car aged$\D t$years is given by$\D V = 12000 e^{-0.2t}.$<br />(i) Write down the value of the car when it was new. <br />(ii) Find the time it takes for the value to decrease to$\D \frac{2}{3}$of the value when it was new. <br /><br />7 (CIE 2017, s, paper 12, question 7)<br />It is given that$\D y = A(10^{bx}),$where$\D A$and$\D b$are constants. The straight line graph obtained when$\D \lg y$is plotted against$\D x$passes through the points (0.5, 2.2) and (1.0, 3.7).<br />(i) Find the value of$\D A$and of$\D b.$<br />Using your values of$\D A$and$\D b,$find<br />(ii) the value of$\D y$when$\D x = 0.6,$<br />(iii) the value of$\D x$when$\D y = 600.$<br /><br />8 (CIE 2018, s, paper 11, question 5)<br />The population,$\D P,$of a certain bacterium$\D t$days after the start of an experiment is modelled by$\D P = 800e^{kt},$where$\D k$is a constant.<br />(i) State what the figure 800 represents in this experiment. <br />(ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of$\D k.$<br />(iii) Calculate the population three days after the start of the experiment. <br /><br />9 (CIE 2018, s, paper 12, question 7)<br />A population,$\D B,$of a particular bacterium,$\D t$hours after measurements began, is given by$\D B =1000e^{\frac{t}{4}}.$<br />(i) Find the value of$\D B$when$\D t = 0.$<br />(ii) Find the time taken for$\D B$to double in size. <br />(iii) Find the value of$\D B$when$\D t = 8.$<br /><br /><br />1.$\D 250;208;4700$<br />2. (i)$\D 900$<br />(ii)$\D 3456$<br />(iii)$\D 591$<br />(iv)$\D 15.8$<br />3. (ii)$\D 2a + b = \ln 3.297$<br />(iii)$\D a = 0.5; b = 0.193$<br />(iv)$\D n = 10; P = 180000$<br />4. (i)$\D 8213$<br />(ii)$\D 27.7$<br />(iii)$\D \pm 67$<br />5. (i)$\D b = 2;A = 0.273,$<br />(ii)$\D 14.9$<br />6. (i)$\D 12 000$<br />(ii)$\D 2$<br />7. (i)$\D b=3,A=5.01$or$\D 10.7$<br />(ii)$\D y=315$or$\D 102.5$<br />(iii)$\D x = 0.693$<br />8. The number of bacteria at the<br />start of the experiment<br />$\D 1.61,100 000$<br />9. (i)$\D 1000,$<br />(ii)$\D 2.77$<br />(iii)$\D 7389$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-87401847684374002432018-12-18T00:47:00.003-08:002018-12-29T00:58:32.848-08:00Coordinate Geometry (CIE)$\def\D{\displaystyle}$<br /><br />1 (CIE 2012, s, paper 12, question 11)<br />The point$\D P$lies on the line joining$\D A(-1, -5)$and$\D B(11, 13)$such that$\D AP = \frac{1}{3} AB.$<br />(i) Find the equation of the line perpendicular to$\D AB$and passing through$\D P.$<br />The line perpendicular to$\D AB$passing through$\D P$and the line parallel to the x-axis passing through$\D B$intersect at the point$\D Q.$<br />(ii) Find the coordinates of the point$\D Q.$<br />(iii) Find the area of the triangle$\D PBQ.<br /><br />2 (CIE 2012, s, paper 21, question 10)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-FY7lSyTrFEA/XBiyWqpBghI/AAAAAAAACKs/r1iGXDd_zzgXqsGpV7EMLgBVbHLOR3MdACEwYBhgL/s1600/coordinate02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="451" data-original-width="633" height="227" src="https://3.bp.blogspot.com/-FY7lSyTrFEA/XBiyWqpBghI/AAAAAAAACKs/r1iGXDd_zzgXqsGpV7EMLgBVbHLOR3MdACEwYBhgL/s320/coordinate02.png" width="320" /></a></div><br />The diagram shows a trapezium\D ABCD$with vertices$\D A(11, 4), B(7, 7), C(-3, 2)$and$\D D.$The side$\D AD$is parallel to$\D BC$and the side$\D CD$is perpendicular to$\D BC.$Find the area of the trapezium$\D ABCD.$<br /><br />3 (CIE 2012, w, paper 11, question 8)<br />The points$\D A(-3, 6), B(5, 2)$and$\D C$lie on a straight line such that$\D B$is the mid-point of$\D AC.$<br />(i) Find the coordinates of$\D C.$<br />The point$\D D$lies on the y-axis and the line$\D CD$is perpendicular to$\D AC.$<br />(ii) Find the area of the triangle$\D ACD.$<br /><br />4 (CIE 2012, w, paper 12, question 5)<br />The line$\D x - 2y = 6$intersects the curve$\D x^2 + xy + 10y + 4y^2 = 156$at the points$\D A$and$\D B.$Find the length of$\D AB.<br /><br />5 (CIE 2012, w, paper 12, question 7)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-EPOJu9HiH-Y/XBiyW7Zgn_I/AAAAAAAACKk/xCwm_itspRo9lNJm0qoe-Rmk4L-SD3nqQCEwYBhgL/s1600/coordinate05.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="571" data-original-width="845" height="216" src="https://4.bp.blogspot.com/-EPOJu9HiH-Y/XBiyW7Zgn_I/AAAAAAAACKk/xCwm_itspRo9lNJm0qoe-Rmk4L-SD3nqQCEwYBhgL/s320/coordinate05.png" width="320" /></a></div><br />The vertices of the trapezium\D ABCD$are the points$\D A(-5, 4), B(8, 4), C(6, 8)$and$\D D.$The line$\D AB$is parallel to the line$\D DC.$The lines$\D AD$and$\D BC$are extended to meet at$\D E$and angle$\D AEB = 90^{\circ}.$<br />(i) Find the coordinates of$\D D$and of$\D E.$<br />(ii) Find the area of the trapezium$\D ABCD.$<br /><br />6 (CIE 2012, w, paper 22, question 12either)<br />The point$\D A(0, 10)$lies on the curve for which$\D \frac{dy}{dx}=e^{-\frac{\pi}{4}}.$&nbsp; The point$\D B,$with x-coordinate$\D -4,$also lies on the curve.<br />(i) Find, in terms of$\D e,$the y-coordinate of$\D B.$<br />The tangents to the curve at the points$\D A$and$\D B$intersect at the point$\D C.$<br />(ii) Find, in terms of$\D e,$the x-coordinate of the point$\D C.$<br /><br />7 (CIE 2012, w, paper 23, question 8)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br />The points$\D A (4, 5), B(-2, 3), C(1, 9)$and$\D D$are the vertices of a trapezium in which$\D BC$is parallel to$\D AD$and angle$\D BCD$is$\D 90^{\circ}.$Find the area of the trapezium. <br /><br />8 (CIE 2013, s, paper 12, question 5)<br />The line$\D 3x + 4y = 15$cuts the curve$\D 2xy = 9$at the points$\D A$and$\D B.$Find the length of the line$\D AB.$<br /><br />9 (CIE 2013, s, paper 21, question 8)<br />The line$\D y = 2x - 8$cuts the curve$\D 2x^2 +y^2- 5xy+ 32= 0$&nbsp; at the points$\D A$and$\D B.$Find the length of the line$\D AB.$<br /><br />10 (CIE 2013, s, paper 22, question 8)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br />The points$\D A(- 6, 2), B(2, 6)$and$\D C$are the vertices of a triangle.<br />(i) Find the equation of the line$\D AB$in the form$\D y = mx + c.$&nbsp; <br />(ii) Given that angle$\D ABC = 90^{\circ},$find the equation of$\D BC.$<br />(iii) Given that the length of$\D AC$is 10 units, find the coordinates of each of the two possible positions of point$\D C.$<br /><br />11 (CIE 2013, s, paper 22, question 9)<br />(a) The graph of$\D y = k(3^x) + c$passes through the points$\D (0, 14)$and$\D (- 2, 6).$Find the value of$\D k$<br />and of$\D c.$<br />(b) The variables$\D x$and$\D y$are connected by the equation$\D y = e^x + 25 - 24e^{-x}.$<br />(i) Find the value of$\D y$when$\D x = 4.$<br />(ii) Find the value of$\D e^x$when$\D y = 20$and hence find the corresponding value of$\D x.$<br /><br />12 (CIE 2013, w, paper 11, question 10)<br />Solutions to this question by accurate drawing will not be accepted.<br />The points$\D A(-3, 2)$and$\D B(1, 4)$are vertices of an isosceles triangle$\D ABC,$where angle$\D B = 90^{\circ}.$<br />(i) Find the length of the line$\D AB.$<br />(ii) Find the equation of the line$\D BC.$<br />(iii) Find the coordinates of each of the two possible positions of$\D C.$<br /><br />13 (CIE 2013, w, paper 23, question 7)<br />The line$\D 4x + y = 16$intersects the curve$\D \frac{4}{x}-\frac{8}{y}=1$at the points$\D A$and$\D B.$The x-coordinate of$\D A$is less than the x-coordinate of$\D B.$Given that the point$\D C$lies on the line$\D AB$such that<br />$\D AC : CB = 1 : 2,$find the coordinates of$\D C.<br /><br />14 (CIE 2013, w, paper 23, question 8)<br />Solutions to this question by accurate drawing will not be accepted.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-OPDbrx0heQY/XBiyW8Y1lqI/AAAAAAAACKw/kk3D0Cm9Eewclrc9OwnEezRqPt-7DmSkACEwYBhgL/s1600/coordinate14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="346" data-original-width="562" height="197" src="https://4.bp.blogspot.com/-OPDbrx0heQY/XBiyW8Y1lqI/AAAAAAAACKw/kk3D0Cm9Eewclrc9OwnEezRqPt-7DmSkACEwYBhgL/s320/coordinate14.png" width="320" /></a></div><br />The diagram shows a quadrilateral\D ABCD,$with vertices$\D A(- 4, 6), B(6, - 4), C(10, 4)$and$\D D.$The angle$\D ADC = 90^{\circ}.$The lines$\D BC$and$\D AD$are extended to intersect at the point$\D X.$<br />(i) Given that C is the midpoint of BX, find the coordinates of D. <br />(ii) Hence calculate the area of the quadrilateral$\D ABCD.$<br /><br />15 (CIE 2014, s, paper 21, question 9)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br />The points$\D A(p,1), B(1, 6), C(4, q) and D(5, 4),$where$\D p$and$\D q$are constants, are the vertices of a kite$\D ABCD.$The diagonals of the kite,$\D AC$and$\D BD,$intersect at the point$\D E.$The line$\D AC$is the perpendicular bisector of$\D BD.$Find<br />(i) the coordinates of$\D E,$<br />(ii) the equation of the diagonal$\D AC,$<br />(iii) the area of the kite ABCD. <br /><br /><br />16 (CIE 2014, s, paper 22, question 8)<br />The line$\D y = x - 5$meets the curve$\D x^2+ y^2+ 2x- 35= 0$at the points$\D A$and$\D B.$Find the exact length of$\D AB.$<br /><br />17 (CIE 2014, s, paper 23, question 6)<br />Find the coordinates of the points of intersection of the curve$\D \frac{8}{x}-\frac{10}{y}=1$and the line$\D x + y = 9.$<br /><br />18 (CIE 2014, s, paper 23, question 9)<br />Solutions to this question by accurate drawing will not be accepted.<br /><br />The points$\D A(2,11), B(-2, 3)$and$\D C(2,-1)$are the vertices of a triangle.<br />(i) Find the equation of the perpendicular bisector of$\D AB.$<br />The line through$\D A$parallel to$\D BC$intersects the perpendicular bisector of$\D AB$at the point$\D D.$<br />(ii) Find the area of the quadrilateral$\D ABCD.$<br /><br />19 (CIE 2014, w, paper 11, question 5)<br />(i) Find the equation of the tangent to the curve$\D y= x^3- \ln x$&nbsp; at the point on the curve<br />where$\D x = 1.$<br />(ii) Show that this tangent bisects the line joining the points$\D (-2,16)$and$\D (12, 2).$<br /><br />20 (CIE 2014, w, paper 11, question 8)<br />The point$\D P$lies on the line joining$\D A(- 2, 3)$and$\D B(10, 19)$such that$\D AP:PB = 1:3.$<br />(i) Show that the x-coordinate of$\D P$is 1 and find the y-coordinate of$\D P.$<br />(ii) Find the equation of the line through$\D P$which is perpendicular to$\D AB.$<br />The line through$\D P$which is perpendicular to$\D AB$meets the y-axis at the point$\D Q.$<br />(iii) Find the area of the triangle$\D AQB.$<br /><br />21 (CIE 2014, w, paper 21, question 6)<br />(i) Calculate the coordinates of the points where the line$\D y = x + 2$cuts the curve$\D x^2+y^2=10.$<br />(ii) Find the exact values of$\D m$for which the line$\D y = mx + 5$is a tangent to the curve$\D x^2+y^2=10.$<br /><br />Answer<br />1. (i)$\D 2x + 3y = 9$<br />(ii)$\D Q(-15; 13)$<br />(iii)$\D 156$<br />2.$\D 55$<br />3. (i)$\D C(13;-2)$<br />(ii)$\D 260$<br />4.$\D 8\sqrt{5}$<br />5. (i)$\D D(3, 8);E(5.4, 9.2)$<br />(ii)$\D 32$<br />6. (i)$\D 14 - 4e$<br />(ii)$\D x = \frac{4}{1-e}$<br />7.$\D 20$<br />8.$\D 1.25$<br />9.$\D 10\sqrt{5}$<br />10. (i)$\D y = 0.5x + 5$<br />(ii)$\D y - 6 = -2(x - 2)$<br />(iii)$\D (0,10),(4,2)$<br />11. (a)$\D k = 9; c = 5$<br />(b)(i)$\D 79.2$<br />(ii)$\D x = \ln 3$<br />12. (i)$\D \sqrt{20}$<br />(ii)$\D y = -2x + 6$<br />(iii)$\D x = 3; y = 0; x = -1; y = 8$<br />13.$\D (4, 0)$<br />14. (i)$\D D(8, 10)$<br />(ii)$\D 100$<br />15.$\D (3,5),y = 2x - 1,15$<br />16.$\D 6\sqrt{2}$<br />17.$\D x = 3; y = 6; x = 24; y = -15$<br />18. (i)$\D y = -0.5x + 7$<br />(ii)$\D 84$<br />19. (i)$\D y = 2x - 1$<br />20. (i)$\D y = 7$<br />(ii)$\D 3x + 4y = 31$<br />(iii)$\D 12.5$<br />21. (i)$\D (1, 3); (-3,-1)$<br />(ii)$\D m = \pm \sqrt{1.5}$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-19872164300906986352018-12-17T19:38:00.000-08:002019-02-07T21:27:23.868-08:00Counting (CIE) Permutation and combination1 (CIE 2012, s, paper 11, question 4)<br /><blockquote>(a) Arrangements containing 5 different letters from the word AMPLITUDE are to be made. Find<br />(i) the number of 5-letter arrangements if there are no restrictions, &nbsp;&nbsp;&nbsp;<br />(ii) the number of 5-letter arrangements which start with the letter A and end with the<br />letter E. <br />(b) Tickets for a concert are given out randomly to a class containing 20 students. No student is given more than one ticket. There are 15 tickets.<br />(i) Find the number of ways in which this can be done. <br /><br />There are 12 boys and 8 girls in the class. Find the number of different ways in which<br />(ii) 10 boys and 5 girls get tickets, <br />(iii) all the boys get tickets. </blockquote><br />2 (CIE 2012, s, paper 22, question 10)<br />(a) A team of 7 people is to be chosen from 5 women and 7 men. Calculate the number of different ways in which this can be done if<br />(i) there are no restrictions, <br />(ii) the team is to contain more women than men. <br />(b) (i) How many different 4-digit numbers, less than 5000, can be formed using 4 of the 6 digits 1, 2, 3, 4, 5 and 6 if no digit can be used more than once? <br />(ii) How many of these 4-digit numbers are divisible by 5? <br /><br />3 (CIE 2012, w, paper 13, question 3)<br />A committee of 7 members is to be selected from 6 women and 9 men. Find the number of different committees that may be selected if<br />(i) there are no restrictions, <br />(ii) the committee must consist of 2 women and 5 men, <br />(iii) the committee must contain at least 1 woman. <br /><br />4 (CIE 2012, w, paper 21, question 9)<br />(a) An art gallery displays 10 paintings in a row. Of these paintings, 5 are by Picasso, 4 by Monet and 1 by Turner.<br />(i) Find the number of different ways the paintings can be displayed if there are no restrictions. <br />(ii) Find the number of different ways the paintings can be displayed if the paintings by each of the artists are kept together. <br /><br />(b) A committee of 4 senior students and 2 junior students is to be selected from a group of 6 senior students and 5 junior students.<br />(i) Calculate the number of different committees which can be selected. <br /><br />One of the 6 senior students is a cousin of one of the 5 junior students.<br />(ii) Calculate the number of different committees which can be selected if at most one of<br />these cousins is included. <br /><br />5 (CIE 2012, w, paper 22, question 5)<br />A 4-digit number is formed by using four of the six digits 2, 3, 4, 5, 6 and 8; no digit may be used more than once in any number. How many different 4-digit numbers can be formed if<br />(i) there are no restrictions, <br />(ii) the number is even and more than 6000? <br /><br />6 (CIE 2013, s, paper 11, question 3)<br />A committee of 6 members is to be selected from 5 men and 9 women. Find the number of different committees that could be selected if<br />(i) there are no restrictions, <br />(ii) there are exactly 3 men and 3 women on the committee, <br />(iii) there is at least 1 man on the committee. <br /><br /><br />7 (CIE 2013, s, paper 12, question 2)<br />A 4-digit number is to be formed from the digits 1, 2, 5, 7, 8 and 9. Each digit may only be used once. Find the number of different 4-digit numbers that can be formed if<br />(i) there are no restrictions, <br />(ii) the 4-digit numbers are divisible by 5, <br />(iii) the 4-digit numbers are divisible by 5 and are greater than 7000. <br /><br />8 (CIE 2013, w, paper 11, question 7)<br />(a) (i) Find how many different 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8 and 9 if each digit may be used only once. <br />(ii) Find how many of these 4-digit numbers are even. <br /><br />(b) A team of 6 people is to be selected from 8 men and 4 women. Find the number of different<br />teams that can be selected if<br />(i) there are no restrictions, <br />(ii) the team contains all 4 women, <br />(iii) the team contains at least 4 men. <br /><br />9 (CIE 2013, w, paper 23, question 2)<br />(i) Find how many different numbers can be formed using 4 of the digits 1, 2, 3, 4, 5, 6 and 7 if no digit is repeated. <br />Find how many of these 4-digit numbers are<br />(ii) odd, <br />(iii) odd and less than 3000. <br /><br />10 (CIE 2014, s, paper 11, question 10)<br />(a) How many even numbers less than 500 can be formed using the digits 1, 2, 3, 4 and 5? Each digit may be used only once in any number. <br /><br />(b) A committee of 8 people is to be chosen from 7 men and 5 women. Find the number of different committees that could be selected if<br />(i) the committee contains at least 3 men and at least 3 women, <br />(ii) the oldest man or the oldest woman, but not both, must be included in the committee. <br /><br />11 (CIE 2014, s, paper 12, question 8)<br />(a) (i) How many different 5-digit numbers can be formed using the digits 1, 2, 4, 5, 7 and 9 if no digit is repeated? <br />(ii) How many of these numbers are even? <br />(iii) How many of these numbers are less than 60 000 and even? <br /><br />(b) How many different groups of 6 children can be chosen from a class of 18 children if the class contains one set of twins who must not be separated? <br /><br />12 (CIE 2014, s, paper 13, question 7)<br />(a) A 5-character password is to be chosen from the letters A, B, C, D, E and the digits 4, 5, 6, 7. Each letter or digit may be used only once. Find the number of different passwords that can be chosen if<br />(i) there are no restrictions, <br />(ii) the password contains 2 letters followed by 3 digits. <br /><br />(b) A school has 3 concert tickets to give out at random to a class of 18 boys and 15 girls. Find the number of ways in which this can be done if<br />(i) there are no restrictions, <br />(ii) 2 of the tickets are given to boys and 1 ticket is given to a girl, <br />(iii) at least 1 boy gets a ticket. <br /><br />13 (CIE 2014, w, paper 11, question 10)<br />(a) (i) Find how many different 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 and 6 if no digit is repeated. <br />(ii) How many of the 4-digit numbers found in part (i) are greater than 6000? <br />(iii) How many of the 4-digit numbers found in part (i) are greater than 6000 and are odd? <br /><br />(b) A quiz team of 10 players is to be chosen from a class of 8 boys and 12 girls.<br />(i) Find the number of different teams that can be chosen if the team has to have equal numbers<br />of girls and boys. <br />(ii) Find the number of different teams that can be chosen if the team has to include the youngest and oldest boy and the youngest and oldest girl. <br /><br />14 (CIE 2014, w, paper 23, question 2)<br />A committee of four is to be selected from 7 men and 5 women. Find the number of different committees that could be selected if<br />(i) there are no restrictions, <br />(ii) there must be two male and two female members. <br /><br />A brother and sister, Ken and Betty, are among the 7 men and 5 women.<br />(iii) Find how many different committees of four could be selected so that there are two male and two female members which must include either Ken or Betty but not both. <br /><br /><h4>Answers</h4>1. (a)15120;210<br />(b)15504;3696;56<br />2. (a)792;196<br />(b)240;48<br />3. 6435;1890;6399<br />4. (a)3628800;17280<br />(b)150;110<br />5. (i)360 (ii)72<br />6. 3003;840;2919<br />7. 360;60;36<br />8. (a)360;120<br />(b)924;28;672<br />9. 840;480;140<br />10. (a)28(b)420,240<br />11. (a)720,240,144<br />(b)9828<br />12. (a)15120,480<br />(b)5456,2295,5001<br />13. (a)360;60;36<br />(b)44352;8008<br />14. 495;210;96<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-50203711470588851722018-12-17T05:54:00.002-08:002018-12-29T01:04:41.705-08:00Calculus CIE (Differentation and Integration)$\def\D{\displaystyle}$<br />1 (CIE 2012, s, paper 21, question 4)<br />(i) Find$\D \frac{d}{dx}(x^2\ln x).$<br />(ii) Hence, or otherwise, find$\D \int x\ln x dx.$<br /><br />2 (CIE 2012, w, paper 12, question 11either)<br />A curve is such that$\D y = \frac{5x^2}{1+x^2}.$<br />(i) Show that$\D \frac{dy}{dx}=\frac{kx}{(1+x^2)^2},$where k is an integer to be found. <br />(ii) Find the coordinates of the stationary point on the curve and determine the nature of this<br />stationary point. <br />(iii) By using your result from part (i), find$\D \int \frac{x}{(1+x^2)^2} dx$&nbsp; and hence evaluate$\D \int_{-1}^{2}\frac{x}{(1+x^2)^2}dx.$<br /><br />3 (CIE 2012, w, paper 13, question 11or)<br />(i) Given that$\D y =\frac{3e^{2x}}{1+e^{2x}},$show that$\D \frac{dy}{dx}=\frac{Ae^{2x}}{(1+e^{2x})^2},$&nbsp; where$\D A$is a constant to be found. <br />(ii) Find the equation of the tangent to the curve$\D y = \frac{3e^{2x}}{1+e^{2x}}$&nbsp; at the point where the curve<br />crosses the y-axis. <br />(iii) Using your result from part (i), find$\D \int \frac{e^{2x}}{(1+e^{2x})^2}dx$&nbsp; and hence evaluate$\D \int_{0}^{\ln 3}\frac{e^{2x}}{(1+e^{2x})^2}dx$<br /><br />4 (CIE 2012, w, paper 22, question 7)<br />(i) Find$\D \frac{d}{dx} (\tan 4x).$<br />(ii) Hence find$\D \int (1 + \sec^2 4x) dx.$<br />(iii) Hence show that$\D \int_{-\frac{\pi}{16}}^{\frac{\pi}{16}}<br />(1 + \sec^2 4x) dx = k(\pi +4),$where$\D k$is a constant to be found. <br /><br />5 (CIE 2013, s, paper 12, question 10)<br />(a) (i) Find$\D \int \sqrt{2x-5}dx.$<br />(ii) Hence evaluate$\D \int_{3}^{15}\sqrt{2x-5}dx.$<br />(b) (i) Find$\D \frac{d}{dx}(x^3\ln x).$<br />(ii) Hence find$\D \int x^2 \ln xdx.$<br /><br />6 (CIE 2013, s, paper 22, question 11)<br />A curve has equation$\D y = 3x +\frac{1}{(x-4)^3}.$<br />(i) Find$\D \frac{dy}{dx}$and$\D \frac{d^2y}{dx^2}.$<br />(ii) Show that the coordinates of the stationary points of the curve are (5, 16) and (3, 8). <br />(iii) Determine the nature of each of these stationary points. <br />iv) Find$\D \int \left(3x+\frac{1}{(x-4)^3}\right)dx.$<br />(v) Hence find the area of the region enclosed by the curve, the line$\D x = 5,$the x-axis and the<br />line$\D x = 6 .$<br /><br />7 (CIE 2013, w, paper 11, question 9)<br />(a) Differentiate$\D 4x^3 \ln(2x +1)$&nbsp; with respect to x. <br />(b) (i) Given that$\D y=\frac{2x}{\sqrt{x+2}},$&nbsp; show that$\D \frac{dy}{dx}=\frac{x+4}{(\sqrt{x+2})^3}.$<br />(ii) Hence find$\D \int \frac{5x+20}{(\sqrt{x+2})^3}dx.$<br />(iii) Hence evaluate$\D \int_{2}^{7}\frac{5x+20}{(\sqrt{x+2})^3}dx.$<br /><br />8 (CIE 2014, s, paper 11, question 5)<br />(i) Given that$\D y= e^{x^2},$&nbsp; find$\D \frac{dy}{dx}.$<br />(ii) Use your answer to part (i) to find$\D \int xe^{x^2} dx.$<br />(iii) Hence evaluate$\D \int_{0}^{2}xe^{x^2}dx.$<br /><br />9 (CIE 2014, s, paper 23, question 10)<br />(i) Given that$\D y=\frac{2x}{\sqrt{x^2+21}},$&nbsp; show that$\D \frac{dy}{dx}=\frac{k}{\sqrt{(x^2+21)^3}},$&nbsp; where$\D k$is a constant to be found. <br />(ii) Hence find$\D \int \frac{6}{\sqrt{(x^2+21)^3}}dx$and evaluate$\D \int_{2}^{10}\frac{6}{\sqrt{(x^2+21)^3}}dx.$<br /><br />10 (CIE 2014, w, paper 13, question 8)<br />(i) Given that$\D f(x) = x \ln x^3 ,$show that$\D f'(x) = 3(1+\ln x).$&nbsp; <br />(ii) Hence find$\D \int (1+\ln x)dx.$<br />(iii) Hence find$\D \int_{1}^{2}\ln x dx$&nbsp; in the form$\D p + \ln q,$where$\D p$and$\D q$are integers. <br /><br />11 (CIE 2014, w, paper 21, question 8)<br />(i) Given that$\D y=\frac{x^2}{2+x^2},$show that$\D \frac{dy}{dx}=\frac{kx}{(2+x^2)^2},$where$\D k$is a constant to be found. <br />(ii) Hence find$\D \int \frac{x}{(2+x^2)^2}dx.$<br /><br /><h3>Answer</h3>1. (i)$\D 2x \ln x + x$<br />(ii)$\D 0.5x^2 \ln x - x^2/4$<br />2. (i)$\D k = 10$<br />(ii)$\D (0,0),$min<br />(iii)$\D \frac{x^2}{2(1+x^2)},$<br />$\D 0.15$<br />3. (i)$\D A = 6$<br />(ii)$\D 2y - 3 = 3x$<br />(iii)$\D \frac{e^{2x}}{2(1+e^{2x})}, 0.2$<br />4. (i)$\D 4 \sec^2 4x$<br />(ii)$\D x + \frac{1}{4}\tan 4x$<br />(iii)$\D k=1/8$<br />5. (a)(i)$\D \frac{1}{3}(2x - 5)^{3/2}$<br />(ii)$\D 124/3$<br />(b)(i)$\D x^2 + 3x^2 \ln x$<br />(ii)$\D \frac{1}{3}(x^3 \ln x - \frac{x^3}{3})$<br />6. (i)$\D y'= 3-3(x-4)^{-4}$<br />$\D y''=12(x-4)^{-5}$<br />(iii)$\D x = 5$, min,$\D x = 3,$max<br />(iv)$\D \frac{3x^2}{2}-\frac{(x-4)^2}{2}$<br />(v)$\D 135/8$<br />7. (a)$\D 12x^2 \ln(2x+1)+8x^3/(2x+1)$<br />(b)(ii)$\D \frac{10x}{\sqrt{x+2}}$<br />(iii)$\D 40/3$<br />8.$\D 2xe^{x^2},0.5e^{x^2},26.8$<br />9. (i)$\D k = 42$<br />(ii)$\D \frac{8}{55}$<br />10. (ii)$\D x \ln x$<br />(iii)$\D -1 + \ln 4$<br />11. (i)$\D k = 4$<br />(ii)$\D \frac{x^2}{4(2+x^2)}$<br /><br />Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-4174715680620870562018-12-15T21:06:00.002-08:002018-12-29T01:06:00.469-08:00Area under curve$\def\D{\displaystyle}<br /><br />1 (CIE 2012, s, paper 11, question 11either)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-D0zqkKKzGG4/XBXbdFCiKqI/AAAAAAAACKU/N_zz5VBZC_gadl5X0HgTgTLcLyvSmb-YgCEwYBhgL/s1600/areacurve_1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="406" data-original-width="869" height="149" src="https://2.bp.blogspot.com/-D0zqkKKzGG4/XBXbdFCiKqI/AAAAAAAACKU/N_zz5VBZC_gadl5X0HgTgTLcLyvSmb-YgCEwYBhgL/s320/areacurve_1.png" width="320" /></a></div>The diagram shows part of the curve\D y = 9x^2 - x^3,$which meets the x-axis at the origin$\D O$and at the point$\D A$. The line$\D y - 2x + 18 = 0$passes through$\D A$and meets the y-axis at the point$\D B.$<br />(i) Show that, for$\D x \ge&nbsp; 0, 9x^2 - x^3 \le&nbsp; 108.$<br />(ii) Find the area of the shaded region bounded by the curve, the line$\D ABand the y-axis. <br /><br />2 (CIE 2012, s, paper 11, question 11or)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-rpehFri0PEw/XBXbe5N_znI/AAAAAAAACKU/1GAujhNHi8QcLHWBIMNgdXzPFKrjqaP3QCEwYBhgL/s1600/areacurve_2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="436" data-original-width="930" height="149" src="https://4.bp.blogspot.com/-rpehFri0PEw/XBXbe5N_znI/AAAAAAAACKU/1GAujhNHi8QcLHWBIMNgdXzPFKrjqaP3QCEwYBhgL/s320/areacurve_2.png" width="320" /></a></div>The diagram shows part of the curve\D y = 2\sin 3x .$The normal to the curve$\D y = 2\sin 3x$at the point where$\D x = \frac{\pi}{9}$meets the y-axis at the point$\D P.$<br />(i) Find the coordinates of$\D P.<br />(ii) Find the area of the shaded region bounded by the curve, the normal and the y-axis. <br /><br />3 (CIE 2012, s, paper 22, question 11either)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-YPVHcwbsWC0/XBXbe52sehI/AAAAAAAACKI/RoatxwySeW8oTPRvHLMW56WxiijfMIhGACEwYBhgL/s1600/areacurve_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="712" height="177" src="https://2.bp.blogspot.com/-YPVHcwbsWC0/XBXbe52sehI/AAAAAAAACKI/RoatxwySeW8oTPRvHLMW56WxiijfMIhGACEwYBhgL/s320/areacurve_3.png" width="320" /></a></div>The diagram shows part of the curve\D y = \sin\frac{1}{2}x.$&nbsp; The tangent to the curve at the point$\D P\left(\frac{3\pi}{2},\frac{\sqrt{2}}{2}\right)$&nbsp; cuts the x-axis at the point$\D Q.$<br />(i) Find the coordinates of$\D Q.$<br />(ii) Find the area of the shaded region bounded by the curve, the tangent and the x-axis. <br /><br />4 (CIE 2012, s, paper 22, question 11or)<br />(i) Given that$\D y = xe^{-x},$find$\D \frac{dy}{dx}$&nbsp; and hence show that$\D \int xe^{-x}dx=-xe^{-x}-e^{-x}+c.&nbsp; <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-0zcV4rIzbGE/XBXbfnUGW3I/AAAAAAAACKM/DIkJB0bofVYxnlUfhnueHJUSdvso9ZqCQCEwYBhgL/s1600/areacurve_4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="318" data-original-width="606" height="167" src="https://4.bp.blogspot.com/-0zcV4rIzbGE/XBXbfnUGW3I/AAAAAAAACKM/DIkJB0bofVYxnlUfhnueHJUSdvso9ZqCQCEwYBhgL/s320/areacurve_4.png" width="320" /></a></div>The diagram shows part of the curve\D y = xe^{-x}$and the tangent to the curve at the point$\D R\left(2,\frac{2}{e^2}\right).<br />(ii) Find the area of the shaded region bounded by the curve, the tangent and the y-axis. <br /><br />5 (CIE 2012, w, paper 11, question 12either)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-FxyyYeYUnGI/XBXbfw0KOiI/AAAAAAAACKM/QjqBS_rj0LY7bSs9zdC-SFHpkKvkS1QlgCEwYBhgL/s1600/areacurve_5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="361" data-original-width="919" height="125" src="https://1.bp.blogspot.com/-FxyyYeYUnGI/XBXbfw0KOiI/AAAAAAAACKM/QjqBS_rj0LY7bSs9zdC-SFHpkKvkS1QlgCEwYBhgL/s320/areacurve_5.png" width="320" /></a></div>The diagram shows part of the graph of\D y = (12 - 6x)(1 + x)^2,$which meets the x-axis at the points$\D A$and$\D B.$The point$\D C$is the maximum point of the curve.<br />(i) Find the coordinates of each of$\D A, B$and$\D C.<br />(ii) Find the area of the shaded region. <br /><br />6 (CIE 2012, w, paper 11, question 12or)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-1vZXKIDJ2Qo/XBXbgAg6QII/AAAAAAAACKQ/a4ivpLe-E2kZA76ZEw9kC0AZBWnipXnewCEwYBhgL/s1600/areacurve_6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="401" data-original-width="1029" height="124" src="https://3.bp.blogspot.com/-1vZXKIDJ2Qo/XBXbgAg6QII/AAAAAAAACKQ/a4ivpLe-E2kZA76ZEw9kC0AZBWnipXnewCEwYBhgL/s320/areacurve_6.png" width="320" /></a></div>The diagram shows part of a curve such that\D \frac{dy}{dx} = 3x^2 - 6x - 9.$Points$\D A$and$\D B$are stationary points of the curve and lines from$\D A$and$\D B$are drawn perpendicular to the x-axis. Given that the curve passes through the point (0, 30), find<br />(i) the equation of the curve, <br />(ii) the x-coordinate of$\D A$and of$\D B,$<br />(iii) the area of the shaded region. <br /><br />7 (CIE 2012, w, paper 13, question 11either)<br />The tangent to the curve$\D y = 5e^x + 3e^{-x}$at the point where$\D x = \ln\frac{3}{5},$meets the x-axis at the point$\D P.$<br />(i) Find the coordinates of$\D P.$<br />The area of the region enclosed by the curve$\D y = 5e^x + 3e^{-x},$the y-axis, the positive x-axis and<br />the line$\D x = a$is 12 square units.<br />(ii) Show that$\D 5e^{2a} - 14e^a - 3 = 0.$<br />(iii) Hence find the value of$\D a.<br /><br />8 (CIE 2012, w, paper 22, question 12or)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MEba0X1RLqw/XBXbgmedFSI/AAAAAAAACKU/zTPC_e0P-0wwlxqcWJau7swdZfxJEphxQCEwYBhgL/s1600/areacurve_8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="542" data-original-width="741" height="234" src="https://1.bp.blogspot.com/-MEba0X1RLqw/XBXbgmedFSI/AAAAAAAACKU/zTPC_e0P-0wwlxqcWJau7swdZfxJEphxQCEwYBhgL/s320/areacurve_8.png" width="320" /></a></div>The diagram shows part of the curve\D y = 8 + e^{-\frac{x}{3}}$&nbsp; crossing the y-axis at$\D P.$The normal to the curve at$\D P$meets the x-axis at$\D Q.$<br />(i) Find the coordinates of$\D Q.$<br />The line through$\D Q,$parallel to the y-axis, meets the curve at$\D R$and$\D OQRS$is a rectangle.<br />(ii) Find$\D \int \left(8+e^{-\frac{x}{3}}\right)dx&nbsp; &nbsp;and hence find the area of the shaded region. <br /><br />9 (CIE 2013, w, paper 13, question 6)<br />Do not use a calculator in this question.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-N03g-hjCzZo/XBXbg2kRRII/AAAAAAAACKQ/381S_xqSq24_4KkwUEhxwahLdegz5qhjgCEwYBhgL/s1600/areacurve_9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="392" data-original-width="896" height="140" src="https://4.bp.blogspot.com/-N03g-hjCzZo/XBXbg2kRRII/AAAAAAAACKQ/381S_xqSq24_4KkwUEhxwahLdegz5qhjgCEwYBhgL/s320/areacurve_9.png" width="320" /></a></div>The diagram shows part of the curve\D y= 4- x^2.$<br />Show that the area of the shaded region can be written in the form$\D \frac{\sqrt{2}}{p},$&nbsp; where$\D pis an integer to<br />be found. <br /><br />10 (CIE 2013, w, paper 21, question 11)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-PobEw7ky69Y/XBXbdNBvkiI/AAAAAAAACKE/Zv_nOUh1VrYtTcXHrf3nUZqLcSv8IX1ugCEwYBhgL/s1600/areacurve_10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="440" data-original-width="745" height="188" src="https://1.bp.blogspot.com/-PobEw7ky69Y/XBXbdNBvkiI/AAAAAAAACKE/Zv_nOUh1VrYtTcXHrf3nUZqLcSv8IX1ugCEwYBhgL/s320/areacurve_10.png" width="320" /></a></div>The diagram shows part of the curve\D y= e^{\frac{x}{3}}.$The tangent to the curve at$\D P(9,e^3),$&nbsp; meets the<br />x-axis at$\D Q.$<br />(i) Find the coordinates of$\D Q.$<br />(ii) Find the area of the shaded region bounded by the curve, the coordinate axes and the tangent<br />to the curve at$\D P.$<br /><br />11 (CIE 2014, s, paper 12, question 4)<br />The region enclosed by the curve$\D y = 2 \sin 3x,$the x-axis and the line$\D x = a ,$where<br />$\D 0&lt;a&lt;1$radian, lies entirely above the x-axis. Given that the area of this region is$\D \frac{1}{3}$&nbsp; square unit,<br />find the value of$\D a.<br /><br />12 (CIE 2014, s, paper 13, question 11)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-2DytCskoJbg/XBXbdIdy71I/AAAAAAAACKM/aeRetgtlIf8890kp9cBbxyltQ1JLySOUwCEwYBhgL/s1600/areacurve_12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="397" data-original-width="937" height="135" src="https://3.bp.blogspot.com/-2DytCskoJbg/XBXbdIdy71I/AAAAAAAACKM/aeRetgtlIf8890kp9cBbxyltQ1JLySOUwCEwYBhgL/s320/areacurve_12.png" width="320" /></a></div>The diagram shows the graph of\D y = \cos 3x + \sqrt{3} \sin 3x,$which crosses the x-axis at$\D A$and has a maximum point at$\D B.$<br />(i) Find the x-coordinate of$\D A.$<br />(ii) Find$\D \frac{dy}{dx}$and hence find the x-coordinate of$\D B.$<br />(iii) Showing all your working, find the area of the shaded region bounded by the curve, the x-axis<br />and the line through$\D Bparallel to the y-axis. <br /><br />13 (CIE 2014, w, paper 13, question 11)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-IH2zYhNjh40/XBXbeZaZAKI/AAAAAAAACKI/bEcbWQOy4fIPL-oPq3vZPr_Z4DxJzxhWwCEwYBhgL/s1600/areacurve_13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="366" data-original-width="884" height="132" src="https://3.bp.blogspot.com/-IH2zYhNjh40/XBXbeZaZAKI/AAAAAAAACKI/bEcbWQOy4fIPL-oPq3vZPr_Z4DxJzxhWwCEwYBhgL/s320/areacurve_13.png" width="320" /></a></div>The diagram shows part of the curve\D y=(x+5)(x-1)^2.$<br />(i) Find the x-coordinates of the stationary points of the curve. <br />(ii) Find$\D \int (x+5)(x-1)^2dx.$<br />(iii) Hence find the area enclosed by the curve and the x-axis. <br />(iv) Find the set of positive values of$\D k$for which the equation$\D (x+5)(x-1)^2=k$&nbsp; has only one real solution. <br /><br />14 (CIE 2014, w, paper 21, question 12)<br />(i) Show that$\D x-2$is a factor of$\D 3x^3-14^2+32.$<br />(ii) Hence factorise$\D 3x^3-14x^2+32completely.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Wdpixwp7ykc/XBXbeOUpl2I/AAAAAAAACKI/NNJIp8CcxT0b0N-igBk1NzGemq4XBo6IQCEwYBhgL/s1600/areacurve_14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="332" data-original-width="926" height="114" src="https://1.bp.blogspot.com/-Wdpixwp7ykc/XBXbeOUpl2I/AAAAAAAACKI/NNJIp8CcxT0b0N-igBk1NzGemq4XBo6IQCEwYBhgL/s320/areacurve_14.png" width="320" /></a></div>The diagram below shows part of the curve\D y =3x-14+\frac{32}{x^2}$&nbsp; cutting the x-axis at the points$\D P$and$\D Q.$<br />(iii) State the x-coordinates of$\D P$and$\D Q.$<br />(iv) Find$\D \int (3x-14+\frac{32}{x^2})dx$&nbsp; and hence determine the area of the shaded region. <br /><br />Answers<br />1. (ii)$\D&nbsp; 628$<br />2. (i)$\D (0, 1.58)$<br />(ii)$\D 0.292$<br />3. (i)$\D x = 2 + \frac{3\pi}{2}$<br />(ii)$\D \frac{3\sqrt{2}}{2}-2$<br />4.&nbsp;$\D (9/e^2) - 1$<br />5. (i)$\D A(-1, 0),B(2, 0),<br />C(1, 24)$<br />(ii)$\D 40.5$<br />6. (i)$\D y = x^3 - 3x^2 - 9x + 30$<br />(ii)$\D x = -1, 3$<br />(iii)$\D 76$<br />7. (i)$\D P(3.49, 0)$<br />(ii)<br />(iii)$\D \ln 3$<br />8. (i)$\D Q(-3, 0)$<br />(ii)$\D 8x - 3e^{-\frac{x}{3}}, A = 3$<br />9.&nbsp;$\D \frac{\sqrt{2}}{3}$<br />10. (i)$\D (6, 0)$<br />(ii)$\D 27.1$<br />11.&nbsp;$\D a =\frac{\pi}{9}$<br />12. (i)$\D x = \frac{5\pi}{18}$<br />(ii)$\D \frac{dy}{dx} =&nbsp; &nbsp;3\sqrt{3} \cos 3x - 3 \sin 3x$<br />(iii)$\D A = \frac{2}{3}$<br />13. (i)$\D x = 1,-3$<br />(ii)$\D \frac{x^4}{4}+x^3- \frac{9x^2}{2}+5x.$<br />(iii)&nbsp;$\D 108$(iv)$\D k &gt; 32$<br />14. (ii)$\D (x - 2)(x - 4)(3x + 4)$<br />(iii)$\D x = 2, 4,$<br />(iv)$\D 2$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-27563441503990162192018-12-15T04:28:00.001-08:002018-12-29T01:07:14.317-08:00Surd for CIE$\def\D{\displaystyle}$<br /><br />1 (CIE 2012, s, paper 12, question 6)<br />You must not use a calculator in this question.<br />(i) Express$\D \frac{8}{\sqrt{3}+1} $in the form$\D a(\sqrt{3}-1),$where$\D a$is an integer. <br />An equilateral triangle has sides of length$\D \frac{8}{\sqrt{3}+1}.$<br />(ii) Show that the height of the triangle is$\D 6 - 2\sqrt{3} .$<br />(iii) Hence, or otherwise, find the area of the triangle in the form$\D p\sqrt{3}&nbsp; - q,$where$\D p$and$\D q$are integers. <br /><br />2 (CIE 2012, s, paper 21, question 2)<br />A cuboid has a square base of side$\D (2 + \sqrt{3} )$cm and a volume of$\D (16 + 9\sqrt{3}&nbsp; )$cm$\D ^3.$Without using a calculator, find the height of the cuboid in the form$\D (a + b\sqrt{3}&nbsp; )$cm, where$\D a$and$\D b$are integers. <br /><br />3 (CIE 2012, w, paper 11, question 7)<br />Do not use a calculator in any part of this question.<br />(a) (i) Show that$\D 3\sqrt{5}&nbsp; - 2 \sqrt{2}$is a square root of$\D 53 - 12\sqrt{10}.$<br />(ii) State the other square root of$\D 53 - 12\sqrt{10}.$<br />(b) Express$\D \frac{6\sqrt{3}+7\sqrt{2}}{4\sqrt{3}+5\sqrt{2}}$in the form$\D a + b \sqrt{6},$where$\D a$and$\D bare integers to be found. <br /><br />4 (CIE 2012, w, paper 12, question 6)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-D-ahroHXY2Y/XBTycHC3gBI/AAAAAAAACJE/P9XaQ6bLXU8iP-GBIIOJto5fLJCBWpjdACLcBGAs/s1600/surd_04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="224" data-original-width="503" height="142" src="https://1.bp.blogspot.com/-D-ahroHXY2Y/XBTycHC3gBI/AAAAAAAACJE/P9XaQ6bLXU8iP-GBIIOJto5fLJCBWpjdACLcBGAs/s320/surd_04.png" width="320" /></a></div>Using\D \sin15^{\circ} =\frac{\sqrt{2}}{4}(\sqrt{3}-1)$&nbsp; and without using a calculator, find the value of$\D \sin\theta$in the form$\D a + b \sqrt{2},$where$\D a$and$\D b$are integers. <br /><br />5 (CIE 2012, w, paper 23, question 3)<br />Without using a calculator, simplify$\D \frac{(3\sqrt{3}-1)^2}{2\sqrt{3}-3},$&nbsp; giving your answer in the form$\D \frac{a\sqrt{3}+b}{3},$where$\D a$and$\D bare integers. <br /><br />6 (CIE 2013, s, paper 11, question 7)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vDD67S-DVY8/XBTycFG1jNI/AAAAAAAACI8/re4rBo3MG6cVFC1LTy06CMDdiyXokCJAACEwYBhgL/s1600/surd_06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="164" data-original-width="838" height="62" src="https://2.bp.blogspot.com/-vDD67S-DVY8/XBTycFG1jNI/AAAAAAAACI8/re4rBo3MG6cVFC1LTy06CMDdiyXokCJAACEwYBhgL/s320/surd_06.png" width="320" /></a></div>Calculators must not be used in this question.<br />The diagram shows a triangle\D ABC$in which angle$\D A = 90^{\circ}.$Sides$\D AB$and$\D AC$are$\D \sqrt{5} - 2$and$\D \sqrt{5} + 1$respectively. Find<br />(i)$\D \tan B$in the form$\D a + b\sqrt{5},$where$\D a$and$\D b$are integers, <br />(ii)$\D \sec^2B$in the form$\D c + d \sqrt{5},$where$\D c$and$\D dare integers. <br /><br />7 (CIE 2013, s, paper 22, question 5) Fig<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-zZy4yn9XIMo/XBTycTP6UeI/AAAAAAAACJI/7sc_q2itjXE4nfDDSa1gwEUSiEin6sw2gCEwYBhgL/s1600/surd_07.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="286" data-original-width="834" height="109" src="https://4.bp.blogspot.com/-zZy4yn9XIMo/XBTycTP6UeI/AAAAAAAACJI/7sc_q2itjXE4nfDDSa1gwEUSiEin6sw2gCEwYBhgL/s320/surd_07.png" width="320" /></a></div>The diagram shows a trapezium\D ABCD$in which$\D AD = 7$cm and$\D AB =(4+\sqrt{5})$cm.$\D AX$is perpendicular to$\D DC$with$\D DX = 2$cm and$\D XC = x$cm. Given that the area of trapezium$\D ABCD$is$\D 15(\sqrt{5}+2)$cm$\D ^2,$obtain an expression for$\D x$in the form$\D a + b \sqrt{5},$where$\D a$and$\D b$are integers. <br /><br />8 (CIE 2013, w, paper 21, question 2)<br />Do not use a calculator in this question.<br />Express$\D \frac{(4\sqrt{5}-2)^2}{\sqrt{5}-1}$&nbsp; in the form$\D p \sqrt{5} + q,$where$\D p$and$\D q$are integers. <br /><br />9 (CIE 2014, s, paper 21, question 2)<br />Without using a calculator, express$\D 6(1+\sqrt{3})^{-2}$&nbsp; in the form$\D a + b \sqrt{3},$where$\D a$and$\D b$are integers to be found. <br /><br />10 (CIE 2014, s, paper 22, question 1)<br />Without using a calculator, express$\D \frac{(2+\sqrt{5})^2}{\sqrt{5}-1}$&nbsp; in the form$\D a + b\sqrt{5},$where$\D a$and$\D b$are constants to be found. <br /><br />11 (CIE 2014, s, paper 23, question 5)<br />Do not use a calculator in this question.<br />(i) Show that$\D (2\sqrt{2}+4)^2-8(2\sqrt{2}+3)=0.$<br />(ii) Solve the equation$\D (2\sqrt{2}+3)x^2-(2\sqrt{2}+4)x+2=0,$giving your answer in the form$\D a + b\sqrt{2} $where$\D a$and$\D b$are integers. <br /><br />12 (CIE 2014, w, paper 21, question 9)<br />Integers$\D a$and$\D b$are such that$\D (a+ 3\sqrt{5} )^2+ a- b\sqrt{5}= 51.$&nbsp; Find the possible values of$\D a$and the corresponding values of$\D b.$<br /><br />Answers<br />1.(i)$\D a=4$<br />(ii)<br />(iii)$\D 16\sqrt{3}-24$<br />2.$\D 4-\sqrt{3}$<br />3.(a)(i)<br />(ii)$\D -3\sqrt{5}+2\sqrt{2}$<br />(b)$\D -1+\sqrt{6}$<br />4.$\D 6-4\sqrt{2}$<br />5.$\D \frac{38\sqrt{3}}{3}$<br />6.(i)$\D 7+3\sqrt{5}$<br />(ii)$\D 95+42\sqrt{5}$<br />7.$\D 4+3\sqrt{5}$<br />8.$\D 17\sqrt{5}+1$<br />9.$\D 6-3\sqrt{3}$<br />10.$\D \frac{29}{4}+\frac{13}{4}\sqrt{5}$<br />11.$\D 2-\sqrt{2}$<br />12.$\D a=-3,2:b=-18,12$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-66200808140976564982018-12-14T07:02:00.002-08:002018-12-16T00:38:03.753-08:00GP Series (Edexcel IGCSE, Further Pure Mathematics)$\def\D{\displaystyle}$<br /><br />1. (Edexcel Further Pure&nbsp; Mathematics 2011, june, paper 2, no.8)<br />The sum of the first and third terms of a geometric series is 100. The sum of the second and third terms is 60<br />(a) Find the two possible values of the common ratio of the series. (5)<br />Given that the series is convergent, find<br />(b) the first term of the series, (2)<br />(c) the least number of terms for which the sum is greater than 159.9 (4)<br /><br />2.&nbsp; (Edexcel Further Pure&nbsp; Mathematics, 2012, jan, Paper 2, no 10)<br />The sum of the first and third terms of a geometric series$\D G$is 104. The sum of the second and third terms of$\D G$is 24 Given that$\D G$is convergent and that the sum to infinity is$\D S,$find<br />(a) the common ratio of$\D G.$(4)<br />(b) the value of$\D S.$(4)<br />The sum of the first and third terms of another geometric series$\D H$is also 104 and the sum of the second and third terms of$\D H$is 24. The sum of the first n terms of H is$\D S_n.$<br />(c) Write down the common ratio of$\D H$. (1)<br />(d) Find the least value of$\D n$for which$\D S_n&gt; S.$&nbsp; (6)<br /><br />3. (Edexcel Further Pure Mathematics, 2012, june, Paper 2, no 6)<br />The first term of a geometric series$\D S$is$\D \sqrt{2}.$The second term of$\D S$is$\D \sqrt{2} − 2.$<br />(a) (i) Find the exact value of the common ratio of$\D S.$<br />(ii) Find the third term of$\D S,$giving your answer in the form$\D a \sqrt{2} + b,$where$\D a$and$\D b$are integers. (5)<br />(b) (i) Explain why the series is convergent.<br />(ii) Find the sum to infinity of$\D S.$(3)<br /><br />4. (Edexcel Further Pure&nbsp; Mathematics, 2013, Jan, Paper 2, no 9)<br />The third and fifth terms of a geometric series$\D S$are 48 and 768 respectively. Find<br />(a) the two possible values of the common ratio of$\D S,$(3)<br />(b) the first term of$\D S.$(1)<br />Given that the sum of the first 5 terms of$\D S$is 615.<br />(c) find the sum of the first 9 terms of$\D S.$(4)<br />Another geometric series$\D T$has the same first term as$\D S.$The common ratio of$\D T$is$\D \frac{1}{r},$where$\D r$is one of the values obtained in part (a). The$\D n^{th}$term of$\D T$is$\D t_n.$Given that$\D t_2 &gt; t_3.$<br />(d) find the common ratio of$\D T.$(1)<br />The sum of the first$\D n$terms of$\D T$is$\D T_n.$<br />(e) Writing down all the numbers on your calculator display, find$\D T_9.$(2)<br />The sum to infinity of$\D T$is$\D T_{\infty}.$Given that$\D T_{\infty} - T_n &gt; 0.002.$<br />(f) find the greatest value of$\D n.$(5)<br /><br /><br /><br />5. (Edexcel Further Pure&nbsp; Mathematics, 2013, june, Paper 2, no 4)<br />The$\D n_{th}$term of a geometric series is$\D t_n$and the common ratio is$\D r,$where$\D r &gt; 0.$Given that$\D t_1 = 1.$<br />(a) write down an expression in terms of$\D r$and$\D n$for$\D t_n.$(1)<br />Given also that$\D t_n + t_{n+1} = t_{n+2}, $<br />(b) show that$\D r = \frac{1+\sqrt{5}}{2}$(4)<br />(c) find the exact value of$\D t_4$giving your answer in the form$\D f + g\sqrt{h},$where$\D f, g$and$\D h$are integers. (3)<br /><br /><br />6. (Edexcel Further Pure&nbsp; Mathematics, 2014, Jan, Paper 2, no 10)<br />The sum of the second and third terms of a convergent geometric series is 7.5. The sum to infinity,$\D S,$of the series is 20. The common ratio of the series is$\D r.$<br />(a) Show that$\D r$is a root of the equation$\D 8r^3 – 8r + 3 = 0.$(4)<br />(b) Show that$\D r =\frac{1}{2}$is a root of this equation. (1)<br />Given that$\D r &lt; 0.6.$<br />(c) show that$\frac{1}{2}$is the only possible value of$\D r.$(4)<br />(d) Find the first term of the series. (2)<br />The sum of the first$\D n$terms of the series is$\D S_n$.<br />(e) Find the least value of$\D n$for which Sn exceeds 99\% of$\D S.$(6)<br /><br />7.(Edexcel Further Pure&nbsp; Mathematics, 2015, june, Paper 2, No 3)<br />Every term of a convergent geometric series is positive. The difference between the third term and the fourth term is twice the fifth term.<br />(a) Show that the common ratio of the series is$\D \frac{1}{2}.$(3)<br />The sum to infinity of this convergent series is 400. Find<br />(b) the first term of the series, (2)<br />(c) the sum of the first 10 terms of the series, writing down all the digits on your calculator display. (2)<br /><br /><br /><br /><br />1.(a)$\D r=1/2,-3$<br />(b)$\D a=80$<br />(c)$\D n=11$<br />2. (a)$\D r=1/5 (r=-3/2)$<br />(b)$\D a=100,S=125$<br />(c)$r'=-3/2$<br />(d)$\D n=7$<br />3. (a)(i)$\D r=1-\sqrt{2} $<br />(ii)$\D 3\sqrt{2}-4$<br />(b)(i)$\D |r|&lt;1$<br />(ii)$\D S=1$<br />4. (a)$\D \pm 4$<br />(b)$\D a=3$<br />(c)$\D r=-4, S_9=157287$<br />(d)$\D r=\frac{1}{4}$<br />(e)$\D T_9=3.999984741$<br />(f)$\D n=5$<br />5. (a)$\D t_n=r^{n-1}$<br />(b)<br />(c)$\D 2+\sqrt{5}$<br />6. (a)(b)(c)<br />(d)$\D a=10$<br />(e)$\D n=7$<br />7. (a)<br />(b)$\D a=200$<br />(c)$S_{10}=399.609375$<br /><br />Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-24807749287571126132018-12-13T06:32:00.000-08:002018-12-29T01:08:50.447-08:00Binomial CIE$\def\D{\displaystyle}$<br /><br />1 (CIE 2012, s, paper 11, question 9)<br />Find the values of the positive constants$\D p$and$\D q$such that, in&nbsp; binomial expansion of$\D ( p + qx)^{10},$the coefficient of$\D x^5$is 252 and the coefficient of$\D x^3$is 6 times the coefficient of$\D x^2.$<br /><br /><br />2 (CIE 2012, s, paper 22, question 6)<br />(a) Find the coefficient of$\D x^3$in the expansion of<br />(i)$\D (1 - 2x)^7,$<br />(ii)$\D (3 + 4x)(1 - 2x)^7.$<br />(b) Find the term independent of$\D x$in the expansion of$\D\left(x+\frac{3}{x^2}\right)^6.$<br /><br />3 (CIE 2012, w, paper 11, question 6)<br />(i) Find the first 3 terms, in descending powers of$\D x,$in the expansion of$\D \left(x+\frac{2}{x^2}\right)^6.$<br />(ii) Hence find the term independent of$\D x$in the expansion of$\D \left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6.$&nbsp; <br /><br />4 (CIE 2012, w, paper 13, question 6)<br />In the expansion of$\D (p + x)^6,$where$\D p$is a positive integer, the coefficient of$\D x^2$is equal to 1.5 times the coefficient of$\D x^3.$<br />(i) Find the value of$\D p.$<br />(ii) Use your value of$\D p$to find the term independent of$\D x$in the expansion of$\D (p + x)^6 \left(1-\frac{1}{x}\right)^2.$<br /><br />5 (CIE 2012, w, paper 22, question 4)<br />(i) Find the coefficient of$\D x^5$in the expansion of$\D (2 - x)^8.$<br />(ii) Find the coefficient of$\D x^5$in the expansion of$\D (l + 2x)(2 - x)^8.$<br /><br />6 (CIE 2013, s, paper 12, question 9)<br />(i) Given that$\D n$is a positive integer, find the first 3 terms in the expansion of$\D\left(1+\frac{1}{2}x\right)^n.$&nbsp; in ascending powers of x. <br />(ii) Given that the coefficient of$\D x^2$in the expansion of$\D(1 - x) \left(1+\frac{1}{2}x\right)^n$is$\D\frac{25}{4},$&nbsp; find the value of$\D n.$<br /><br />7 (CIE 2013, s, paper 21, question 7)<br />(i) Find the first four terms in the expansion of$\D (2+ x)^6$&nbsp; in ascending powers of$\D x.$<br />(ii) Hence find the coefficient of$\D x^3$in the expansion of&nbsp;$\D(1+3x)(1-x)(2+x)^6.$<br /><br />8 (CIE 2013, w, paper 21, question 6)<br />(a) (i) Find the coefficient of$\D x^3$in the expansion of$(1-2x)^6.$<br />(ii) Find the coefficient of$\D x^3$in the expansion of$\D \left(1+\frac{x}{2}\right)(1-2x)^6.$<br />(b) Expand$\D \left(2\sqrt{x}+\frac{1}{\sqrt{x}}\right)^4$in a series of powers of$\D x$with integer coefficients. <br /><br />9 (CIE 2013, w, paper 23, question 6)<br />The expression$\D 2x^3 + ax^2 + bx + 21$has a factor$\D x + 3$and leaves a remainder of 65 when divided by$\D x - 2.$<br />(i) Find the value of$\D a$and of$\D b$. <br />(ii) Hence find the value of the remainder when the expression is divided by$\D 2x + 1.$<br /><br />10 (CIE 2014, s, paper 13, question 5)<br />(i) The first three terms in the expansion of$\D (2 - 5x)^6 ,$in ascending powers of$\D x,$are$\D p + qx + rx^2 .$Find the value of each of the integers$\D p, q$and$\D r.$<br />(ii) In the expansion of$\D (2 - 5x)^6 (a + bx)^3 ,$the constant term is equal to 512 and the coefficient of$\D x$is zero. Find the value of each of the constants$\D a$and$\D b.$<br /><br />11 (CIE 2014, s, paper 21, question 6)<br />(a) Find the coefficient of$x^5$in the expansion of$\D (3-2x)^8.$<br />(b) (i) Write down the first three terms in the expansion of$\D (1+2x)^6$in ascending powers of$\D x.$<br />(ii) In the expansion of$\D (1+ax)(1+2x)^6,$&nbsp; the coefficient of$\D x^2$is 1.5 times the coefficient of$\D x.$Find the value of the constant$\D a.$<br /><br />12 (CIE 2014, s, paper 22, question 5)<br />(i) Find and simplify the first three terms of the expansion, in ascending powers of$\D x,$of$\D (1-4x)^5.$<br />(ii) The first three terms in the expansion of$\D (1-4x)^5(1+ax+bx^2)$&nbsp; are$\D 1- 23x+ 222x^2$.&nbsp; Find the value of each of the constants$\D a$and$\D b.$<br /><br />13 (CIE 2014, w, paper 11, question 6)<br />(i) Given that the coefficient of$\D x^2$in the expansion of$\D(2+ px)^6$is 60, find the value of the positive constant$\D p.$<br />(ii) Using your value of$\D p,$find the coefficient of$\D x^2$in the expansion of$\D (3- x) (2 +px)^6.$<br /><br />14 (CIE 2014, w, paper 13, question 9)<br />(a) Given that the first 3 terms in the expansion of$\D (5-qx)^p$are$\D 625- 1500x +rx^2,$find the value of each of the integers$\D p, q$and$\D r.$<br />(b) Find the value of the term that is independent of$\D x$in the expansion of$\D \left(2x+\frac{1}{4x^3}\right)^{12}.$&nbsp; <br /><br />15 (CIE 2015, s, paper 11, question 3)<br />(i) Find the first 4 terms in the expansion of$\D (2+x^2)^6$in ascending powers of$\D x.$<br />(ii) Find the term independent of$\D x$in the expansion of$\D (2+x^2)^6\left(1-\frac{3}{x^2}\right)^2.$&nbsp; <br /><br />16 (CIE 2015, s, paper 22, question 7)<br />In the expansion of$\D (1+2x)^n$, the coefficient of$\D x^4$is ten times the coefficient of$\D x^2.$Find the value of the positive integer,$\D n.$<br /><br />17 (CIE 2015, w, paper 13, question 8)<br />(a) Given that the first 4 terms in the expansion of$\D (2 +kx)^8$&nbsp; are$\D 256+ 256x+ px^2+ qx^3,$find the value of$\D k,$of$\D p$and of$\D q.$<br />(b) Find the term that is independent of$\D x$in the expansion of$\D \left(x-\frac{2}{x^2}\right)^9.$<br /><br />18 (CIE 2015, w, paper 21, question 2)<br />(i) Find, in the simplest form, the first 3 terms of the expansion of$\D (2 -3x)^6,$&nbsp; in ascending powers of$\D x.$<br />(ii) Find the coefficient of$\D x^2$in the expansion of$\D (1+ 2x) (2- 3x)^6.$<br /><br />1.$\D p = 2/3; q = 3/2$<br />2. (a)$\D -280;-504$(b)$\D 135$<br />3. (i)$\D x^6 + 12x^3 + 60 + \cdots$<br />(ii)$\D 72$<br />4. (i)$\D p = 2,$(ii)$\D -80$<br />5. (i)$\D -448$(ii)$\D 1792$<br />6. (i)$\D 1+n(x/2)+\frac{n(n-1)}{2}<br />(x/2)^2$<br />(ii)$\D n = 10$<br />7. (i)$\D 64 + 192x + 240x^2 + 160x^3$<br />(ii)$\D 64$<br />8. (a)(i)$\D -160$(ii)$\D -130$<br />(b)$\D 16x^2 + 32x + 24 + \frac{8}{x}+\frac{1}{x^2}$<br />9. (i)$\D a = 5; b = 4$(ii)$\D 20$<br />10. (i)$\D 64 - 960x + 6000x^2$<br />(ii)$\D a = 2; b = 10$<br />11. (a)$\D -48384$<br />(b)(i)$\D 1 + 12x + 60x^2$<br />(ii)$\D -4$<br />12. (i)$\D 1 - 20x + 160x^2$<br />(ii)$\D a = -3; b = 2$<br />13. (i)$\D p = 1/2$(ii)$\D 84$<br />14. (a)$\D p = 4; q = 3; r =<br />1350$(b)$\D 1760$<br />15. (i)$\D 64 + 192x^2 + 240x^4<br />+ 160x^6,$(ii)$\D 1072$<br />16.$\D n = 8$<br />17. (a)$\D k = 1/4; p = 112; q = 28$<br />(b)$\D -672$<br />18. (i)$\D 64 - 576x + 2160x^2$<br />(ii)$\D 1008$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-58789088255358626562018-12-12T05:30:00.000-08:002018-12-29T01:09:45.383-08:00Kinematics (Application in Differentation)$\def\D{\displaystyle}$<br /><br />1 (CIE 2012, s, paper 21, question 9)<br />A particle moves in a straight line so that,$\D t$s after passing through a fixed point$\D O,$its velocity,$\D v$ms$\D^{-1},$is given by$\D v = 2t - 11 +\frac{6}{t+1}.$Find the acceleration of the particle when it is at instantaneous rest. <br /><br />2 (CIE 2012, w, paper 13, question 7)<br />A particle$\D P$moves along the x-axis such that its distance,$\D x$m, from the origin$\D O$at time$\D t$s is given by$\D x = \frac{t}{t^2+1}$for$\D t\ge 0.$<br />(i) Find the greatest distance of$\D P$from$\D O.$<br />(ii) Find the acceleration of$\D P$at the instant when$\D P$is at its greatest distance from$\D O.$<br /><br />3 (CIE 2012, w, paper 21, question 11either)<br />A particle travels in a straight line so that,$\D t$s after passing through a fixed point$\D O,$its displacement,$\D s$m, from$\D O$is given by$\D s = t^2 - 10t + 10\ln(l + t),$where$\D t &gt; 0.$<br />(i) Find the distance travelled in the twelfth second. <br />(ii) Find the value of$\D t$when the particle is at instantaneous rest. <br />(iii) Find the acceleration of the particle when$\D t = 9.$<br /><br />4 (CIE 2012, w, paper 21, question 11or)<br />A particle travels in a straight line so that,$\D t$s after passing through a fixed point$\D O,$its velocity,$\D v$cms$\D^{-1},$is given by$\D v = 4e^{2t} - 24t.$<br />(i) Find the velocity of the particle as it passes through$\D O.$<br />(ii) Find the distance travelled by the particle in the third second. <br />(iii) Find an expression for the acceleration of the particle and hence find the stationary value of<br />the velocity. <br /><br />5 (CIE 2012, w, paper 22, question 10)<br />The acceleration,$\D a$m s$\D^{-2},$of a particle,$\D t$s after passing through a fixed point$\D O,$is given by$\D a = 4 - 2t,$for$\D t &gt; 0.$The particle, which moves in a straight line, passes through$\D O$with a velocity of 12 m s$\D^{-1}.$<br />(i) Find the value of$\D t$when the particle comes to instantaneous rest. <br />(ii) Find the distance from$\D O$of the particle when it comes to instantaneous rest. <br /><br />6 (CIE 2013, s, paper 12, question 12)<br />A particle P moves in a straight line such that,$\D t$s after leaving a point$\D O,$its velocity$\D v$m s$\Delta^{-1}$is given by$\D v = 36t -3t^2$for$\D t&gt;0.$<br />(i) Find the value of$\D t$when the velocity of$\D P$stops increasing. <br />(ii) Find the value of$\D t$when$\D P$comes to instantaneous rest. <br />(iii) Find the distance of$\D P$from$\D O$when$\D P$is at instantaneous rest. <br />(iv) Find the speed of$\D P$when$\D P$is again at$\D O.$<br /><br />7 (CIE 2013, w, paper 23, question 9)<br />A particle travels in a straight line so that,$\D t$s after passing through a fixed point$\D O,$its velocity,$\D v$ms$\D^{-1},$is given by$\D v = 3 + 6 \sin 2t .$<br />(i) Find the velocity of the particle when$\D t=\frac{\pi}{4}.$<br />(ii) Find the acceleration of the particle when$\D t = 2.$<br />The particle first comes to instantaneous rest at the point$\D P.$<br />(iii) Find the distance$\D OP.$<br /><br />8 (CIE 2014, s, paper 13, question 8)<br />A particle moves in a straight line such that, t s after passing through a fixed point$\D O,$its velocity,$\D v$ms$\D^{-1} ,$is given by$\D v = 5 - 4e^{-2t}.$<br />(i) Find the velocity of the particle at$\D O.$<br />(ii) Find the value of$\D t$when the acceleration of the particle is 6ms$\D^{-2} .$<br />(iii) Find the distance of the particle from$\D O$when$\D t = 1.5.$<br />(iv) Explain why the particle does not return to$\D O.$<br /><br />9 (CIE 2014, w, paper 21, question 7)<br />A particle moving in a straight line passes through a fixed point$\D O.$The displacement,$\D x$metres, of the particle,$\D t$seconds after it passes through$\D O,$is given by$\D x = t + 2 \sin t.$<br />(i) Find an expression for the velocity,$\D v$ms$\D^{-1}&nbsp; ,$at time$\D t.$<br />When the particle is first at instantaneous rest, find<br />(ii) the value of$\D t,$<br />(iii) its displacement and acceleration. <br /><br />10 (CIE 2014, w, paper 23, question 8)<br />A particle moving in a straight line passes through a fixed point$\D O.$The displacement,$\D x$metres, of the particle,$\D t$seconds after it passes through$\D O,$is given by$\D x = 5t - 3 \cos 2t + 3.$<br />(i) Find expressions for the velocity and acceleration of the particle after$\D t$seconds. <br />(ii) Find the maximum velocity of the particle and the value of$\D t$at which this first occurs. <br />(iii) Find the value of$\D t$when the velocity of the particle is first equal to 2 ms$\D^{-1}$and its acceleration at this time. <br /><br />11 (CIE 2015, s, paper 12, question 6)<br />A particle moves in a straight line such that its displacement,$\D x$m, from a fixed point$\D O$is given by$\D x= 10 \ln(t^2+4)-4t.$<br />(i) Find the initial displacement of the particle from$\D O.$<br />(ii) Find the values of$\D t$when the particle is instantaneously at rest. <br />(iii) Find the value of$\D t$when the acceleration of the particle is zero. <br /><br />12 (CIE 2015, s, paper 21, question 6)<br />A particle$\D P$is projected from the origin$\D O$so that it moves in a straight line. At time$\D t$seconds after projection, the velocity of the particle,$\D v$ms$\D^{-1},$is given by$\D v= 2t^2-14t+12.$<br />(i) Find the time at which$\D P$first comes to instantaneous rest. <br />(ii) Find an expression for the displacement of$\D P$from$\D O$at time$\D t$seconds. <br />(iii) Find the acceleration of$\D P$when$\D t = 3.$<br /><br />Answers<br />1. 11/6<br />2. (i)1=2(ii)a = -0.5<br />3. (i)13.8(ii)t = 4(iii)1.9<br />4. (i)v = 4(ii)638(iii)-1.18<br />5. (i)t = 6 (ii)s = 72<br />6. (i)t = 6 (ii)t = 12<br />(iii)s = 864(iv)324<br />7. (ai)9 (ii)-7.84(iii)11.1<br />8. (i)1; t = 0:144; s = 5:6<br />(iv)V is always positive<br />9. (i)v = 2 cos t + 1<br />(ii)t = 2:09<br />(iii)$\D t =-\sqrt{3}$<br />10. (i)v = 5 + 6 sin 2t<br />a = 12 cos 2t<br />(ii)$\D t=\frac{\pi}{4}$; v = 11<br />(iii)$\D t=\frac{7\pi}{12}; a = -6\sqrt{3}$<br />11. (i)10 ln 4 (ii)t = 1; 4<br />(iii)t = 2<br />12. (i)1<br />(ii)$\D 2t^3/3-14t^2/2+12t$<br />(iii)-2<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-15433296468457618702018-12-12T04:13:00.002-08:002018-12-29T01:10:43.344-08:00Vector$\def\lvec#1{\overrightarrow{#1}}\def\D{\displaystyle} $<br />$\newcommand{\iixi}{\left(\begin{array}{c}#1\\#2\end{array}\right)}$<br /><br />1 (CIE 2012, s, paper 12, question 12either)<br />EITHER<br />At 12 00 hours, a ship has position vector$\D 54\mathbf{i}+16<br />$km relative to a lighthouse, where$\D \mathbf{i}$is<br />a unit vector due East and$\D\mathbf{j}$is a unit vector due North. The ship is travelling with a speed of<br />20 km$\D h^{-1}$in the direction$\D 3\mathbf{i} +4\mathbf{j}.$<br />(i) Show that the position vector of the ship at 15 00 hours is$\D(90\mathbf{i} + 64\mathbf{j})km. <br />(ii) Find the position vector of the ship t hours after 12 00 hours. <br />A speedboat leaves the lighthouse at 14 00 hours and travels in a straight line to intercept the ship.<br />Given that the speedboat intercepts the ship at 16 00 hours, find<br />(iii) the speed of the speedboat, <br />(iv) the velocity of the speedboat relative to the ship, <br />(v) the angle the direction of the speedboat makes with North. <br /><br /><br />2 (CIE 2012, s, paper 12, question 12or)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-QCDI1l3X2aQ/XA5edAbcRXI/AAAAAAAACIk/jZbJCbMHIrwF0WspM8U5N8RsqYmvItziACLcBGAs/s1600/vector2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="373" data-original-width="789" height="151" src="https://1.bp.blogspot.com/-QCDI1l3X2aQ/XA5edAbcRXI/AAAAAAAACIk/jZbJCbMHIrwF0WspM8U5N8RsqYmvItziACLcBGAs/s320/vector2.png" width="320" /></a></div>2 (CIE 2012, s, paper 12, question 12or)<br />The position vectors of points\D A$and$\D B$relative to an origin$\D O$are$\D a$and$\D b$respectively. The point$\D P$is such that$\D \lvec{OP} = 54\lvec{OB}$. The point$\D Q$is such that$\D \lvec{AQ} = 13\lvec{AB}$. The point$\D R$lies on$\D OA$such that$\D RQP$is a straight line where$\D \lvec{OR} =\lambda\lvec{OA}$and$\D \lvec{QR} = \mu \lvec{PR} .$<br />(i) Express$\D \lvec{OQ}$and$\D\lvec{PQ}$in terms of$\D\mathbf{a}$and$\D\mathbf{b}.$<br />(ii) Express$\lvec{QR}$in terms of$\D \lambda, \mathbf{a}$and$\D \mathbf{b}.$<br />(iii) Express$\D \lvec{QR}$in terms of$\D\mu , \mathbf{a}$and$\D \mathbf{b}.$<br />(iv) Hence find the value of$\D\lambda$and of$\D \mu.$<br /><br /><br />3 (CIE 2012, s, paper 21, question 8)<br />Relative to an origin O, the position vectors of the points A and B are$\D 2\mathbf{i} – 3\mathbf{j}$and 11i + 42j<br />respectively.<br />(i) Write down an expression for<br />$\D \lvec{AB.}$<br />The point C lies on AB such that<br />$\D \lvec{AC} = \frac{1}{3}\lvec{AB}.$<br />(ii) Find the length of$D\lvec{OC.}$<br />The point$\D D$lies on$\D \lvec{OA}$such that$\D \lvec{DC}$&nbsp; is parallel to$\D \lvec{OB.}$<br />(iii) Find the position vector of$\D D.$<br /><br /><br />4 (CIE 2012, w, paper 12, question 1)<br />It is given that$\D\mathbf{a} = \iixi 43<br />, \mathbf{b} = \iixi{-1}{2}$and$\D \mathbf{c} = \iixi{21}{2}.$<br />(i) Find$\D |\mathbf{a} + \mathbf{b} + \mathbf{c}|.$<br />(ii) Find λ and μ such that&nbsp;$\D\lambda \mathbf{a}+ \mu \mathbf{b} = \mathbf{c}.$<br /><br />5 (CIE 2012, w, paper 13, question 5)<br />A pilot flies his plane directly from a point A to a point B, a distance of 450 km. The bearing of B from A is 030°. A wind of 80 km h$\D^{-1}$is blowing from the east. Given that the plane can travel at 320 km h$\D^{-1}$in still air, find<br />(i) the bearing on which the plane must be steered, <br />(ii) the time taken to fly from A to B. <br /><br /><br />6 (CIE 2012, w, paper 21, question 7)<br />In this question$\D\iixi 10$is a unit vector due east and$\D\iixi 01$is a unit vector due north. At 12 00 a coastguard, at point O, observes a ship with position vector$\D\iixi{16}{12}$km relative to O. The ship is moving at a steady speed of 10kmh$\D^{-1}$on a bearing of 330°.<br />(i) Find the value of p such that$\D\iixi{-5}{p}$kmh$\D^{-1}$represents the velocity of the ship. <br />(ii) Write down, in terms of$\D t,$the position vector of the ship, relative to$\D O, t$hours after 12 00.<br /><br />(iii) Find the time when the ship is due north of O. <br />(iv) Find the distance of the ship from$\D O$at this time. <br /><br /><br />7 (CIE 2012, w, paper 22, question 9)<br />A plane, whose speed in still air is 420 km h$\D^{-1},$travels directly from$\D A$to$\D B,$a distance of 1000 km. The bearing of$\D B$from$\D A$is 230° and there is a wind of 80 km h$\D^{-1}$from the east.<br />(i) Find the bearing on which the plane was steered. <br />(ii) Find the time taken for the journey. <br /><br /><br />8 (CIE 2012, w, paper 23, question 4)<br />The points$\D X, Y$and$\D Z$are such that$\D\lvec{XY} = 3\lvec{YZ} .$The position vectors of$\D X$and$\D Z,$relative to an origin$\D O,$are$\D\iixi{4}{-27}$and$\D\iixi{20}{-7}$respectively. Find the unit vector in the direction$\D\lvec{OY}.$<br /><br /><br />9 (CIE 2013, s, paper 11, question 9)<br />The figure shows points$\D A, B$and$\D C$with position vectors$\mathbf{a,b}$and$\D \mathbf{c}$respectively, relative to an origin$\D O.$The point$\D P$lies on$\D AB$such that$\D AP:AB = 3:4.$The point$\D Q$lies on$\D OC$such that$\D OQ:QC = 2:3.$<br />(i) Express$\D\lvec{AP}$in terms of$\mathbf{a}$and$\mathbf{b}$and hence show that$\D\lvec{OP} =\frac{1}{4}(\mathbf{a} +3\mathbf{b}).$<br />(ii) Find$\D\lvec{PQ}$in terms of$\D\mathbf{a,b}$and$\D\mathbf{c}.$<br />(iii) Given that$\D 5\lvec{PQ} = 6\lvec{BC},$find$\D\mathbf{c}$in terms of$\D\mathbf{a}$and$\D\mathbf{b}.$<br /><br /><br />10 (CIE 2013, s, paper 21, question 10)<br />A plane, whose speed in still air is 240 kmh$\D^{-1},$flies directly from$\D A$to$\D B,$where$\D B$is 500 km from$\D A$on a bearing of 032°. There is a constant wind of 50 kmh$\D^{-1}$blowing from the west.<br />(i) Find the bearing on which the plane is steered. <br />(ii) Find, to the nearest minute, the time taken for the flight. <br /><br /><br />11 (CIE 2013, s, paper 22, question 4)<br />The position vectors of the points$\D A$and$\D B,$relative to an origin$\D O,$are$\D 4\mathbf{i}- 21\mathbf{j}$and$\D 22\mathbf{i} - 30\mathbf{j}$respectively. The point$\D C$lies on$\D AB$such that$\D \lvec{AB} = 3\lvec{AC.}$<br />(i) Find the position vector of$\D C$relative to$\D O.$<br />(ii) Find the unit vector in the direction$\lvec{OC.}$<br /><br />Answers<br />1. (ii)$\D 54\mathbf{i} + 16\mathbf{j} + (12\mathbf{i} + 16\mathbf{j})t$<br />(iii) 64.8<br />(iv)$\D 39\mathbf{i}+24\mathbf{j}$(v)51.9<br />2. (i)$\D\frac{2}{3}\mathbf{a}-\frac{11}{12}\mathbf{b}$<br />(ii)$\D\lambda\mathbf{a}- (2/3)\mathbf{a}- (1/3)\mathbf{b}$<br />(iii)$\D\frac{\mu}{1-\mu}(\frac{2}{3}\mathbf{a}-\frac{11}{12}\mathbf{b})$<br />(iv)$\D \mu=\frac{4}{15},\lambda=\frac{10}{11}$3. (i)$\D 9\mathbf{i}+45\mathbf{j}$(ii)13<br />(iii)$\D\frac{4}{3}\mathbf{i}-2\mathbf{j}$<br />4. (i)25<br />(ii)$\D\lambda = 4,\mu=-5$<br />5. (i)Bearing 043;(ii) 1.65<br />6. (i)$\D 5\sqrt{3}$<br />(ii)$\D\frac{16-5t}{12+8.66t}$<br />(iii)1512 (iv)39.7<br />7. (i)bearing 223<br />(ii)2h 5min<br />8.$\D\frac{1}{5}\iixi{4}{-3}$<br />9. (i)$\D \frac{1}{4}(\mathbf{a}+3\mathbf{b})$<br />(ii)$\D\frac{2}{5}\mathbf{c}-\frac{1}{4}\mathbf{a}-\frac{3}{4}\mathbf{b}$<br />(iii)$\D \mathbf{c} = \frac{9\mathbf{b}-5\mathbf{a}}{16}$<br />10. (i) 022(ii)1h54m<br />11. (i)10i-24j<br />(ii)$\D \frac{1}{13}(5\mathbf{i}-12\mathbf{j})$<br /><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-6065271567081862912018-12-08T07:43:00.000-08:002018-12-29T01:14:38.509-08:00Matrices$\def\D{\displaystyle}$<br />$\newcommand{\iixii}{\left(\begin{array}{cc}#1&amp;#2\\#3&amp;#4\end{array}\right)}$<br />$\newcommand{\matrixa}{\left(\begin{array}{cc}#1\end{array}\right)}$<br /><div><br /></div>1 (CIE 2012, s, paper 21, question 1)<br />(i) Given that$\D A= \iixii{4}{-3}{2}{5},$find the inverse matrix$\D A^{-1}.$<br />(ii) Use your answer to part (i) to solve the simultaneous equations<br />$\D 4x - 3y = -10,$<br />$\D 2x + 5y = 21. $<br /><br />2 (CIE 2012, s, paper 22, question 4)<br />In a competition the contestants search for hidden targets which are classed as difficult, medium<br />or easy. In the first round, finding a difficult target scores 5 points, a medium target 3 points and<br />an easy target 1 point. The number of targets found by the two contestants, Claire and Denise, are<br />shown in the table.<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Target&nbsp; Difficult&nbsp; Medium&nbsp; Easy<br />Contestant<br />Claire&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;4&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;7<br />Denise&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 5&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1<br /><br />In the second round, finding a difficult target scores 8 points, a medium target 4 points and an<br />easy target 2 points. In the second round Claire finds 2 difficult, 5 medium and 2 easy targets<br />whilst Denise finds 4 difficult, 3 medium and 6 easy targets.<br />(i) Write down the sum of two matrix products which, on evaluation, would give the total score<br />for each contestant. <br />(ii) Use matrix multiplication and addition to calculate the total score for each contestant. <br /><br />3 (CIE 2012, w, paper 12, question 2)<br />(i) Find the inverse of the matrix<br />$\D \iixii{2}{-1}{-1}{1.5}.$<br />(ii) Hence find the matrix A such that$\D \iixii{2}{-1}{-1}{1.5}A=\iixii{1}{6}{-0.5}{4}.$<br /><br />4 (CIE 2012, w, paper 21, question 5)<br />It is given that$\D A=\matrixa{ 4&amp; -2\\<br />8&amp; -3}, B = \matrixa{2&amp; 0&amp; 4\\<br />5&amp; -1&amp; 4}$and$\D C =&nbsp; \matrixa{5\\-2\\3}.$<br />(i) Calculate$\D ABC.$<br />(ii) Calculate$\D A^{-1} B.$<br /><br />5 (CIE 2012, w, paper 23, question 2)<br />(i) Given that$\D A = \iixii{7}{8}{4}{6}$, find the inverse matrix,$\D A^{-1}.$<br />(ii) Use your answer to part (i) to solve the simultaneous equations<br />$\D 7x + 8y = 39,$<br />$\D 4x + 6y = 23.$<br /><br />6 (CIE 2013, s, paper 11, question 6)<br />(i) Given that$\D A=\iixii{2}{-1}{3}{5}$, find$\D A^{-1}.$<br />(ii) Using your answer from part (i), or otherwise, find the values of$\D a, b, c$and$\D d$such that<br />$A\iixii{a}{b}{c}{-1}=\iixii{7}{5}{17}{d}.$ <br /><br />7 (CIE 2013, s, paper 12, question 8)<br />(a) Given that the matrix$\D A =\iixii{4}{2}{3}{-5},$<br />&nbsp;find<br />(i)$\D A^2,$<br />(ii)$\D 3A + 4I,$where$\D I$is the identity matrix. <br />(b) Find the inverse matrix of$\D \iixii{6}{1}{-9}{3}.$<br />Hence solve the equations<br />$\D 6x+y=5$<br />$\D -9x+3y=\frac{3}{2}.$<br /><br />8 (CIE 2013, w, paper 11, question 11)<br />(a) It is given that the matrix<br />$\D A=\iixii{2}{3}{4}{1}.$<br />(i) Find$\D A + 2I.$<br />(ii) Find$\D A^2.$<br />(iii) Using your answer to part (ii) find the matrix$\D B$such that$\D A^2B = I.$<br />(b) Given that the matrix$\D C=\iixii{x}{-1}{x^2-x+1}{x-1},$<br />&nbsp;show that$\D \det C\not=0.$<br /><br />9 (CIE 2013, w, paper 13, question 7)<br />It is given that$\D A=\iixii{2t}{2}{t^2-t+1}{t}.$<br />(i) Find the value of$\D t$for which$\D \det A = 1.$<br />(ii) In the case when$\D t = 3,$find$\D A^{-1}$and hence solve<br />$\D 3x + y = 5,$<br />$\D 7x + 3y = 11.$<br /><br />10 (CIE 2014, s, paper 11, question 6)<br />Matrices$\D A$and$\D B$are such that$\D A =\matrixa{-1&amp;4\\7&amp;6\\4&amp;2}.$<br />and$\D B =\iixii{2}{1}{3}{5}.$<br />(i) Find$\D AB.$<br />(ii) Find$\D B^{-1} .$<br />(iii) Using your answer to part (ii), solve the simultaneous equations<br />$\D 4x+2y=-3$<br />$\D 6x+10y=-22.$<br /><br />11 (CIE 2014, s, paper 12, question 6)<br />(a) Matrices$\D X, Y$and$\D Z$are such that$\D X=\iixii{2}{3}{1}{2}, Y=\matrixa{1&amp;3\\4&amp;5\\6&amp;7}$and$\D Z=\matrixa{1&amp;2&amp;3}.$<br />&nbsp;Write down all the matrix products which are possible using any two of these matrices. Do not<br />evaluate these products. <br />(b) Matrices$\D A$and$\D B$are such that$\D A=\iixii{5}{-2}{-4}{1}$and$\D AB= \iixii{3}{9}{-6}{-3}.$Find the matrix$\D B.$<br /><br />12 (CIE 2014, s, paper 23, question 3)<br />In a motor racing competition, the winning driver in each race scores 5 points, the second and third<br />placed drivers score 3 and 1 points respectively. Each team has two members. The results of the drivers in one team, over a number of races, are shown in the table below.<br />Driver&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1st place&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2nd place&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;3rd place<br />Alan&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 4<br />Brian&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 4&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0<br />(i) Write down two matrices whose product under matrix multiplication will give the number of<br />points scored by each of the drivers. Hence calculate the number of points scored by Alan and by<br />Brian. <br />(ii) The points scored by Alan and by Brian are added to give the number of points scored by the team. Using your answer to part (i), write down two matrices whose product would give the number of points scored by the team. <br /><br />13 (CIE 2014, w, paper 11, question 7)<br />Matrices$\D A$and$\D B$are such that$\D \iixii{3a}{2b}{-a}{b}$and$\D B=\iixii{-a}{b}{2a}{2b},$where$\D a$and$\D b$are non-zero constants.<br />(i) Find$\D A^{-1}.$&nbsp; <br />(ii) Using your answer to part (i), find the matrix$\D X$such that$\D XA = B.$<br /><br />14 (CIE 2014, w, paper 13, question 5)<br />(a) A drinks machine sells coffee, tea and cola. Coffee costs \$0.50, tea costs \$0.40 and cola costs<br />\$0.45. The table below shows the numbers of drinks sold over a 4-day period.<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Coffee Tea Cola<br />Tuesday&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 12&nbsp; &nbsp; &nbsp; &nbsp;2&nbsp; &nbsp; &nbsp;1<br />Wednesday&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 9&nbsp; &nbsp; &nbsp; &nbsp; 3&nbsp; &nbsp; &nbsp;0<br />Thursday&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;8&nbsp; &nbsp; &nbsp; &nbsp; 5&nbsp; &nbsp; &nbsp;1<br />Friday&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;11&nbsp; &nbsp; &nbsp; &nbsp;2&nbsp; &nbsp; &nbsp;0<br />(i) Write down 2 matrices whose product will give the amount of money the drinks machine<br />took each day and evaluate this product. <br />(ii) Hence write down the total amount of money taken by the machine for this 4-day period. <br />(b) Matrices $\D X$ and $\D Y$ are such that $\D X = \iixii{2}{4}{-5}{1}$<br />and $\D XY = I,$ where $\D I$ is the identity matrix. Find the matrix $\D Y.$ <br /><br /><h3></h3><div><h3>Answers</h3>$\newcommand{\dfrac}{\displaystyle\frac{#1}{#2}}$<br />1.(i) $\D \frac{1}{26}\iixii{5}{3}{-2}{4}$<br />(ii) $\D x=0.5,y=4$<br />2(i) $\D \matrixa{4&amp;1&amp;7\\2&amp;5&amp;1} \matrixa{5\\3\\1} +\matrixa{2&amp;5&amp;2\\4&amp;3&amp;6}\matrixa{8\\4\\1}$<br />(ii) Claire=70, Denise=82<br />3(i) $\D \frac{1}{2}\matrixa{1.5&amp;1\\1&amp;2}$<br />(ii) $\D \matrixa{.5&amp;6.5\\0&amp;7}$<br />4(i) $\D \matrixa{10\\59}$<br />(ii) $\D \matrixa{1&amp;-0.5&amp;-1\\1&amp;-1&amp;-4}$<br />5(i) $\D \frac{1}{10}\matrixa{6&amp;-8\\-4&amp;7}$<br />(ii) $\D x=5,y=0.5$<br />6(i) $\D A^{-1}\dfrac{1}{13}\matrixa{5&amp;1\\-3&amp;2}$<br />(ii) $\D a=4,b=2,c=-1,d=1$<br />7 (a)(i) $\D \matrixa{22&amp;-2\\-3&amp;31}$<br />(ii) $\D \matrixa{16&amp;6\\9&amp;-11}$<br />(b)(i) $\D \dfrac{1}{27}\matrixa{3&amp;-1\\9&amp;6}$<br />(ii) $\D x=0.5,y=2$<br />8(a)(i)$\D \matrixa{4&amp;3\\4&amp;3}$<br />(ii) $\D A^2=\matrixa{16&amp;9\\12&amp;13}$<br />(iii) $\D \dfrac{1}{100}\matrixa{13&amp;-9\\-12&amp;16}$<br />9(i) $\D t=\dfrac{3}{2}$<br />(ii) $\D A=\matrixa{6&amp;2\\7&amp;3}, A^{-1}=\dfrac{1}{4}\matrixa{3&amp;-2\\-7&amp;6}$<br />$\D x=2,y=-1$<br />10 (i) $\D \matrixa{10&amp;19\\32&amp;37\\14&amp;14}$<br />(ii) $\D B^{-1}=\dfrac{1}{7}\matrixa{5&amp;-1\\-3&amp;2}$<br />$\D&nbsp; &nbsp;x=0.5,y=-2.5$<br />11(a) $\D YX,ZY$<br />(b) $\D \matrixa{3&amp;-1\\6&amp;-7}$<br />12(i) $\D \matrixa{3&amp;1&amp;4\\1&amp;3&amp;0},\matrixa{5\\3\\1}, \matrixa{22\\17}$<br />(ii) $\D (1,1),\matrixa{22\\17}$<br />13(i) $\D A^{-1} =\dfrac{1}{5ab}\matrixa{b&amp;-2b\\ a&amp;3a}$<br />(ii) $\D X=\matrixa{0&amp;1\\4/5&amp;2/5}$<br />14(a)(i) $\D \matrixa{0.5&amp;0.4&amp;0.45} \matrixa{12&amp;9&amp;8&amp;11\\ 2&amp;3&amp;5&amp;2\\ 1&amp;0&amp;1&amp;0} =\matrixa{7.25&amp;5.70&amp;6.45&amp;6.30}$<br />(ii) 25.70<br />(b) $\D \dfrac{1}{22}\matrixa{1&amp;-4\\5&amp;2}$</div><div><br /></div><div><br /></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0tag:blogger.com,1999:blog-5969775204143489576.post-33651681568817835912018-12-06T22:51:00.000-08:002018-12-26T06:02:31.146-08:00AP Sequence and Binomial Expansion$\def\D{\displaystyle}$<br />Let $\D a_0,a_1,a_2,\ldots,a_n$ be the&nbsp; coefficients of the expansion of $\D (1+x)^n$, and define $\D u_k=\frac{a_{k-1}}{a_{k-1}+a_{k}}, k=1,2,\ldots,n.$<br />Show that $\D u_{k+1}-u_k= \frac{1}{n+1},$ for $\D k=1,2,\ldots, n-1.$<br />Hence deduce that $\D u_k+u_{k+2}=2u_{k+1}$ for $\D k=1,2,\ldots,n-2.$<br /><br /><h3>, Proof:&nbsp;</h3>Since $\D a_k=^nC_k,$<br />\begin{eqnarray}<br />a_k&amp;=&amp; \frac{n(n-1)\cdots (n-k+1)}{1\times2\times \cdots \times k}\\<br />a_{k+1}&amp;=&amp;\frac{n(n-1)\cdots (n-k+1)(n-k)}{1\times2\times \cdots \times k\times (k+1)}.<br />\end{eqnarray}<br />$\D (2)\div (1):\frac{a_{k+1}}{a_k}=\frac{n-k}{k+1},$ for $\D k=0,1,\ldots,(n-1).$<br />Moreover $\D<br />\frac{1}{u_k}= \frac{a_k+a_{k+1}}{a_k}$$\D&nbsp; =1+\frac{a_{k+1}}{a_k}$$\D =1+\frac{n-k}{k+1}$$\D =\frac{k+1+n-k}{k+1} =\frac{n+1}{k+1},$ and hence $\D<br />u_k=\frac{k+1}{n+1}.$<span style="white-space: pre;"> </span><br />Now $\D u_{k+1}-u_k=\frac{(k+1)+1}{n+1}-\frac{k+1}{n+1}=\frac{1}{n+1}.$<br />Thus $\D u_1,u_2,\ldots,u_n$ is AP with common difference $\D \frac{1}{n+1}.$<br />Therefore $\D u_{k+2}-u_{k+1}=u_{k+1}-u_k$ implies<br />&nbsp;$\D u_k+u_{k+2}=2u_{k+1}.$<br /><div><br /><h3><a href="https://target-maths.blogspot.com/2018/11/problem-study-binomial-theorem_28.html">See also:</a></h3></div>Dr Shwe Kyawhttp://www.blogger.com/profile/04125478369062343808noreply@blogger.com0