Thursday, December 27, 2018

Trigonometry (Selected Problems)

$\def\D{\displaystyle}$
Question (1): In $\Delta ABC$, if $\cot A+\cot B+\cot C=\sqrt 3$, then $\Delta ABC$ is equilateral.

$\D \begin{array}{|rl|}\hline
\cot(A+B)&=\D\frac{\cot A\cot B-1}{\cot A+\cot B}\\ \hline
\end{array}$

Proof:
\[\cot C=\cot(180-(B+C))=-\cot(A+B)=\frac{1-\cot A\cot B}{\cot A+\cot B}.\]
Let $\cot A=x,\cot B=y.$ Hence,
\begin{eqnarray*}
x+y+\frac{1-xy}{x+y}&=&\sqrt 3\\
(x^2+2xy+y^2) +(1-xy)&=&\sqrt 3x+\sqrt 3y\\
y^2+(x-\sqrt 3)y+(x^2-\sqrt 3x+1)&=&0\qquad \qquad (1)
\end{eqnarray*}
For real  solutions, $b^2-4ac\ge 0.$ Thus
\begin{eqnarray*}
(x-\sqrt 3)^2-4(1)(x^2-\sqrt 3x+1)&\ge 0\\
3x^2-2\sqrt 3x+1&\le&0\\
(\sqrt 3x-1)^2&\le&0\\
\sqrt 3x-1&=&0.
\end{eqnarray*}
Thus $x=1/\sqrt 3$. By (1), $y=1/\sqrt 3$. Therefore
\[\cot A=\cot B=\frac{1}{\sqrt 3}\Longrightarrow A=B=60^{\circ}.\]

Question (2): In $\Delta ABC$, if $\sin^2 A+\sin^2 B+\sin^2 C= 2$, then $\Delta ABC$ is a right triangle.
$\D\begin{array}{|rl|}\hline
\sin^2A&=\D \frac{1-\cos2A}{2}\\
\cos A+\cos B&=\D2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\\
\sin^2A&=1-\cos^2A\\
\sin A&=\sin(180^{\circ}-A)\\
\hline \end{array}$

Proof:
\begin{eqnarray*}
\sin^2A+\sin^2B&=&\frac{1-\cos2A}{2}+\frac{1-\cos2B}{2} \\
&=&1-\frac{1}{2}(\cos2A+\cos2B)\\
&=&1-\frac{1}{2}\left(2\cos \frac{2A+2B}{2}\cos \frac{2A-2B}{2}\right)\\
\sin^2A+\sin^2B &=&1-\cos(A+B)\cos(A-B)\cdots (1)\\
\sin^2C&=&\sin^2(180^{\circ}-C)=\sin^2(A+B)\\
\sin^2C&=&1-\cos^2(A+B)\cdots (2)
\end{eqnarray*}
(1)+(2):
$\D\sin^2A+\sin^2B+\sin^2C$\begin{eqnarray*}
&=&2-\cos(A+B)(\cos(A+B)+\cos(A-B))\\
2&=&2-\cos(A+B)\left(\cos A\cos B-\sin A\sin B \right.\\
&&\left.\qquad \qquad +\cos A\cos B+\sin A\sin B\right)\\
0&=&-2\cos(A+B)\cos A\cos B
\end{eqnarray*}
Hence, $\D \cos(A+B)=0$ or \(\D \cos A=0\) or \(\D\cos B=0.\)
Thus  \(\D A+B=90^{\circ}\), ie \(\D C=90^{\circ}\) or \(\D A=90^{\circ}\) or \(\D B=90^{\circ}.\)

Wednesday, December 26, 2018

AP GP Series (IB Standard Level)

$\def\D{\displaystyle}$
1.) In an arithmetic sequence, $\D u_1 = 2$ and $\D u_3 = 8.$
(a) Find $\D d.$
(b) Find $\D u_{20}.$
(c) Find $\D S_{20}.$ (Total 6 marks)

2.) In an arithmetic sequence $\D u_1 = 7, u_{20} = 64$ and $\D u_n = 3709.$
(a) Find the value of the common difference.
(b) Find the value of $\D n.$ (Total 5 marks)

3.) Consider the arithmetic sequence 3, 9, 15, ..., 1353.
(a) Write down the common difference.
(b) Find the number of terms in the sequence.
(c) Find the sum of the sequence. (Total 6 marks)

4.) An arithmetic sequence, $\D u_1, u_2, u_3, \ldots ,$ has $\D d = 11$ and $\D u_{27} = 263.$
(a) Find $u_1.$
(b) (i) Given that $\D u_n = 516,$ find the value of $\D n.$
(ii) For this value of $\D n,$ find $\D S_n.$ (Total 6 marks)

5.) The first three terms of an infinite geometric sequence are 32, 16 and 8.
(a) Write down the value of $\D r.$
(b) Find $\D u_6.$
(c) Find the sum to infinity of this sequence. (Total 5 marks)

6.) The $\D n^{th}$ term of an arithmetic sequence is given by $\D u_n = 5 + 2n.$
(a) Write down the common difference.
(b) (i) Given that the $\D n^{th} $ term of this sequence is 115, find the value of $\D n.$
(ii) For this value of $\D n,$ find the sum of the sequence. (Total 6 marks)

7.) In an arithmetic series, the first term is $\D -7$ and the sum of the first 20 terms is 620.
(a) Find the common difference.
(b) Find the value of the $\D 78 ^{th}$ term. (Total 5 marks)

8.) In a geometric series, $\D u_1 =
\frac{1}{81}$ and $\D u_4 =\frac{1}{3}.$
(a) Find the value of $\D r.$
(b) Find the smallest value of $\D n$ for which $\D S_n > 40.$ (Total 7 marks)

9.) (a) Expand $\D \sum_{r=4}^{7} 2^r$ as the sum of four terms.
(b) (i) Find the value of $\D \sum_{r=4}^{30} 2^r.$
(ii) Explain why $\D \sum_{r=4}^{\infty} 2^r$ cannot be evaluated. (Total 7 marks)

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)

11.) Consider the arithmetic sequence 2, 5, 8, 11, ....
(a) Find $\D u_{101}.$
(b) Find the value of $\D n$ so that $\D u_n = 152.$ (Total 6 marks)

12.) Consider the infinite geometric sequence $\D 3000, - 1800, 1080, -648, … .$
(a) Find the common ratio.
(b) Find the 10 th term.
(c) Find the exact sum of the infinite sequence. (Total 6 marks)

13.) Consider the infinite geometric sequence $\D 3, 3(0.9), 3(0.9)^2, 3(0.9)^3, … .$
(a) Write down the 10 th term of the sequence. Do not simplify your answer.
(b) Find the sum of the infinite sequence. (Total 5 marks)

14.) In an arithmetic sequence $\D u_{21} = -37$ and $\D u_4 = -3.$
(a) Find
(i) the common difference;
(ii) the first term.
(b) Find $\D S_{10}.$ (Total 7 marks)

15.) Let $\D u_n = 3 - 2n.$
(a) Write down the value of $\D u_1, u_2,$ and $\D u_3.$
(b) Find $\D \sum_{n=1}^{20} (3-2n)$ (Total 6 marks)

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.
(a) Calculate the number of seats in the 20th row.
(b) Calculate the total number of seats. (Total 6 marks)

17.) A sum of \$ 5000 is invested at a compound interest rate of 6.3 \% per annum.
(a) Write down an expression for the value of the investment after $\D n$ full years.
(b) What will be the value of the investment at the end of five years?
(c) The value of the investment will exceed \$ 10 000 after $\D n$ full years.
(i) Write down an inequality to represent this information.
(ii) Calculate the minimum value of $\D n.$ (Total 6 marks)

18.) Consider the infinite geometric sequence 25, 5, 1, 0.2, … .
(a) Find the common ratio.
(b) Find
(i) the 10th term;
(ii) an expression for the n th term.
(c) Find the sum of the infinite sequence. (Total 6 marks)

19.) The first four terms of a sequence are 18, 54, 162, 486.
(a) Use all four terms to show that this is a geometric sequence.
(b) (i) Find an expression for the $\D n$ th term of this geometric sequence.
(ii) If the $\D n$ th term of the sequence is 1062 882, find the value of $\D n.$ (Total 6 marks)

20.) (a) Write down the first three terms of the sequence $\D u_n = 3n,$ for $\D n\ge 1.$
(b) Find
(i) $\D \sum_{n=1}^{20} 3n$
(ii) $\D \sum_{n=21}^{100} 3n$ (Total 6 marks)

21.) Consider the infinite geometric series 405 + 270 + 180 +....
(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the fifteenth term of this series.
(c) Find the exact value of the sum of the infinite series. (Total 6 marks)

22.) (a) Consider the geometric sequence $\D -3, 6, -12, 24, ….$
(i) Write down the common ratio.
(ii) Find the 15th term.
Consider the sequence $\D x - 3, x +1, 2x + 8,\ldots.$
(b) When $\D x = 5,$ the sequence is geometric.
(i) Write down the first three terms.
(ii) Find the common ratio.
(c) Find the other value of $\D x$ for which the sequence is geometric.
(d) For this value of $\D x,$ find
(i) the common ratio;
(ii) the sum of the infinite sequence. (Total 12 marks)

Answers 

1 (a) $\D =3$
(b) $\D u_{20}=59$
(c) $\D S_{20}=610$
2 (a) $\D d=3$
(b) $\D n=1235$
3(a) $\D d=6$
(b)$\D n=226$
(c) $\D S_{226}=153228$
4 (a) $\D -23$
(b)(i) $\D 50$
(ii) 12325
5 (a) $\D r=\frac{1}{2}$
(b) $\D u_6=-1$
(c) $\D S=64$
6 (a) $\D d=2$
(b)(i) $\D n=55$
(ii) $\D S_{55}=3355$
7 (a) $\D d=4$
(b) $\D u_{78}=301$
8 (a) $\D r=3$
(b) $\D n=8$
9 (a) $\D 2^4+2^5+2^6+2^7$
(b)(i) 2147483632
(ii) $\D r\ge 1$
10 $\D u_1=-11,d=3$
11 $\D u_{101}=302,n=51$
12 $\D r=-0.6$
(b) $\D u_{10}=-30.2$
(c) 1875
13 (a) $\D u_{10}=3(0.9)^9$
(b) $\D S=30$
14 (a)(i) $\D d=-2,$
(ii) $\D u_1=3$
(b) $\D S_{10}=-60$
15 (a) $\D 1,-1,-3$
(b) $\D S_{20}=-360$
16 $\D u_{20}=53$
(b) $\D S_{20}=680$
17 (a) $\D 5000(1.063)^n$
(b) (a) $\D 6786$
(c) (i) $\D 5000(1.063)^n>10000$
(ii) 12 years
18 (a) $\D r=1/5$
(b) (i) 0.0000128
(ii) $\D u_n=25(.2)^n-1$
(c) $\D S=31.25$
19 (a)
(b) (i)$\D u_n=18\times 3^{n-1}$
(ii) $\D n=11$
20 (a) 3,6,9
(b) (i) 630
(ii) 14520
21 (a) $\D r=2/3$
(b) $\D u_{15}=1.39$
(c) $\D S=1215$
22 (a) (i) $\D r=-2$
(ii) $\D u_{15}=-49152$
(b) (i) 2,6,18
(ii) $\D r=3$
(c)  $\D x=-5$
(d) (i) $\D r=0.5$
(ii) $\D S=-16$

Tuesday, December 25, 2018

Graph (CIE)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 11, question 1)
(i) Sketch the graph of $\D y = |2x - 5|,$ showing the coordinates of the points where the graph meets the coordinate axes. [2]
(ii) Solve $\D |2x - 5| = 3 .$ [2]

2 (CIE 2012, s, paper 12, question 7)
(i) Sketch the graph of $\D y = |x^2 - x - 6|,$ showing the coordinates of the points where the curve meets the coordinate axes. [3]
(ii) Solve $|x^2 - x - 6| = 6.$ [3]

3 (CIE 2012, s, paper 21, question 3)
The diagram shows a sketch of the curve $\D y = a\sin(bx) + c$ for $\D 0^{\circ}\le  x \le  180^{\circ}.$ Find the
values of $\D a, b$ and $\D c.$ [3]
(b) Given that $\D f(x) = 5\cos3x + 1,$ for all $\D x,$ state
(i) the period of $\D f,$ [1]
(ii) the amplitude of $\D f.$ [1]

4 (CIE 2012, w, paper 11, question 1)
(i) Sketch the graph of $\D y = |3 + 5x|,$ showing the coordinates of the points where your graph meets the coordinate axes. [2]
(ii) Solve the equation $\D |3 + 5x| = 2.$ [2]

5 (CIE 2012, w, paper 12, question 9)
(a) (i) Using the axes below, sketch for $\D 0\le   x \le  \pi,$ the graphs of $\D y = \sin 2x$ and $\D y = 1 + \cos 2x.$ [4]
(ii) Write down the solutions of the equation $\D \sin 2x - \cos 2x = 1,$ for $\D 0 \le x \le  \pi.$ [2]
(b) (i) Write down the amplitude and period of $\D 5 \cos 4x - 3.$ [2]
(ii) Write down the period of $\D 4 \tan 3x.$ [1]

6 (CIE 2012, w, paper 13, question 4)
(i) On the axes below sketch, for $\D 0\le   x \le  \pi,$ the graphs of $\D y = \tan x$ and $\D y = 1 + 3\sin 2x.$ [3]
Write down
(ii) the coordinates of the stationary points on the curve $\D y = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi,$ [2]
(iii) the number of solutions of the equation $\D \tan x = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi.$ [1]

7 (CIE 2012, w, paper 21, question 3)
(i) On the grid below draw, for $\D 0^{\circ} \le  x \le  360^{\circ},$ the graphs of $\D y = 3 \sin 2x$ and $\D y = 2 + \cos x.$ [4]
(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  x \le  360^{\circ}. $ [1]

8 (CIE 2013, s, paper 11, question 1)
On the axes below sketch, for $\D 0 \le  x \le  2\pi,$ the graph of
(i) $\D y = \cos x - 1,$ [2]
(ii) $\D y = \sin 2x.$ [2]


(iii) State the number of solutions of the equation $\D \cos x - \sin 2x = 1,$ for $\D 0 \le x \le  2\pi.$ [1]

9 (CIE 2013, s, paper 21, question 2)
The velocity-time graph represents the motion of a particle moving in a straight line.
(i) Find the acceleration during the first 5 seconds. [1]
(ii) Find the length of time for which the particle is travelling with constant velocity. [1]
(iii) Find the total distance travelled by the particle. [3]

10 (CIE 2013, s, paper 21, question 4)
(i) Sketch the graph of $\D y = |4x - 2|$, showing the coordinates of the points where the graph meets the axes. [3]
(ii) Solve the equation $\D |4x - 2| = x.$ [3]

11 (CIE 2013, s, paper 22, question 3)
(i) Write down the letter of each graph which does not represent a function. [2]
(ii) Write down the letter of each graph which represents a function that does not have an inverse. [2]
(b)
The diagram shows the graph of a function $\D y = f(x).$ On the same axes sketch the graph of $\D y = f^{-1}(x).$

12 (CIE 2013, s, paper 22, question 10)
(a) The function $\D f$ is defined, for $\D 0^{\circ} \le x\le  360^{\circ},$ by $\D f(x) = 1 + 3 \cos 2x.$
(i) Sketch the graph of $\D y = f(x)$ on the axes below. [4]
(ii) State the amplitude of $\D f.$ [1]
(iii) State the period of $\D f.$ [1]
(b) Given that $\D \cos x = p ,$ where $\D 270^{\circ} < x < 360^{\circ},$ find  cosec $\D x$ in terms of $\D p.$ [3]

13 (CIE 2013, w, paper 11, question 1)
The diagram shows the graph of $\D y = a \sin(bx) + c$ for $\D 0 \le  x \le  2\pi,$ where $\D a, b$ and $\D c$ are positive integers. State the value of $\D a,$ of $\D b$ and of $\D c.$ [3]

14 (CIE 2013, w, paper 11, question 8)
(i) On the grid below, sketch the graph of $\D y = |(x - 2) (x + 3)|$ for $\D - 5 \le  x \le  4,$ and state the coordinates of the points where the curve meets the coordinate axes. [4]
(ii) Find the coordinates of the stationary point on the curve $\D y = |(x - 2) (x + 3)| .$ [2]
(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. [1]

15 (CIE 2013, w, paper 23, question 4)
(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 $\D B.$ [2]
(ii) Given that the point $\D \left(\frac{\pi}{8},7\right)$  also lies on the graph, find the value of $\D C.$ [1]
(b) Given that $\D f (x) = 8 - 5 \cos 3x,$ state the period and the amplitude of $\D f.$ [2]

16 (CIE 2014, s, paper 11, question 9a)
(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.
Find the distance travelled by the particle $\D P.$ [2]
(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.
On the axes below, plot the corresponding velocity-time graph for the particle $\D Q.$ [3]

(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 $\D s = 4t - 16 \ln(t+1)+ 13.$
(i) Find the value of $\D t$ for which the particle $\D R$ is instantaneously at rest. [3]
(ii) Find the value of $\D t$ for which the acceleration of the particle $\D R$ is 0.25ms$\D ^{-1}.$ [2]

17 (CIE 2014, s, paper 11, question 9b)
18 (CIE 2014, s, paper 12, question 3)
(i) Sketch the graph of $\D y = |(2x+1)(x-2)|$ for $\D -2\le  x\le   3,$ showing the coordinates of the points where the curve meets the x- and y-axes. [3]
(ii) Find the non-zero values of $\D k$ for which the equation $\D |(2x+1)(x-2)| = k$ has two solutions only.
[2]

19 (CIE 2014, s, paper 21, question 3)
(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. [2]
(ii) Find the set of values of k for which ^x - 4h^x + 2h = k has four solutions. [3]

20 (CIE 2014, w, paper 11, question 2)
(a) On the axes below, sketch the curve $\D y = 3 \cos 2x - 1$ for  $\D 0^{\circ}\le  x \le  180^{\circ}.$ [3]
(b) (i) State the amplitude of $\D 1 - 4 \sin 2x.$ [1]
(ii) State the period of $\D 5 \tan 3x + 1.$ [1]

21 (CIE 2014, w, paper 13, question 1)
The diagram shows the graph of $\D y = a \cos bx + c$ for $\D 0^{\circ} \le x \le  360^{\circ},$ where $\D a, b$ and $\D c$ are
positive integers. State the value of each of $\D a, b$ and $\D c.$ [3]
22 (CIE 2014, w, paper 13, question 2)
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 $\D AB.$ [5]

Answers


1. (i)
(ii) $\D x = 1, 4$
2. (i)
(ii) $\D x = 0, 1$
3. (a) $\D a = 3, b = 8, c = 7$
(b) $\D 2\pi/3, 5$
4. (i)
(ii) $\D x = -1=5,-1$
5. (ai)
(ii) $\D x = \pi/4, \pi/2$
(b)(i)Amp=5,Period= $\D \pi/2$
(ii)Period= $\D \pi/3$
6. (i)
(ii) $\D (\pi/4, 4), (3\pi/4,-2)$
(iii) $\D 3$
7. (i)
(ii) 4
8. (ii)
(iii) 3
9. 3.2,15,312
10. (ii) 2/5
11. (ai) A,E
(ii) C,D
(b)
12. (a)(i)
(ii)3
(iii)180
(b) $\D \frac{-1}{\sqrt{1-p^2}}$
13. $\D a = 3, b = 2, c = 1$
14. (i)
(ii) $\D (-0.5, 25/4)$
(iii) $\D k > 25/4$
15. (a)(i) $\D A = 3,B = 2$
(ii) $\D C = 4$
(b) 120,5
16. (a) 480
17. (b)
(c)  3,7
18. (i)
(ii) $\D k > 25/8$
19. (i)
(ii) $\D 0 < k < 9$
20. (a)
(b) $\D 4,\pi/3$
21. $\D a = 3, b = 2, c = 4$
22. $\D 2\sqrt{17}$

Sunday, December 23, 2018

Straight Line Equations (CIE)

$\def\D{\displaystyle}\newcommand{\vcol}[2]{\begin{array}{|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}\newcommand{\vicol}[2]{\begin{array}{|c|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}$1 (CIE 2012, s, paper 11, question 7)
The table shows values of variables $\D x$ and $\D y.$
$\D \vicol{x& 1& 3& 6& 10& 14}{y& 2.5& 4.5& 0 &–20 &–56}$

(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. [4]
(ii) Use your graph to find the value of $\D A$ and of $\D B.$ [4]

2 (CIE 2012, s, paper 22, question 7)
The table shows experimental values of variables $\D x$ and $\D y.$
$\D \vcol{x& 5& 30& 150& 400}{y& 8.9& 21.9& 48.9& 80.6}$
(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. [4]
(ii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [4]

3 (CIE 2012, w, paper 11, question 10)
The table shows values of the variables $\D x$ and $\D y.$
$\D\vicol{ x^{\circ}& 10& 30 &45 &60 &80}{ y &11.2& 16 &19.5& 22.4& 24.7}$
(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. [4]
(ii) Use your graph to find the value of $\D A$ and of $\D B.$ [3]
(iii) Estimate the value of $\D y$ when $\D x = 50.$ [2]
(iv) Estimate the value of $\D x$ when $\D y = 12.$ [2]

4 (CIE 2012, w, paper 22, question 8)

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).
(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. [5]
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
(ii) the gradient, [1]
(iii) the intercept on the vertical axis? [1]

5 (CIE 2012, w, paper 23, question 9)
The table shows experimental values of two variables $\D x$ and $\D y.$
$\D \vcol{ x& 1& 2& 3& 4}{y& 9.41 &1.29& – 0.69& – 1.77}$
It is known that $\D x$ and $\D y$ are related by the equation $\D y = \frac{a}{x^2}+bx,$  where $\D a$ and $\D b$ are constants.
(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. [1]
(ii) On the grid opposite, draw this straight line graph. [3]
(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [3]
(iv) Estimate the value of $\D y$ when $\D x$ is 3.7. [2]

6 (CIE 2013, s, paper 11, question 2)
Variables $\D x$ and $\D y$ are such that $\D y= Ab^x,$  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 $\D b.$ [5]

7 (CIE 2013, s, paper 22, question 1)

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 $\D x.$ [4]

8 (CIE 2013, w, paper 13, question 10)
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 $\D t.$
$\D \vcol{s& 2& 4& 6& 8}{t& 25.00& 6.25& 2.78& 1.56}$
(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. [1]
(ii) Draw this straight line graph on the grid below. [3]
(iii) Use your graph to find the value of $\D k$ and of $\D n.$ [4]
(iv) Estimate the value of $\D s$ when $\D t = 4.$ [2]

9 (CIE 2013, w, paper 21, question 8)
The table shows experimental values of two variables $\D x$ and $\D y.$
$\D \vcol{x& 2 &4& 6& 8}{y& 9.6& 38.4& 105& 232}$
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.
(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. [1]
(ii) Draw this straight line graph on the grid below. [2]
(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [3]
(iv) Estimate the value of $\D x$ for which $\D 2y = 25x.$ [2]

10 (CIE 2014, s, paper 11, question 8)
The table shows values of variables $\D V$ and $\D p.$
$\D \vcol{ V &10& 50& 100& 200}{p& 95.0& 8.5& 3.0& 1.1}$
(i) By plotting a suitable straight line graph, show that $\D V$ and $\D p$ are related by the equation $\D p = kV^n ,$
where $\D k$ and $\D n$ are constants. [4]
Use your graph to find
(ii) the value of $\D n,$ [2]
(iii) the value of $\D p$ when $\D V = 35.$ [2]

11 (CIE 2014, s, paper 13, question 10)
The table shows experimental values of $\D x$ and $\D y.$
$\D \vcol{x& 1.50 &1.75& 2.00& 2.25}{y& 3.9& 8.3 &19.5& 51.7}$
(i) Complete the following table.
$\D \vcol{x^2&\qquad &\qquad &\qquad &\qquad}{\lg y&&&&}$
[1]
(ii) By plotting a suitable straight line graph on the graph paper, show that $\D x$ and $\D y$ are related by the equation $\D y= Ab^{x^2},$  where $\D A$ and $\D b$ are constants. [2]
(iii) Use your graph to find the value of $\D A$ and of $\D b.$ [4]
(iv) Estimate the value of $\D y$ when $\D x = 1.25.$ [2]

12 (CIE 2014, s, paper 22, question 10)
Two variables $\D x$ and $\D y$ are connected by the relationship $\D y = Ab^x ,$ where $\D A$ and $\D b$ are constants.
(i) Transform the relationship $\D y = Ab^x$ into a straight line form. [2]
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.

(ii) Use the graph to determine the value of $\D A$ and the value of $\D b,$ giving each to 1 significant figure. [4]
(iii) Find $\D x$ when $\D y = 220.$ [2]

13 (CIE 2014, w, paper 11, question 9)
The table shows experimental values of variables $\D x$ and $\D y.$
$\D \vicol{x& 2& 2.5& 3 &3.5& 4}{y& 18.8& 29.6& 46.9& 74.1 &117.2}$
(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. [4]
(ii) Use your graph to find the value of $\D a$ and of $\D b.$ [4]

14 (CIE 2014, w, paper 23, question 6)
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.

(i) Find the equation of this line in the form $\D \ln y= m3^x+ c,$  where $\D m$ and $\D c$ are constants to be found. [3]
(ii) Find $\D y$ when $\D x = 0.5.$ [2]
(iii) Find $\D x$ when $\D y = 2000.$ [3]

Answers


1. (i) $\D y/x = A + Bx$
$\D \vicol{x& 1& 3& 6& 10& 14}{y/x& 2.5& 1.5& 0& -2& -4}$
(ii) $\D B = -0.5;A = 3$
2. (i) $\D \ln y = ln a + b ln x$
(ii) $\D b = 0.5; a = 4$
(iii) 32 to 49
3. (i) $\D \vicol{\sin x& 0.17& 0.5& 0.71& 0.87& 0.98}{\sqrt{y}& 3.35& 4 &4.42& 4.73& 4.97}$
(ii) $\D A = 2;B = 3$
(iii) $\D y = 20.5$
(iv) $\D x = 14.5$
4. (i) $\D y = 1000\sqrt{x}$
(ii) $\D m = 1$
(iii) $\D c = 6$
5. (i) $\D x^3$
(ii) $\D \vcol{x^3& 1& 8& 27& 64}{x^2y& 9.41 &5.16& -6.21& -28.32}$
(iii) $\D a = 10; b = -0.6$
(iv) $\D -1.48$
6. $\D b = e;A = e^2$
7. $\D y = (5x^2 - 2)^2$
8. (i) $\D \lg s$
(ii) $\D \vcol{\lg s& 0.3 &0.6& 0.78& 0.9}{lg t& 1.4& 0.8& 0.44& 0.19}$
(iii) $\D n = -2; k = 100$
(iv) $\D s = 4.9$
9. (i) $\D x^2$
(ii) $\D \vcol{x^2& 4& 16& 36& 64}{\frac{y}{x}& 4.8& 9.6& 17.5& 29}$
(iv) $\D 4.8$
10. (i)
(ii) $\D n = 1.5$
(iii) $\D 15$
11. (i) $\D \vcol{x^2& 2.25& 3.06& 4& 5.06}{\lg y& 0.59& 0.92 &1.29& 1.71}$
(ii)
(iii) $\D b = 2.5;A = 0.5$
(iv) $\D 2.1$
12. (i) $\D \log y = \log A + x \log b$
(ii) $\D 0.5$ (iii) $\D 4.4$
13. (i)
(ii) $\D b = 2.5; a = 3$
14. (i) $\D \ln y = 4(3^x) + 3$
(ii) $\D y = 20500$
(iii) $\D x = 0.127$

Saturday, December 22, 2018

Area of Sector (CIE, IGCSE Edexcel)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 12, question 8)

The figure shows a circle, centre $\D  O,$ 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.
(i) Find the perimeter of the shaded region. [3]
(ii) Find the area of the shaded region. [4]

2 (CIE 2012, s, paper 21, question 11)

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.
(i) Show that the length of $\D AB$ is approximately 10.1 cm. [1]
(ii) Find the perimeter of the shaded region. [5]
(iii) Find the area of the shaded region. [4]

3 (CIE 2012, w, paper 12, question 8)
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 $\D C.$
(i) Find the area of the shaded region. [3]
(ii) Find the perimeter of the shaded region. [4]

4 (CIE 2012, w, paper 13, question 9)
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.
(i) Find the perimeter of $\D ABCD.$ [4]
(ii) Find the area of $\D ABCD.$ [4]

5 (CIE 2012, w, paper 21, question 8)

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.
(i) Given that the perimeter of the shaded region is 20 cm, express $\D y$ in terms of $\D x.$ [2]
(ii) Given that the area of the shaded region is 16cm$\D^2,$ express $\D y^2$ in terms of $\D x^2.$ [2]
(iii) Find the value of $\D x$ and of $\D y.$ [4]

6 (CIE 2013, s, paper 11, question 8)

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.
(i) Show that $\D \theta = 1.855$ radians, correct to 3 decimal places. [2]
(ii) Calculate the perimeter of the shaded region. [4]
(iii) Calculate the area of the shaded region. [3]

7 (CIE 2013, s, paper 22, question 6)
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.
(i) Show that the perimeter of the segment is $\D r\left(\frac{3+\pi}{3}\right).$ [2]
(ii) Given that the perimeter of the segment is 26 cm, find the value of $\D r$ and the area of the
segment. [5]

8 (CIE 2013, w, paper 13, question 8)
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.
(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. [2]
(ii) Find the perimeter of the shaded region. [4]
(iii) Find the area of the shaded region. [3]

9 (CIE 2013, w, paper 21, question 10)
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.
(i) Find the perimeter of the shaded region. [5]
(ii) Find the area of the shaded region. [4]

10 (CIE 2014, s, paper 12, question 7)
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.
(i) Show that $\D \theta = 1.696,$ correct to 3 decimal places. [2]
(ii) Find the perimeter of the shaded region. [3]
(iii) Find the area of the shaded region. [3]

11 (CIE 2014, s, paper 23, question 1)

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$\D ^2 .$
(i) Find the value of $\D r.$ [2]
(ii) Hence find the perimeter of the sector. [2]

12 (CIE 2014, w, paper 13, question 6)
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 $\D OC = CB.$
(i) Show that $\D OC = 9.65$ cm correct to 3 significant figures. [2]
(ii) Find the perimeter of the shaded region. [3]
(iii) Find the area of the shaded region. [3]

13 (CIE 2014, w, paper 21, question 11)
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
(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, [5]
(ii) the perimeter of $\D PQSR,$ [3]
(iii) the area of $\D PQSR.$ [2]

Answers
1. (i) $\D 55.6 $
(ii) $\D 68.5$
2. (ii) $\D 54.3$
(iii) $\D 187$
3. (i) $\D 86.6$
(ii) $\D 55.5$
4. (i) $\D 73.9,$
(ii) $\D 231$
5. (i) $\D y = 3x - 20$
(ii) $\D y^2 = x^2 -32$
(iii) $\D x = 9; y = 7$
6. (ii) $\D P = 54.6$
(iii) $\D A = 115.25$
7. (ii) $\D r = 12.7;A = 14.6$
8. (ii) $\D 15.9 $
(iii) $\D 14.4$
9. (i) $\D 74.1 $
(ii) $\D 422$
10.  $\D P=48.7,A=178.5$
11.  $\D 25; 90$
12.  $\D P = 22.8;A = 19.4$
13.   (ii) $\D P = 19.8;A = 25$

Logarithms (CIE, Myanmar Grade 9)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 11, question 8)
(a) Find the value of $\D x$ for which $\D 2\lg x - \lg(5x + 60) = 1 .$ [5]
(b) Solve $\D \log_5 y = 4\log_y 5 .$ [4]

2 (CIE 2012, w, paper 11, question 3)
Given that $\D p = \log_q 32,$ express, in terms of $\D p,$
(i) $\D \log_q 4,$ [2]
(ii) $\D \log_q 16q.$ [2]

3 (CIE 2012, w, paper 12, question 4)
Given that $\D \log_a pq = 9$ and $\D \log_a p^2q = 15,$ find the value of
(i) $\D \log_a p$ and of $\D \log_a q,$ [4]
(ii) $\D \log_p a + \log_q a.$ [2]

4 (CIE 2013, s, paper 11, question 4)
(i) Given that $\D \log_4 x=\frac{1}{2},$  find the value of $\D x.$ [1]
(ii) Solve $\D 2\log_4y- \log_4 (5y-12)=\frac{1}{2}.$ [4]

5 (CIE 2013, w, paper 13, question 2)
Solve $\D 2 \lg y - \lg(5y+60)=1.$ [5]

6 (CIE 2013, w, paper 21, question 4)
Given that $\D \log_p X= 5$ and $\D \log_p Y= 2,$ find
(i) $\D \log_p X^2,$ [1]
(ii) $\D \log_p\frac{1}{X},$  [1]
(iii) $\D \log_{XY} p .$ [2]

7 (CIE 2014, s, paper 22, question 6)
(a) (i) State the value of $\D u$ for which $\D \lg u = 0.$ [1]
(ii) Hence solve $\lg |2x + 3| = 0.$ [2]
(b) Express $\D 2 \log_315- (\log_a5) (\log_3a),$ where $\D a > 1,$ as a single logarithm to base 3. [4]

8 (CIE 2014, s, paper 23, question 2)
Using the substitution $\D u= \log_3 x,$ solve, for $\D x,$ the equation $\D \log_3x -12 \log_x3= 4 .$ [5]

9 (CIE 2014, w, paper 13, question 7)
Solve the equation $\D 1+ 2 \log_5 x= \log_5(18x-9).$ [5]

10 (CIE 2014, w, paper 21, question 3)
Solve the following simultaneous equations.
$\D \begin{array}{rcl}
\log_2(x+3)&=&2+\log_2y\\
\log_2(x+y)&=&3
\end{array}$ [5]

Answers
1. (a) $\D x = 60$
(b) $\D y = 25,\frac{1}{25}$
2. (i) $\D 2p/5$
(ii) $\D 1 + 4p/5$
3. (i) $\D \log_a p = 6,\log_a q = 3$
(ii) $\D 0.5$
4. (i) $\D 2$
(ii) $\D y = 4, 6$
5. $\D 60$
6. $\D 10;-5;1/7$
7. (a) $\D 1;x = -1,-2$
(b) $\log_3 45$
8. $\D 729; 1/9$
9. $\D x = 3/5; 3$
10. $\D x = 5.8; y = 2.2$

Selected Trigonometric Examples

$\def\D{\displaystyle}$

Example 1 

Prove that $\D \sin^4x+\cos^4x=\frac{1}{4}(3+\cos 4x).$

$\D \begin{array}{|rl|}\hline (x+y)^2&=x^2+y^2+2xy \\ \sin^2 x&=\frac{1-\cos 2x}{2} \\ \sin^2 x+\cos^2 x&=1\\ 2\sin x\cos x&=\sin 2x\\ \hline \end{array} $

Proof: 

\begin{eqnarray*}
\left(\sin^2x+\cos^2x\right)^2&=&\sin^4x+\cos^4x+2\sin^2x\cos^2x\\
1^2&=&\sin^4x+\cos^4x+\frac{1}{2}(2\sin x\cos x)^2\\
1&=&\sin^4x+\cos^4x+\frac{1}{2}\sin^22x\\
&=&\sin^4x+\cos^4x+\frac{1}{2}\times \frac{1-\cos4x}{2}\\
\sin^4x+\cos^4x&=&1-\left( \frac{1}{4}-\frac{\cos4x}{4}\right) \\
&=&\frac{3}{4}+\frac{\cos4x}{4}
\end{eqnarray*}
Hence $\D \sin^4x+\cos^4x=\frac{1}{4}\left(3+\cos 4x\right).$

Example 2 

If $\D \sin x+\cos x=a,$ then show that $\D \sin^6x+\cos^6x=\frac{1}{4}\left(4-3\left(a^2-1\right)^2\right).$

$\D\begin{array}{|rcl|}\hline
(x+y)^3&=&x^3+y^3+3xy(x+y)\\
(x+y)^2&=&x^2+y^2+2xy\\
1&=&\sin^2x+\cos^2x\\ \hline
\end{array}$

Proof: 

\begin{eqnarray*}
a^2&=&\left(\sin x+\cos x\right)^2\\
&=&\sin^2x+\cos^2x+2\sin x\cos x\\
&=&1+2\sin x\cos x\\
\sin x\cos x&=&\frac{a^2-1}{2}
\end{eqnarray*}
\begin{eqnarray*}
\left(\sin^2x+\cos^2x\right)^3
&=&\left(\sin^2x\right)^3+\left(\cos^2x\right)^3\\
&&+3\sin^2x\cos^2x\left(\sin^2x+\cos^2x\right)\\
1^3&=&\sin^6x+\cos^6x+3(\sin x\cos x)^2\times 1\\
1&=&\sin^6x+\cos^6x+3\left(\frac{a^2-1}{2}\right)^2\\
\sin^6x+\cos^6x&=&1-\frac{3}{4}(a^2-1)^2\\
&=&\frac{1}{4}[4-3(a^2-1)^2]
\end{eqnarray*}

Example 3 

If $\D \cos x-\sin x=\sqrt{2}\sin x$, show that $\D cos x+\sin x=\sqrt{2}\cos x.$

Proof: 

$\D \begin{array}{lrll}
&\cos x&=\sin x+\sqrt{2}\sin x&\cdots (1)\\
(1)\times \sqrt{2}:&  \sqrt{2}\cos x&=\sqrt{2}\sin x+2\sin x&\cdots (2)\\
(2)-(1):& \sqrt{2}\cos x-\cos x&=\sin x &
\end{array}$
Hence $\D \sqrt{2}\cos x=\sin x+\cos x.$

Example 4

Prove that \[\frac{\cos 3x+\sin 3x}{\cos x-\sin x}=1+2\sin 2x.\]
$\D \begin{array}{|rl|}\hline
\cos x-\cos y&\D =-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\\
\sin x+\sin y&=\D \quad 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\\ \hline
\end{array}$

Proof:

$\D \begin{array}{rll}
\cos 3x-\cos x&=-2\sin \frac{3x+x}{2}\sin \frac{3x-x}{2}\\
&=-2\sin 2x\sin x&\cdots (1)\\
\sin 3x+\sin x&=2\sin \frac{3x+x}{2}\cos\frac{3x-x}{2}\\
&=2\sin2x\cos x&\cdots (2)
\end{array}$

(1)+(2): $\D \cos 3x+\sin 3x-(\cos x-\sin x) =2\sin 2x(\cos x-\sin x).$
\[\div (\cos x-\sin x): \frac{\cos 3x+\sin 3x}{\cos x-\sin x}-1=2\sin x.\]
Therefore \[\frac{\cos 3x+\sin 3x}{\cos x-\sin x}=1+2\sin x.\]

Friday, December 21, 2018

Indices (CIE, Myanmar Grade 9)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 12, question 2)
Using the substitution $\D u = 2^x,$ find the values of $\D x$ such that $\D 2^{2x+2} = 5 (2^x) - 1 .$ [5]

2 (CIE 2012, s, paper 21, question 5)
(a) Solve the equation $\D 3^{2x} = 1000,$ giving your answer to 2 decimal places. [2]
(b) Solve the equation $\D \frac{36^{2y-5}}{6^{3y}}=\frac{6^{2y-1}}{216^{y+6}}.$ [4]

3 (CIE 2012, w, paper 11, question 4)
Using the substitution $\D u = 5^x,$ or otherwise, solve
$\D 5^{2x+1} = 7(5^x) - 2.$ [5]

4 (CIE 2012, w, paper 22, question 6)
(i) Given that $\D \frac{2^{x-3}}{8^{2y-3}}=16^{x-y},$  show that $\D 3x + 2y = 6.$ [2]
(ii) Given also that $\D \frac{5^y}{125^{x-2}}=25,$ find the value of $\D x$ and of $\D y.$ [4]

5 (CIE 2012, w, paper 23, question 11)
(a) Solve $\D \left( 2^{x-2}\right)^\frac{1}{2}=100,$  giving your answer to 1 decimal place. [3]
(b) Solve $\D \log_y 2 = 3 - \log_y 256.$ [3]
(c) Solve $\D \frac{6^{5z-2}}{36^z}= \frac{216^{z-1}}{36^{3-z}}.$

6 (CIE 2013, s, paper 22, question 2)
(a) Solve the equation $\D 3^{p+1} = 0.7 ,$ giving your answer to 2 decimal places. [3]
(b) Express $\D \frac{y\times (4x^3)^2}{\sqrt{8y^3}}$  in the form $\D 2^a \times  x^b \times  y^c,$ where $\D a, b$ and $\D c$ are constants. [3]

7 (CIE 2013, w, paper 21, question 5)
Solve the simultaneous equations
$\D \begin{array}{rcl}
\frac{4^x}{256^y}&=&1024,\\
3^{2x}\times 9^y&=&243.
\end{array}$ [5]

8 (CIE 2014, s, paper 12, question 5)
(i) Given that $\D 2^{5x}\times 4^y= \frac{1}{8},$  show that $\D 5x + 2y = -3.$ [3]
(ii) Solve the simultaneous equations $\D 2^{5x}\times 4^y= \frac{1}{8}$ and $\D 7^x\times 49^{2y}=1.$ [4]

9 (CIE 2014, s, paper 13, question 2)
Given that $\D 2^{4x}\times 4^y\times 8^{x-y}=1$ and $\D 3^{x+y}= \frac{1}{3},$  find the value of $\D x$ and of $\D y.$ [4]

10 (CIE 2014, s, paper 21, question 11)
(a) Solve $\D 2^{x^2-5x}=\frac{1}{64}.$ [4]
(b) By changing the base of $\D \log_{2a} 4  ,$ express $\D (\log_{2a} 4)(1+\log_a 2)$ as a single logarithm to base $\D a.$ [4]

11 (CIE 2014, w, paper 11, question 4)
(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. [2]
(ii) Hence solve the equation $\D 5^{2x+1}-5^{x+1}+2=2(5^x).$ [4]

12 (CIE 2014, w, paper 13, question 10)
Solve the following simultaneous equations.
$\D \begin{array}{rcl}
\frac{5^x}{25^{3y-2}}&=&1\\
\frac{3^x}{27^{y-1}}&=&81
\end{array}$ [5]

Answers

1. $\D x = 0,-2$
2. (a) $\D 3.14$
(b) $\D y = -4.5$
3. $\D x = 0;-0.569$
4. (ii) $\D x = 14/9; y = 2/3$
5. (a) $\D x = 15.3$
(b) $\D y = 8$
(c) $\D z = 3.5$
6. (a) $\D p = -1.32$
(b) $\D a = 5/2; b = 6; c = -1/2$
7. $\D x = 3; y = -0.5$
8. $\D x = -2/3; y = 1/6$
9. $\D x = -1/8; y = -7/8$
10. (a) $\D x = 2; 3$
(b) $\D \log_a 4$
11. (i) $\D 5y^2 - 7y + 2 = 0$
(ii) $\D x = 0; y = 1; x = -0.569; y =2/5$
12. (a) $\D x = 6; y = 5/3$

Tuesday, December 18, 2018

Application of exponent

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 12, question 9)
Variables $\D N$ and $\D x$ are such that $\D N = 200 + 50e^{\frac{x}{100}}.$
(i) Find the value of $\D N$ when $\D x = 0.$ [1]
(ii) Find the value of $\D x$ when $\D N = 600.$ [3]
(iii) Find the value of $\D N$ when $\D \frac{dN}{dx}=45.$ [4]

2 (CIE 2014, w, paper 21, question 5)
The number of bacteria $\D B$ in a culture, $\D t$ days after the first observation, is given by
\[B= 500 +400e^{0.2t}.\]
(i) Find the initial number present. [1]
(ii) Find the number present after 10 days. [1]
(iii) Find the rate at which the bacteria are increasing after 10 days. [2]
(iv) Find the value of $\D t$ when $\D B = 10000.$ [3]

3 (CIE 2014, w, paper 23, question 4)
The profit \$ $\D P$ made by a company in its nth year is modelled by \[P=1000e^{an+b}.\]
In the first year the company made \$2000 profit.
(i) Show that $\D a + b = \ln 2.$ [1]
In the second year the company made \$3297 profit.
(ii) Find another linear equation connecting $\D a$ and $\D b.$ [2]
(iii) Solve the two equations from parts (i) and (ii) to find the value of $\D a$ and of $\D b.$ [2]
(iv) Using your values for $\D a$ and $\D b,$ find the profit in the 10th year. [2]

4 (CIE 2016, w, paper 21, question 4)
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
(i) the number of bacteria when $\D t = 10,$ [1]
(ii) the value of $\D t$ when the number of bacteria reaches 7500, [3]
(iii) the rate at which the number of bacteria is decreasing after 8 days. [3]

5 (CIE 2017, march, paper 12, question 11)
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).
(i) Find the value of $\D A$ and of $\D b.$ [6]
(ii) Find the value of $\D y$ when $\D x = 2.$ [2]

6 (CIE 2017, march, paper 22, question 2)
The value, $\D V$ dollars, of a car aged $\D t$ years is given by $\D V = 12000 e^{-0.2t}.$
(i) Write down the value of the car when it was new. [1]
(ii) Find the time it takes for the value to decrease to $\D \frac{2}{3}$ of the value when it was new. [2]

7 (CIE 2017, s, paper 12, question 7)
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).
(i) Find the value of $\D A$ and of $\D b.$ [5]
Using your values of $\D A$ and $\D b,$ find
(ii) the value of $\D y$ when $\D x = 0.6,$ [2]
(iii) the value of $\D x$ when $\D y = 600.$ [2]

8 (CIE 2018, s, paper 11, question 5)
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.
(i) State what the figure 800 represents in this experiment. [1]
(ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of $\D k.$ [3]
(iii) Calculate the population three days after the start of the experiment. [2]

9 (CIE 2018, s, paper 12, question 7)
A population, $\D B,$ of a particular bacterium, $\D t$ hours after measurements began, is given by $\D B =1000e^{\frac{t}{4}}.$
(i) Find the value of $\D B$ when $\D t = 0.$ [1]
(ii) Find the time taken for $\D B$ to double in size. [3]
(iii) Find the value of $\D B$ when $\D t = 8.$ [1]


1. $\D 250;208;4700$
2. (i) $\D 900$
(ii) $\D 3456$
(iii) $\D 591$
(iv) $\D 15.8$
3. (ii) $\D 2a + b = \ln 3.297$
(iii) $\D a = 0.5; b = 0.193$
(iv) $\D n = 10; P = 180000$
4. (i) $\D 8213$
(ii) $\D 27.7$
(iii) $\D \pm 67$
5. (i) $\D b = 2;A = 0.273,$
(ii) $\D 14.9$
6. (i) $\D 12 000$
(ii) $\D 2$
7. (i) $\D b=3,A=5.01$ or $\D 10.7$
(ii) $\D y=315$ or $\D 102.5$
(iii) $\D x = 0.693$
8. The number of bacteria at the
start of the experiment
$\D 1.61,100 000$
9. (i) $\D 1000,$
(ii) $\D 2.77$
(iii) $\D 7389$

Coordinate Geometry (CIE)

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 12, question 11)
The point $\D P$ lies on the line joining $\D A(-1, -5)$ and $\D B(11, 13)$ such that $\D AP = \frac{1}{3} AB.$
(i) Find the equation of the line perpendicular to $\D AB$ and passing through $\D P.$ [5]
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 $\D Q.$
(ii) Find the coordinates of the point $\D Q.$ [2]
(iii) Find the area of the triangle $\D PBQ.$ [2]

2 (CIE 2012, s, paper 21, question 10)
Solutions to this question by accurate drawing will not be accepted.


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 $\D ABCD.$ [9]

3 (CIE 2012, w, paper 11, question 8)
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 $\D AC.$
(i) Find the coordinates of $\D C.$ [2]
The point $\D D$ lies on the y-axis and the line $\D CD$ is perpendicular to $\D AC.$
(ii) Find the area of the triangle $\D ACD.$ [5]

4 (CIE 2012, w, paper 12, question 5)
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 $\D AB.$ [7]

5 (CIE 2012, w, paper 12, question 7)
Solutions to this question by accurate drawing will not be accepted.


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 $\D AEB = 90^{\circ}.$
(i) Find the coordinates of $\D D$ and of $\D E.$ [6]
(ii) Find the area of the trapezium $\D ABCD.$ [2]

6 (CIE 2012, w, paper 22, question 12either)
The point $\D A(0, 10)$ lies on the curve for which $\D \frac{dy}{dx}=e^{-\frac{\pi}{4}}.$  The point $\D B,$ with x-coordinate $\D -4,$ also lies on the curve.
(i) Find, in terms of $\D e,$ the y-coordinate of $\D B.$ [5]
The tangents to the curve at the points $\D A$ and $\D B$ intersect at the point $\D C.$
(ii) Find, in terms of $\D e,$ the x-coordinate of the point $\D C.$ [5]

7 (CIE 2012, w, paper 23, question 8)
Solutions to this question by accurate drawing will not be accepted.

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. [8]

8 (CIE 2013, s, paper 12, question 5)
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 $\D AB.$ [6]

9 (CIE 2013, s, paper 21, question 8)
The line $\D y = 2x - 8$ cuts the curve $\D 2x^2 +y^2- 5xy+ 32= 0$  at the points $\D A$ and $\D B.$ Find the length of the line $\D AB.$ [7]

10 (CIE 2013, s, paper 22, question 8)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A(- 6, 2), B(2, 6)$ and $\D C$ are the vertices of a triangle.
(i) Find the equation of the line $\D AB$ in the form $\D y = mx + c.$  [2]
(ii) Given that angle $\D ABC = 90^{\circ},$ find the equation of $\D BC.$ [2]
(iii) Given that the length of $\D AC$ is 10 units, find the coordinates of each of the two possible positions of point $\D C.$ [4]

11 (CIE 2013, s, paper 22, question 9)
(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 $\D k$
and of $\D c.$ [3]
(b) The variables $\D x$ and $\D y$ are connected by the equation $\D y = e^x + 25 - 24e^{-x}.$
(i) Find the value of $\D y$ when $\D x = 4.$ [1]
(ii) Find the value of $\D e^x$ when $\D y = 20$ and hence find the corresponding value of $\D x.$ [4]

12 (CIE 2013, w, paper 11, question 10)
Solutions to this question by accurate drawing will not be accepted.
The points $\D A(-3, 2)$ and $\D B(1, 4)$ are vertices of an isosceles triangle $\D ABC,$ where angle $\D B = 90^{\circ}.$
(i) Find the length of the line $\D AB.$ [1]
(ii) Find the equation of the line $\D BC.$ [3]
(iii) Find the coordinates of each of the two possible positions of $\D C.$ [6]

13 (CIE 2013, w, paper 23, question 7)
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
$\D AC : CB = 1 : 2,$ find the coordinates of $\D C.$ [8]

14 (CIE 2013, w, paper 23, question 8)
Solutions to this question by accurate drawing will not be accepted.

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 $\D X.$
(i) Given that C is the midpoint of BX, find the coordinates of D. [7]
(ii) Hence calculate the area of the quadrilateral $\D ABCD.$ [2]

15 (CIE 2014, s, paper 21, question 9)
Solutions to this question by accurate drawing will not be accepted.

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
(i) the coordinates of $\D E,$ [2]
(ii) the equation of the diagonal $\D AC,$ [3]
(iii) the area of the kite ABCD. [3]


16 (CIE 2014, s, paper 22, question 8)
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 $\D AB.$ [6]

17 (CIE 2014, s, paper 23, question 6)
Find the coordinates of the points of intersection of the curve $\D \frac{8}{x}-\frac{10}{y}=1$ and the line $\D x + y = 9.$ [6]

18 (CIE 2014, s, paper 23, question 9)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A(2,11), B(-2, 3)$ and $\D C(2,-1)$ are the vertices of a triangle.
(i) Find the equation of the perpendicular bisector of $\D AB.$ [4]
The line through $\D A$ parallel to $\D BC$ intersects the perpendicular bisector of $\D AB$ at the point $\D D.$
(ii) Find the area of the quadrilateral $\D ABCD.$ [6]

19 (CIE 2014, w, paper 11, question 5)
(i) Find the equation of the tangent to the curve $\D y= x^3- \ln x$  at the point on the curve
where $\D x = 1.$ [4]
(ii) Show that this tangent bisects the line joining the points $\D (-2,16)$ and $\D (12, 2).$ [2]

20 (CIE 2014, w, paper 11, question 8)
The point $\D P$ lies on the line joining $\D A(- 2, 3)$ and $\D B(10, 19)$ such that $\D AP:PB = 1:3.$
(i) Show that the x-coordinate of $\D P$ is 1 and find the y-coordinate of $\D P.$ [2]
(ii) Find the equation of the line through $\D P$ which is perpendicular to $\D AB.$ [3]
The line through $\D P$ which is perpendicular to $\D AB$ meets the y-axis at the point $\D Q.$
(iii) Find the area of the triangle $\D AQB.$ [3]

21 (CIE 2014, w, paper 21, question 6)
(i) Calculate the coordinates of the points where the line $\D y = x + 2$ cuts the curve $\D x^2+y^2=10.$ [4]
(ii) Find the exact values of $\D m$ for which the line $\D y = mx + 5$ is a tangent to the curve $\D x^2+y^2=10.$ [4]

Answer
1. (i)$\D 2x + 3y = 9$
(ii) $\D Q(-15; 13)$
(iii) $\D 156$
2. $\D 55$
3. (i) $\D C(13;-2)$
(ii) $\D 260$
4. $\D 8\sqrt{5}$
5. (i) $\D D(3, 8);E(5.4, 9.2)$
(ii) $\D 32$
6. (i) $\D 14 - 4e$
(ii) $\D x = \frac{4}{1-e}$
7. $\D 20$
8. $\D 1.25$
9. $\D 10\sqrt{5}$
10. (i) $\D y = 0.5x + 5$
(ii) $\D y - 6 = -2(x - 2)$
(iii) $\D (0,10),(4,2)$
11. (a) $\D k = 9; c = 5$
(b)(i) $\D 79.2$
(ii) $\D x = \ln 3$
12. (i) $\D \sqrt{20}$
(ii) $\D y = -2x + 6$
(iii) $\D x = 3; y = 0; x = -1; y = 8$
13. $\D (4, 0)$
14. (i) $\D D(8, 10)$
(ii) $\D 100$
15. $\D (3,5),y = 2x - 1,15$
16. $\D 6\sqrt{2}$
17. $\D x = 3; y = 6; x = 24; y = -15$
18. (i) $\D y = -0.5x + 7$
(ii) $\D 84$
19. (i) $\D y = 2x - 1$
20. (i) $\D y = 7$
(ii) $\D 3x + 4y = 31$
(iii) $\D 12.5$
21. (i) $\D (1, 3); (-3,-1)$
(ii) $\D m = \pm \sqrt{1.5}$