## Saturday, January 12, 2019

$\def\D{\displaystyle}$
1 (Edexcel, Further Pure Math, 2013 jan, Paper 2, No 2)
Using the identities $\D \sin (A + B) = \sin A \cos B + \cos A \sin B \\ \cos (A + B) = \cos A \cos B – \sin A \sin B \\ \tan A=\D\frac{\sin A}{\cos A}$
(a) show that $\D \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$ (3)
(b) Hence show that
(i) $\D \tan 105^{\circ}=\frac{1+\sqrt{3}}{1-\sqrt{3}}$
(ii) $\D \tan 15^{\circ}=\frac{\sqrt{3}-1}{1+\sqrt{3}}$ (4)

## Friday, January 4, 2019

### Probability (Edexcel Mathematic B)

$\def\D{\displaystyle}$
1.[Edexcel 2012,winter,mathematic B, Paper 01, no 21]
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.

(a) Complete the tree diagram representing these two events. (2)
(b) Find the probability that both sweets are red. Give your answer as a simplified fraction.
..............................................................(2)

2[Edexcel 2012,winter, mathematic B, Paper 02, no 5]

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.
(a) Write down the probability that Mr Driver will choose the direct road to the factory at road junction $\D A.$ (1)
(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)
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.
(c) Find the probability that Mr Driver will not arrive. (3)

3. [Edexcel 2012, Summer, Mathematics, Paper 01, No. 26]
A bag contains 3 red balls and 5 black balls. Two balls are to be taken at random, without replacement, from the bag.
(a) Complete the probability tree diagram.

(b) Find the probability that the two balls taken are of the same colour.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2)

4. [Edexcel 2012, Summer, mathematic B, Paper 02, no 5]
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.
(a) Draw a tree diagram to represent this information. (4)
(b) Calculate the probability that one particular day, Iftekhar will not buy a newspaper. (3)

5. [Edexcel 2013, winter, Mathematics, Paper 01, No. 25]
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.
(a) Calculate the probability that the disc is either brown or green.
..................................................(2)
This disc is then returned to the bag. Two discs are now to be chosen at random from the bag without replacement.
(b) Calculate the probability that one disc will be brown and one disc will be green.
..................................................(3)

6. [Edexcel 2013, Winter, mathematic B, Paper 02, no 6]

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.
(a) Calculate how many students took part in the survey. (4)
One of these students is to be chosen at random.
(b) Calculate the probability that this student took more than 30 minutes to travel to school. (2)
A similar survey was carried out on Tuesday and the results were compared with those of Monday’s survey.
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
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.
(c) Calculate the probability that this student took more than 30 minutes to travel to school. (3)

7. [Edexcel 2013, Summer, Mathematics B, Paper 01, No. 22]
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 $\D \frac{2}{5}$.  Given that there are 60 counters in the bag,
(a) find the number of blue counters in the bag.
.............................................................. (2)
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 $\D \frac{1}{2}.$
(b) Find the value of $\D x.$
$\D x =$ ..............................................................(2)

8. [Edexcel 2013, Summer, Mathematics B, Paper 02, No. 8]
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.
(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)
Given that
18 students passed both Mathematics and Science
17 students passed both English and Science
21 students passed Mathematics only
22 students passed English only
37 students passed Science only
(b) show all this information on Figure 4. (3)

(c) Find the value of $\D x.$ (2)
(d) Find the value of
(i) n$\D (M \cup S)$
(ii) n$\D (M \cap E \cap S' )$ (2)
A student is to be chosen at random from the 120 who took examinations in Mathematics, English and Science.
(e) Given that this student passed the Science examination, find the probability that the student also passed the English examination. (3)

9. [Edexcel 2013, Summer, Mathematics B, Paper 01R, No. 11]
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
(a) the first ball taken out of the bag is red,
..............................................................(1)
(b) the first two balls taken out of the bag are both red.
..............................................................(2)

10. [Edexcel 2014, winter, Mathematics B, Paper 01, No. 15]
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
(a) Write down the probability that the ball is either white or black.
..............................................................(1)
(b) Find the probability that the ball is neither white nor black.
..............................................................(2)

11. [Edexcel 2014, winter, Mathematics B, Paper 02, No. 6]
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 $\D \frac{5}{6}.$
(a) Complete the probability tree diagram.

Calculate the probability that on a school day,
(b) it will be raining and the bus will be late, (2)
(c) the bus will be late. (3)

12. [Edexcel 2014, winter, Mathematics, Paper 02R, No. 7]
A sports club has 80 members. For the three activities Swimming $\D (S),$ Cycling $\D (C)$ and Running $\D (R),$
8 members take part in all three activities,
3 members do not take part in any of the three activities,
22 members take part in only Swimming,
23 members take part in Swimming and Cycling,
19 members take part in Swimming and Running,
14 members take part in Cycling and Running.
(a) Using this information place the number of members in the appropriate subsets of the Venn diagram. (3)

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,
(b) form an equation in $\D x.$ (1)
(c) Solve your equation to find the value of $\D x.$ (2)

13. [Edexcel 2014, Summer, mathematic B, Paper 01, no 27]
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.
(a) Complete the probability tree diagram.

Calculate the probability that the archer will
(b) hit the target with his first shot but miss the target with his second shot,
.......................................................(2)
(c) hit the target at least once if he takes both shots.
.......................................................(3)

14. [Edexcel 2014, Summer, mathematic B, Paper 02, no 6]
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.
$\D\begin{array}{|c|c|}\hline \mbox{Number (n) of people}&\mbox{Number of houses with n}\\ \mbox{living in a house}&\mbox{people living in the house}\\ \hline 1& 2\\ 2& 3\\ 3 &1\\ 4 &4\\ 5 &3\\ 6 &6\\ 7 &8\\ 8 &2\\ 9 &1\\ \hline \end{array}$
(a) Find
(i) the modal number of people living in a house,
(ii) the median number of people living in a house,
(iii) the mean number of people living in a house.
(5)
Two houses in the street are chosen at random.
(b) Calculate the probability that 4 people live in one of the houses and 2 people live in the other of the houses. (2)
One of the people living in the street is chosen at random.
(c) Find the probability that this person lives in a house in which at least 5 people live. (2)

15. [Edexcel 2014, Summer, mathematic B, Paper 02R, no 5]
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.
(a) Complete the probability tree diagram. (3)

(b) Find the probability that the two counters taken are of different colours. (3)

Answer
1.(a) $\D \frac{4}{13},\frac{7}{13}$
$\D \frac{3}{13},\frac{6}{13}$
(b) $\D \frac{3}{14}\times\frac{2}{13}$
2(a) $\D \frac{1}{3}$
(b) $\D \frac{1}{3}\times\frac{1}{3}+\frac{1}{3}\times\frac{1}{2}$
(c) $D \frac{1}{3}$
3(a) $\D 5/8,3/7,4/7$
(b) $\D \frac{13}{28}$
(b) $\D \frac{4}{15}$
5(a) $\D \frac{13}{20}$
(b) $\D \frac{22}{99}$
6(a) 166
(b) $\D \frac{28}{83}$
(c) $\D \frac{9}{26}$
7(a) $D x=24$
(b) 12
8 (a) $\D 25-x$
(c) 10
(d) (i) 98(ii) 15
(e) $\D \frac{17}{64}$
9 (a) 3/7
(b) 3/17
10 (a) 0.94
(b) 0.06
(b) 2/35
(c) 37/210
12 (a) 3,22,15,11,6
(b) 80
(c) 5
(d) Swimming
(e)(i)39(ii) 34(iii) 8/39
(b) 3/10
(c) 9/10
14 (a) (i) 7 (ii) 6 (iii) 5.3
(b) 24/870
(c) 132/159
15 (a) 3/10,
6/9,3/9,
7/9,2/9
(b) 7/15

## Wednesday, January 2, 2019

### Set (CIE)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 21, question 6)
By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly.
(i) $\D A\cap B = (A'\cap B)'$  

(ii) $\D X\cap Y = X'\cup Y'$ 

(iii) $\D (P\cap Q)\cup (Q\cap R) = Q\cap (P\cup R)$ 

3 (CIE 2012, w, paper 13, question 1)
(a) On the Venn diagrams below, shade the region corresponding to the set given below each Venn diagram.


(b) It is given that sets $\D E, B, S$ and $\D F$ are such that
$\D E$ = {students in a school},
$\D B =$ {students who are boys},
$\D S =$ {students in the swimming team},
$\D F =$ {students in the football team}.
Express each of the following statements in set notation.
(i) All students in the football team are boys. 
(ii) There are no students who are in both the swimming team and the football team. 

4 (CIE 2012, w, paper 21, question 2)
(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.
(i) There are 11 prime numbers between 10 and 50. 
(ii) 18 is not a multiple of 5. 
(iii) All multiples of 10 are multiples of 5. 
(b) (i) In the Venn diagram below shade the region that represents $\D (A'\cap B) \cup (A\cap B').$ 

(ii) In the Venn diagram below shade the region that represents $\D Q\cap (R\cup S' ).$ 

5 (CIE 2013, s, paper 12, question 1)

The Venn diagram shows the universal set $\D E$, the set $\D A$ and the set $\D B.$ Given that $\D n(B ) = 5, n(A') = 10$
and $\D n(E) = 26,$ find
(i) $\D n(A \cap B),$ 
(ii) $\D n(A),$ 
(iii) $\D n(B' \cap A).$ 

6 (CIE 2013, s, paper 21, question 9)
It is given that $\D x \in R$ and that 
$\D E = {x : − 5 < x < 12},$
$\D S = {x : 5x + 24 > x^2},$
$\D T = {x : 2x + 7 > 15}.$
Find the values of $\D x$ such that
(i) $\D x \in S ,$ 
(ii) $\D x \in S\cup T ,$ 
(iii) $\D x \in (S\cap T )'.$ 

7 (CIE 2013, w, paper 11, question 4)
The sets $\D A$ and $\D B$ are such that
$\D A=\{ x:\cos x=\frac{1}{2} , 0^{\circ}\le x\le 620^{\circ},\}$
$\D B=\{ x:\tan x=\sqrt{3} , 0^{\circ}\le x\le 620^{\circ},\}$
(i) Find $\D n(A).$ 
(ii) Find $\D n(B).$ 
(iii) Find the elements of $\D A \cup B.$ 
(iv) Find the elements of $\D A \cap B.$ 

8 (CIE 2013, w, paper 23, question 5)
(a) (i) In the Venn diagram below shade the region that represents $\D (A\cup B)'$ 

(ii) In the Venn diagram below shade the region that represents $\D P\cap Q\cap R'.$ 

(b) Express, in set notation, the set represented by the shaded region. 

(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 $\D x.$
You may use the Venn diagram below to help you. 

9 (CIE 2014, s, paper 11, question 3)
(a) On the Venn diagrams below, shade the regions indicated.


(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 $\D x\in R.$
(i) Find $\D n(P).$ 
(ii) Find $\D n(Q).$ 

10 (CIE 2014, s, paper 12, question 2)
(a) On the Venn diagrams below, draw sets $\D A$ and $\D B$ as indicated.
(i)
(ii)

(b) The universal set $\D E$ and sets $\D P$ and $\D Q$ are such that $\D n(E) = 20, n(P \cup Q) = 15, n(P) = 13$  and $\D n(P \cap Q) = 4.$ Find
(i) $\D n(Q),$ 
(ii) $\D n(P \cup Q)',$ 
(iii) $\D n(P \cap Q').$ 

11 (CIE 2014, s, paper 23, question 4)
(a) Illustrate the following statements using the Venn diagrams below.
(i) $\D A \cup B = A$ (ii) $\D A \cap B \cap C = \emptyset.$ 

(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.
(i) 50 is not a cube number. 
(ii) 64 is both a square number and a cube number. 
(iii) There are 90 integers between 1 and 100 inclusive which are not square numbers. 

12 (CIE 2014, w, paper 13, question 3)
The universal set $\D E$ is the set of real numbers. Sets $\D A, B$ and $\D C$ are such that
$\D A = \{x:x^2+5x+6=0\},$
$\D B = \{x:(x-3)(x+2)(x+1)=0\},$
$\D C = \{x:x^2+x+3=0\}.$
(i) State the value of each of $\D n(A), n(B)$ and $\D n(C).$ 
(ii) List the elements in the set $\D A \cup B.$ 
(iii) List the elements in the set $\D A \cap B.$ 
(iv) Describe the set $\D C'.$ 

13 (CIE 2014, w, paper 21, question 1)
(a) On each of the Venn diagrams below shade the region which represents the given set.

(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 $\D x.$ 

### Algebra (CIE)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 12, question 4)
Solve the simultaneous equations
$\D 5x + 3y = 2$ and $\D \frac{2}{x}-\frac{3}{y}=1.$


2 (CIE 2012, w, paper 22, question 1)
Solve the equation $\D |7x + 5| = |3x – 13|.$ 

3 (CIE 2012, w, paper 23, question 1)
Solve the equation $\D |5x + 7| = 13.$ 

4 (CIE 2015, s, paper 22, question 5)
Solve the simultaneous equations
$\D \begin{array}{rcl} 2x^2+3y^2&=&7y,\\x+y&=&4. \end{array}$


5 (CIE 2016, w, paper 21, question 1)
Solve the equation $\D |4x - 3 |= x.$ 

6 (CIE 2017, march, paper 22, question 1)
Solve the equation $\D |5 - 3x |= 10.$ 

7 (CIE 2017, s, paper 22, question 1)
Solve $\D |5x + 3 |= |1 - 3x |.$ 

8 (CIE 2017, w, paper 21, question 4)
Solve the following simultaneous equations for $\D x$ and $\D y,$ giving each answer in its simplest surd form.
$\D \begin{array}{rcl} \sqrt{3}x + y& =& 4\\ x - 2y &=& 5 \sqrt{3} \end{array}$


9 (CIE 2017, w, paper 22, question 1)
If $\D z = 2 + \sqrt{3}$ find the integers $\D a$ and $\D b$ such that $\D az^2 + bz = 1 + \sqrt{3}.$  

10 (CIE 2017, w, paper 22, question 3)
Solve the inequality $\D |3x - 1|> 3 + x.$ 

11 (CIE 2017, w, paper 23, question 2)
Solve the equation $\D |3x - 1| = |5 + x| .$  

12 (CIE 2018, s, paper 11, question 1)
Solve the equations
$\D \begin{array}{rcl} y - x &=& 4,\\ x^2 + y^2 - 8x - 4y - 16 &=& 0. \end{array}$


### Answers

1.  $\D x = \frac{1}{5},y=\frac{1}{3}$:
$\D x = 4; y = -6$
2. $\D x = 0.8;-4.5$
3. $\D 1.2,-4$
4. $\D x = \frac{4}{3}, \frac{8}{3}$
$\D x = 3; y = 1$
5. $\D x = 1; 0.6$
6. $\D -5/3; 5$
7. $\D x = -2; x = -0.25$
8. $\D x = 2 +\sqrt{3},y=1-2\sqrt{3}$
9. $\D a = 1; b = -3$
10. $\D x > 2; x < -.5$
11. $\D x = -1$
12.  $\D x = 4; y = 8$
$\D x = -2; y = 2$

### Function (CIE)

$\newcommand{\D}{\displaystyle}$
1 (CIE 2012, s, paper 12, question 10)
(a) It is given that $\D f(x) =\frac{1}{2+x}$  for $\D x \not= -2, x\in R.$
(i) Find $\D f ″(x).$ 
(ii) Find $\D f^{-1} (x).$ 
(iii) Solve $\D f^2(x) = -1.$ 
(b) The functions g, h and k are defined, for $\D x\in R,$ by
\begin{eqnarray*}
g(x)&=&\frac{1}{x+5},x\not=-5\\
h(x)&=&x^2-1,\\
k(x)&=&2x+1.
\end{eqnarray*}
Express the following in terms of g, h and/or k.
(i) $\D \frac{1}{(x^2-1)+5}$ 
(ii) $\D \frac{2}{x+5}+1$ 

2 (CIE 2012, s, paper 21, question 12or)
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 $\D g : x \mapsto 5(x - 4)^2 + q.$
(i) Find the value of $\D p$ and of $\D q.$ 
(ii) Find the range of the function g. 
(iii) Sketch the graph of the function on the axes provided. 
(iv) Given that the function $\D h$ is defined by $\D h : x \mapsto \ln x,$ where $\D x > 0,$ solve the equation $\D gh(x) = 12.$ 

3 (CIE 2012, w, paper 11, question 9)
A function g is such that $\D g(x) = \frac{1}{2x-1}$  for $\D 1 \le x \le 3.$
(i) Find the range of $\D g.$ 
(ii) Find $\D g^{-1}(x).$ 
(iii) Write down the domain of $\D g^{-1}(x).$ 
(iv) Solve $\D g^2(x) = 3.$ 

4 (CIE 2012, w, paper 23, question 12either)
(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. 
The functions f and g are defined by
\begin{eqnarray*}
f:x&\mapsto& 4x^2+32x+55 \mbox{ for } x>-4\\
g:x&\mapsto&\frac{1}{x}\mbox{ for }x>0.
\end{eqnarray*}
(ii) Find $\D f^{-1}(x).$ 
(iii) Solve the equation $\D fg(x) = 135.$ 

5 (CIE 2012, w, paper 23, question 12or)
The functions h and k are defined by
\begin{eqnarray*}
h:x&\mapsto& \sqrt{2x-7} \mbox{ for } x>c\\
k:x&\mapsto&\frac{3x-4}{x-2}\mbox{ for }x>2.
\end{eqnarray*}

(i) State the least possible value of c. 
(ii) Find $\D h^{-1}(x).$ 
(iii) Solve the equation $\D k(x) = x.$ 
(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. 

6 (CIE 2013, s, paper 21, question 11)
A one-one function f is defined by $\D f(x)= (x- 1)^2- 5$ for $\D x \ge k .$
(i) State the least value that k can take. 
For this least value of k
(ii) write down the range of f, 
(iii) find $\D f^{-1}(x),$ 
(iv) sketch and label, on the axes below, the graph of $\D y = f(x)$ and of $\D y= f^{-1}(x),$ 
(v) find the value of x for which $\D f(x)= f^{-1}(x).$  

7 (CIE 2013, w, paper 11, question 12)
(a) A function f is such that $\D f (x)= 3x^2- 1$ for $\D - 10 \le x \le 8.$
(i) Find the range of f. 
(ii) Write down a suitable domain for f for which $f^{-1}$ exists. 
(b) Functions g and h are defined by $\D g(x)= 4e^x- 2$ for $\D x \in R,$ $h(x) = \ln 5x$ for $\D x > 0.$
(i) Find $\D g^{-1} (x).$ 
(ii) Solve $\D gh(x) = 18.$ 

8 (CIE 2013, w, paper 13, question 5)
For $\D x\in R,$ the functions f and g are defined by
\begin{eqnarray*}
f(x)&=&2x^3,\\
g(x)&=&4x-5x^2.
\end{eqnarray*}
(i) Express $\D f^2\left(\frac{1}{2}\right)$ as a power of 2. 
(ii) Find the values of x for which f and g are increasing at the same rate with respect to x. 

9 (CIE 2014, s, paper 21, question 12)
The functions f and g are defined by
\begin{eqnarray*}
f(x)&=&\frac{2x}{x+1}\mbox{ for } x>0,\\
g(x)&=&\sqrt{x+1}\mbox{ for } x>-1.
\end{eqnarray*}
(i) Find $\D fg(8)$. 
(ii) Find an expression for $\D f^2(x),$  giving your answer in the form $\D \frac{ax}{bx+c},$  where a, b and c are integers to be found. 
(iii) Find an expression for $\D g^{-1}(x),$ stating its domain and range. 
(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. 

10 (CIE 2014, s, paper 22, question 11)
The functions f and g are defined, for real values of x greater than 2, by
\begin{eqnarray*}
f(x)&=&2^x-1,\\
g(x)&=&x(x+1).
\end{eqnarray*}
(i) State the range of f. 
(ii) Find an expression for $\D f^{-1} (x),$ stating its domain and range. 
(iii) Find an expression for $\D gf (x)$ and explain why the equation $\D gf (x) = 0$ has no solutions. 

11 (CIE 2014, s, paper 23, question 12)
The function f is such that $\D f(x) = \sqrt{x-3}$ for $\D 4\le x\le 28.$
(i) Find the range of f. 
(ii) Find $\D f^2 (12).$ 
(iii) Find an expression for $\D f^{-1} (x).$ 
The function g is defined by  $\D g(x)=\frac{120}{x}$  for $\D x\ge 0.$
(iv) Find the value of x for which $\D gf (x) = 20.$ 

12 (CIE 2014, w, paper 21, question 4)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=&\sqrt{x-1}-3 \mbox{ for } x>1,\\
g(x)&=& \frac{x-2}{2x-3} \mbox{ for }x>2.
\end{eqnarray*}
(i) Find $\D gf(37).$ 
(ii) Find an expression for $\D f^{-1} (x).$ 
(iii) Find an expression for $\D g^{-1} (x) .$  

13 (CIE 2014, w, paper 23, question 7)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=& \frac{2}{x}+1 \mbox{ for }x>1,\\
g(x)&=&x^2+2.
\end{eqnarray*}

Find an expression for
(i) $\D f^{-1}(x),$ 
(ii) $\D gf(x),$ 
(iii) $\D fg(x).$ 
(iv) Show that $\D ff(x)=\frac{3x+2}{x+2}$ and solve $\D ff(x)=x.$ 

14 (CIE 2015, s, paper 11, question 8)
It is given that
\begin{eqnarray*}
f(x)&=&3e^{2x} \mbox{ for }x\ge 0,\\
g(x)&=&(x+2)^2+5 \mbox{ for } x\ge 0.
\end{eqnarray*}
(i) Write down the range of f and of g. 
(ii) Find $\D g^{-1},$ stating its domain. 
(iii) Find the exact solution of $\D gf(x) = 41.$ 
(iv) Evaluate $\D f'(\ln 4).$ 

Answers

1.(a)(i) $\D 2(2+x)^{-3}$
(ii) $\D \frac{1-2x}{x}$
(iii) $\D x=-\frac{7}{3}$
(b) $\D gh,kg$
2. (i) $p=-40,q=-8$
(ii) $g(x)>-8$
(iii)
(iv) $\D x=e^2,x=e^6$
3.(i) $\D 0.2\le x\le 1$
(ii) $\D g^{-1}(x)=\frac{1+x}{2x}$
(iii) $\D 0.2\le x\le 1$
(iv) $x+1.25$
4(i) $\D (2x+8)^2-9$
(ii) $\D f^{-1}=\frac{\sqrt{x+9}-8}{2}$
(iii) $\D x=0.5$
5(i) 3.5 (ii) $\D h^{-1}(x)=\frac{x^2+7}{2}$
(iii) $\D x=4$ (iv) 5-4/x
6(i)1 (ii) $\D f\ge -5$
(iii) $\D 1+\sqrt{x+5}$ (v)4
7(a)(i) $\D -1\le y\le 299$
(ii) $\D x\ge 0$
(b)(i) $\D \ln\left(\frac{x+2}{4}\right)$
(ii) $\D x=1$
8(i) $\D 2^{-5}$ (ii) $\D x=1/3,-2$
9(i) $\D 3/2$
(ii) $\D 4x/(3x+1)$
(iii) $\D g^{-1}(x)=x^2-1$
D: $x>0$ R:$\D g^{-1}(x)>-1$
(iv)
10(i) $\D f(x)>3$
(ii) $\D f^{-1}(x)=\log_2(x+1)$
$x>3,y>2$
(iii) no solution
11(i) $\D 3<f<7$
(ii) $\D 2+\sqrt{2}$
(iii) $\D f^{-1}(x)=(x-2)^2+3$
(iv) $\D x=19$
12(i) $\D 1/3$
(ii) $\D (x+3)^2+1$
(iii)$\D \frac{3x-2}{2x-1}$
13(i) $\D 2/(x-1)$
(ii) $\D gf(x)=(2/x+1)^2+2$
(iii)$\D fg(x)=2/(x^2+2)+1$
(iv) $\D x=2$

## 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) $\D S=-16$