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19
Sample Examination
Papers
Paper 1 Sample A
Paper 1 is a non-calculator paper. Your exam paper will have instructions
on the first page, some of which are reproduced here.
Full marks are not necessarily awarded for a correct answer with no working.
Answers must be supported by working and/or explanations. Where an
answer is incorrect, some marks may be given for a correct method, provided
this is shown by written working. You are therefore advised to show all
working.
It is important that you remember to show work because examiners will
award marks for correct work leading to the final solution. Also, if your
final answer is incorrect, you will not end up losing all the marks.
Section A
Answer all the questions in the spaces provided. Working may be
continued below the lines, if necessary.
Section B
Answer all the questions on the answer sheets provided. Please start each
question on a new page.
Section A
1 [Maximum mark: 6]
Given that logb a 5 0.74 and logb(a 2 1) 5 0.65, find the value of the
following expression:
logb(a4 2 1) 2 2 logb(a2 1 1) 1 logb(a3 1 a) 2 logb(a 1 1)
Give your answer to 2 decimal places.
2 [Maximum mark: 6]
Find the volume of the solid obtained by rotating the region under the
curve
1    
y 5  ______
x 2 1 1
______
from 0 to √
  e 3 2 1  
about the y-axis.
3 [Maximum mark: 6]
Find out where the normal line to the curve x 2 2 xy 1 y 2 5 3 at the
point (21, 1) intersects the curve a second time.
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4 [Maximum mark: 8]
Flaws appear randomly in a roll of textile at an average of 2 per metre
length.
a) Find the probability that more than 3 flaws appear in a randomly
chosen metre from this material.
b) Find the probability that more than 3 flaws appear in a randomly
chosen 2-metre piece from this fabric.
c) Two pieces of 1 metre each are chosen at random. Find the
probability the total number of flaws is 3.
5 [Maximum mark: 5]
3    , find the value of cosec t 1 cos 2t.
3p   , t , 2p and cos t 5 ____
  ___
If  ___
2
√
  10  
6 [Maximum mark: 6]
When P(x) 5 ax 5 1 3x 2 2 2x 1 b is divided by (x 1 2) the remainder
is 247, while if it is divided by (x 2 1) the remainder is 4. Find the
values of a and b.
7 [Maximum mark: 4]
Simplify the following expression and write your answer in the form
a 1 bi where a and b are real numbers.
(2 1 3i)2
________
  

 
 
3 2 2i
8 [Maximum mark: 5]
Find the value of k such that the following is a convergent geometric
series whose sum to infinity is 12.
n
i 2 1
     4k ∑

i 5 1
9 [Maximum mark: 6]
The following graph is the graph of a function of the form
y 5 a sin(b(x 2 c)) 1 d.
Determine the values of a, b, c and d.
y
4
3
2
( 5π
3 , 1)
1
( 7π
3 , 1)
O
x
1
2
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Sample Examination Papers
10 [Maximum mark: 8]
Graph the rational function
2
2x 1 3x 
   
.
y 5  _________
x 2 1 x 2 2
Show clearly all x- and y-intercepts and asymptotes.
Section B
11 [Maximum mark: 13]
The position vectors of the points A, B and C are
a 5 4i 2 9j 2 k, b 5 i 1 3j 1 5k and c 5 4i 2 7j 2 5k.
_ __›
a) Find the unit vector parallel to AB
  .___
_ __›
›
 . 
b) Find the angle between the vectors AB  and AC
c) Find a set of parametric equations for plane ABC.
d) Find the area of triangle ABC.
[3 marks]
[3 marks]
[3 marks]
[4 marks]
12 [Maximum mark: 21]
a) A and B are two events such that P(A) 5 _ 13 , P(B) 5  _16 , P(B/A) 5  _25 .
Calculate the probability that
(i) both A and B occur
(ii) either A or B occurs, but not both
(iii) A occurs, knowing that B has occurred.
[10 marks]
b) An industrial company has f female and m male employees. The
employees arrive in the morning at random, and we will assume
that the probability that the first employee to arrive any day will be
a female is
f
_____
.
      
f1m
Mr Guard plans to watch the employees arrive on four consecutive
days. If females arrive first every day on all four days, he will
conclude that f . m; if males arrive first every day of the four days,
he will conclude that f , m; otherwise, he will conclude that f 5 m.
Ms Reception on the other hand wants to watch for 7 days. If
females arrive first on 6 or 7 days, she will conclude that f . m;
if males arrive first on 6 or 7 days, she will conclude that f , m;
otherwise, she will conclude that f 5 m.
(i) If f 5 m, who, if any, is more likely to be wrong?
m  , what is the probability that Ms Reception will
(ii) If f 5  __
2
wrongly conclude that f 5 m?
[11 marks]
13 [Maximum mark: 14]
a) A sequence of real numbers {un} is defined by
un 1 1 5 2un cos  2 un 2 1; n . 1, u0 5 1, u1 5 cos .
Prove by mathematical induction that un 5 cos n.
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[8 marks]
b) The kth term vk of the series 12 1 30 1 58 1 … is given by
vk 5 5k 2 1 3k 1 4.
n
Find ∑
    vk, and express your answer in terms of n.
[6 marks]

k 5 1
n
n(n 1 1)(2n 1 1)
 
.
Hint: use the fact that ∑
    i 2 5 _______________
 
   
6

i 5 1
[IBO, 1978]
14 [Maximum mark: 12]
From a fixed point A on a circle with centre O and radius a, a
perpendicular is dropped to the tangent at P to the circle.
O
θ
A
a
P
Q
a) Given the central angle  as shown, prove that the area of triangle
APQ is
_ 1  a2|sin  |(1 2 cos ).
2
[7 marks]
b) As P moves along the circle, find the maximum value of the area of
triangle APQ.
[5 marks]
Paper 1 Sample B
Section A
1 [Maximum mark: 6]
12 and 2  _49 are the second and fifth terms of a geometric sequence.
a) Find the sum of the first n terms of this sequence.
b) Find the sum to infinity of this sequence.
2 [Maximum mark: 4]
The data {0, 11, 12, 12, 14, 16, 17, 18, 19, 20, 21} are represented by the
box-plot below.
0
0 is considered an outlier because it is more than 1.5 IQR (interquartile
range) below the first quartile. Show that this is true.
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19
Sample Examination Papers
3 [Maximum mark: 8]
The diagram below shows the shaded region A, in the first quadrant
that is enclosed by the curve y 2 5 8(2 2 x). Find the ratio of the
volume of the solid formed when A is rotated through 2p radians
around the x-axis to the volume of the solid formed when A is rotated
through 2p radians around the y-axis.
y
4
A
O
x
2
4 [Maximum mark: 6]
P (x) 5 x 3 1 bx 2 1 x 1 c is divisible by (x 2 2) but leaves a remainder
of 235 when divided by (x 1 3). Find b and c.
5 [Maximum mark: 6]
Consider the function
2
f (x) 5 e 2x 2x .
a) Find the maximum value of this function.
b) Find the x-coordinate of the points of inflexion.
6 [Maximum mark: 6]
The diagram shows the graph of a function y 5 f (x), which passes
through the points A(23, 0), B(21, 22), C (1, 0) and D (4, 0).
y
8
6
4
A
6
4
2
O
2
2
B
4
D
C
2
4
6 x
a) Graph the function g(x) 5 f (x 1 2) clearly indicating the
coordinates of the images of A, B, C and D.
b) Graph the function h(x) 5 f (2x 1 2) clearly indicating the
coordinates of the images of A, B, C and D.
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7 [Maximum mark: 6]
The lines L1 and L2 have the following equations:
2
1
y11
x 2 
5 
 
  
L1: r 5 21 1  3 ; L 2:  _____
 5 _____
 5 z 1 5
2
23
24
22
a) Find the point of intersection between these lines.
b) Find the cosine of the acute angle between the lines.
8 [Maximum mark: 6]
Z1
Z2
The cube roots of 1 are 1, z1 5 a 1 bi, and z2 5 a 2 bi.
a) Find the values of a and b.
b) In the Argand diagram, join the points corresponding to the roots.
Find the area of the triangle formed by the points as vertices.
9 [Maximum mark: 6]
Find all solutions to the equation
cos 4 1 sin2 2 5 _ 14 
in the interval [0, 2p].
10 [Maximum mark: 6]
3n 2 1 
4n 
, n  1.
 
The sum of the first n terms of a sequence is  ________
2
a) Find the first three terms of the sequence.
b) Find an expression for the nth term of the sequence, giving your
answer in terms of n.
Section B
11 [Maximum mark: 20]
Consider the plane P with equation 2x 1 3y 2 z 5 11 and the line L
with equation
y 2 1 _____
z 2 3. 
x 2 2  
  
 5  _____
 5     
 _____
5
21
2
a) Show that the point A(1, 3, 8) lies on the line L.
[3 marks]
b) Find the coordinates of point B, the intersection between line L and
plane P.
[3 marks]
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19
Sample Examination Papers
c) Find an equation of a line M containing A and perpendicular to P.
[4 marks]
d) Find the coordinates of the point C, the intersection of line M
and P.
[2 marks]
e) Hence or otherwise, find the point D symmetric to A about plane P.
[4 marks]
f) Find a set of parametric equations of the line through B and D.
[4 marks]
12 [Maximum mark: 20]
Consider the complex number z 5 cos  1 i sin .
a) Show that z 1 __
 1z 5 2 cos  and z 2 __
 1z 5 2i sin .
[2 marks]
1
b) Show that zn 1 __
  n  5 2 cos(n), and find a similar expression
z
 1n  .
[4 marks]
for z n 2 __
z 1
c) Hence, show that sin5  5 __
 16
   (sin 5 2 5 sin 3 1 10 sin ), and
[8 marks]
find a similar expression for cos5 .
d) Hence, find 4 sin2 ___
 3    2 32 sin5   d.
[6 marks]
2
13 [Maximum mark: 20]
A function f is defined by
( 
(  )
)
2
   
.
f (x) 5 __________
  x 2 1 x 2 2 
x 2 2x 2 3
a) What is the largest possible domain of f ? Find its derivative f (x).
Show that f (x) has a constant sign over its domain.
[6 marks]
b) Write down the equations of the asymptotes of the curve
y 5 f (x).
[3 marks]
c) Use the information developed so far to sketch the graph of f (x).
[3 marks]
d) Find the real numbers P, Q and R such that the following is true for
all values of x in the domain of f :
Q
  R    
.
[4 marks]
f (x) 5 P 1 _____
      1 _____
x11 x23
e) Use the expression above to find
f (x)dx.
[4 marks]
Paper 1 Sample C
Section B
1 [Maximum mark: 5]
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When the function f(x) 5 6x 4 1 11x 3 2 22x 2 1 ax 1 6 is divided by
(x 1 1), the remainder is 220.
Find the value of a.
2 [Maximum mark: 5]
A bag contains 2 red balls, 3 blue balls and 4 green balls. A ball is
chosen at random from the bag and is not replaced. A second ball is
chosen. Find the probability of choosing one green ball and one blue
ball in any order.
80
3 [Maximum mark: 6]
The 80 applicants for a Sports Science course were
required to run 800 metres and their times were
recorded. The results were used to produce the
following cumulative frequency graph.
Estimate
a) the median
[2 marks]
b) the interquartile range.
[4 marks]
70
Cumulative frequency
60
50
40
30
20
10
0
0
4 [Maximum mark: 6]
120
130
140
Time (seconds)
150
160
Find the coordinates of the point where the line with the vector equation
4
2
 
 
        1 g 21 
        intersects the plane with the equation
r 5  
22 
 
2
3
2x 1 3y 2 z 5 2.
(  ) (  )
5 [Maximum mark: 7]
a) Express the complex number 8i in polar form.
[3 marks]
b)The cube root of 8i which lies in the first quadrant is denoted by z.
Express z
(i) in polar form
[2 marks]
(ii) in Cartesian form.
[2 marks]
6 [Maximum mark: 7]
Find the equation of the line that is tangent to the curve 3x 2 1 4y 2 5 7
where x 5 1 and y > 0.
7 [Maximum mark: 6]
Find the value of x satisfying the equation
(3x)(42x 1 1) 5 6x 1 2
Give your answer in the form ___
 ln a  where a, b  .
ln b
8 [Maximum mark: 6]
a) The independent events A and B are such that P(A) = 0.4 and
P(A ∪ B) 5 0.88. Find P(B).
[4 marks]
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19
Sample Examination Papers
b) Find the probability that either A occurs or B occurs, but
not both.
[2 marks]
9 [Maximum mark: 6]
The area of the enclosed region shown in the diagram is defined by
y > x 2 1 2, y  ax 1 2, where a > 0
y
2
a
0
x
The region is rotated 360° about the x-axis to form a solid of
revolution. Find, in terms of a, the volume of this solid of revolution.
10 [Maximum mark: 6]
The diagram below shows the graph of equation y1 5 f (x), 0  x  4.
y
0
1
2
3
x
4
∫ 
x
Copy the axes below and sketch the graph of y2 5   f (t) dt, marking
0
clearly the points of inflexion.
y
0
940
1
2
3
4
x
Section B
11 [Maximum mark: 17]
{ 
The probability density function of the random variable X is given by
k     
, for 0  x  1
f (x) = _______
  _____
√
4 
– x 2  
    
 otherwise
   0
a) Find the value of the constant k.
[5 marks]
__
6(2  2 √ 3   )
  p 
 
 
b) Show that E(X) 5 _________
c) Find the median of X.
[7 marks]
[5 marks]
12 [Maximum mark: 16]
a)Find the root of the equation e 2 2 2x 5 2e 2x giving the answer as a
logarithm.
[4 marks]
b)The curve y 5 e 2 2 2x 2 2e 2x has a minimum point. Find the
coordinates of this minimum.
[7 marks]
c) The curve y 5 e 2 2 2x 2 2e 2x is shown below.
y
A
0
B
x
C
Write down the coordinates of the points A, B and C.
[3 marks]
d)Hence state the set of values of k for which the equation
[2 marks]
e 2 2 2x 2 2e 2x 5 k has two distinct roots.
13 [Maximum mark: 13]
a)Show that the following system of equations will have a unique
solution when a ≠ 21.
x 1 3y 2 z 50
3x 1 5y 2 z 50
x 2 5y 1 (2 2 a)z 59 2 a 2
b) Given that a ≠ 21, state the solution in terms of a.
c) Hence, solve
x 1 3y 2 z 50
3x 1 5y 2 z 50
x 2 5y 1 z 58
[5 marks]
[6 marks]
[2 marks]
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Sample Examination Papers
14 [Maximum mark: 14]
a) Using mathematical induction, prove that
n
∑
   (r 1 1)
  
2r 2 1 5 n(2n).
[7 marks]
r 5 1
b)The first three terms of a geometric sequence are also the first,
eleventh and sixteenth terms of an arithmetic sequence.
The terms of the geometric sequence are all different.
The sum to infinity of the geometric sequence is 18.
(i)Find the common ratio of the geometric sequence, clearly
showing all working.
[4 marks]
(ii) Find the common difference of the arithmetic sequence.
[3 marks]
Paper 2 Sample A
Paper 2 is a GDC paper. Your exam paper will have some instructions on
the first page, some of which are reproduced here.
Full marks are not necessarily awarded for a correct answer with no working.
Answers must be supported by working and/or explanations. Where an
answer is incorrect, some marks may be given for a correct method, provided
this is shown by written working. You are therefore advised to show all
working.
It is important that you remember to show work because examiners will
award marks for correct work leading to the final solution. Also, if your
final answer is incorrect, you will not end up losing all the marks.
Specific to GDC papers: If you use a GDC to arrive at your conclusion,
you need to show work leading to what you entered into your GDC. For
example, if you are to find the area of a certain region under a curve
between two points a and b, then you set up the integral leading to the
solution but not necessarily the symbolic manipulation required.
Example
Find the area enclosed by the curve f (x) 5 2x 3 2 9x 2 1 x 1 12 and the
x-axis.
Suggested answer
To find the area of this region, we observe that the function intersects the
x-axis at three different points: at x 5 21, x 5 1.5 and x 5 4.
∫   |2x 2 9x 1 x 1 12|dx  39.1.
4
Therefore, the area of the region is
 
21
3
2

Section A
1 [Maximum mark: 5]
For what values of x is the following inequation true?
27x 2 2 27x 1 4 > 0
942
2 [Maximum mark: 6]
In triangle ABC, BC 5 6, AC 5 7 and A 5 30°. Find all possible
values of AB.
3 [Maximum mark: 6]
An experiment can result in one or both of events A or B with the
following probabilities:
Find:
a) P(A  B)
A
A
B
0.34
0.46
B
0.15
0.05
b) P(A|B).
c) Are A and B independent? Justify.
4 [Maximum mark: 4]
In a binomial experiment with n trials, the probability of success
p 5 0.6 and P(x , 2) 5 0.1792. Find the value of n.
5 [Maximum mark: 7]
x
Consider the function f (x) 5 _______
  2e  x . 
1 1 3e 21
a) Find f (x).
b) Find the exact domain of f 21(x).
6 [Maximum mark: 5]
A part of a track is shown in the diagram. The radius of the inner circle
is 60 m and the width of the track is 3 m. The length of the inner arc is
20p and the outer arc is 21p. Find the area of the track.
θ
60 m
20π
21π
7 [Maximum mark: 6]
__
Consider the complex number z 5 5 √ 3  2 5i.
a) Express z in the form re i, presenting your answer in exact form.
b) Find the fifth roots of the complex number and sketch them in an
Argand diagram.
8 [Maximum mark: 7]
The figure below is that of the function
f (x) 5 cos x ln x, 0 < x < 2p.
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Sample Examination Papers
y
O
x
Find the ratio of the shaded area below the x-axis to the shaded area
above the x-axis.
9 [Maximum mark: 9]
The continuous random variable has the following pdf:
{
______
mx √4 2 x 2   x  [0, 2]
f (x) 5
0
otherwise
a) Find the value of m.
b) Find the ratio of the area between the mean and median to that
between the mean and mode.
10 [Maximum mark: 5]
Solve the initial value problem
dy
e x sin 2y  ___ 5 cos y (e 2x 2 x), y (0) 5 0.
dx
Section B
11 [Maximum mark: 18]
A function is defined by
f (x) 5 _ 13  x 3 2 x 2 2 3x 1 9, x  {R}.
a) Find the points where the graph of this function intersects the
x-axis.
[3 marks]
b) Find the point in the first quadrant where the normal to the
curve at (0, 9) meets the curve again.
[4 marks]
c) Find the local maximum and minimum of the function in the
interval [22, 5].
[3 marks]
d) Sketch the graph of the function.
[3 marks]
e) Find the area enclosed by the function and the line connecting
the maximum to the minimum points.
[5 marks]
12 [Maximum mark: 20]
a) Write down the expanded form of (1 1 x)n using binomial
coefficients. Include the term containing x r where 0 < r < n.
[4 marks]
5
7
b) Calculate the coefficient of x in the expansion of (1 1 x) (1 1 x)11.
[4 marks]
944
c) Calculate the coefficient of x r in the expansion of the identity
(1 1 x)m(1 1 x)n 5 (1 1 x)m 1 n where 0 < r < n, and 0 < r < m.
[4 marks]
d) Hence, show that
n
m 1 n  ).
   
            
 m   )( 
   
 n    1      
 m               
  n     1      
 m               
  n     1 … 1 (      
 m
(      
r   )(  0   )5 (    r   
0 r ) ( 1 )( r 2 1 ) ( 2 )( r 2 2 )
[4 marks]
e) By considering n 5 m 5 r, show that
2
2
2
 2n
(      n0   ) 1 (      n1   ) 1 … 1 (      nn  ) 5 (       
n   ).
[4 marks]
13 [Maximum mark: 22]
The amount of salt extracted in a large salt mine is modelled by a
normal distribution with a mean of 1.5 cubic metres per hour of
production time, and a standard deviation of 0.375 cubic metres.
a) (i) Find the probability that in a randomly chosen hour of
production, the output is between 1.2 cubic metres and
1.875 cubic metres.
[3 marks]
(ii) In 10% of the production hours, the output is considered
low. How many cubic metres are considered low?
[3 marks]
b) The production process can be adjusted to meet production
demand. So, the mean and the standard deviation can be altered.
The management would like to see that the production exceeds 2
cubic metres at most 10% of the time, and falls short of 0.5 cubic
metres 5% of the time. Find the values of the required mean
and standard deviation.
[8 marks]
Because of the hard nature of the extraction process, the machines
used in the process occasionally stop and have to be restarted. The
number of stoppages per hour of production is modelled by a Poisson
distribution with a mean of 3 stoppages.
c) (i) Find the probability that the machines stop at least 4 times in
each of three successive hours of production.
[5 marks]
(ii) Find the probability that the machines stop 20 times during a
randomly chosen 8-hour shift.
[3 marks]
Paper 2 Sample B
Section A
1 [Maximum mark: 7]
A pizza producer packs half-baked pizzas in boxes, freezes them and
distributes them to consumers. For a pizza to fit in a box, the diameter
must not exceed 30 centimetres. All pizzas with diameter larger than
30 cm have to be re-done. To comply with the label, pizzas must not
be smaller than 27 cm in diameter. It is found that 4% of the pizzas
are too large while 1% are too small. Assuming the diameters of
these pizzas to be normally distributed, find the mean and standard
deviation.
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Sample Examination Papers
2 [Maximum mark: 5]
An infinite geometric series converges to 24. The sum of the first three
terms is 208/9.
Find the sum of the first 6 terms.
3 [Maximum mark: 6]
Consider the function
2
f (x) 5 33x 2x .
a) Find the maximum value of this function.
b) Find the coordinates of the points of inflexion.
4 [Maximum mark: 5]
Solve the following inequation:
|x 2 1| 1 3
__________
 , 2
 
 
|x 1 1| 2 2
5 [Maximum mark: 6]
Consider the function
______
).
f (x) 5 sin( √
  4 2 x 2   
a) Find the domain and range of the function.
b) For what values of x does this function have an extreme value?
6 [Maximum mark: 7]
The probability density function of a random variable is
f (x) 5
{
k(2 2 log3 (4x 2 1 1))
0
2a < x < a
otherwise
__
a) Show that a 5 √
  2  .
b) Find the value of k correct to 3 decimal places.
7 [Maximum mark: 7]
The number of defects per square metre of fabric is known to follow a
Poisson distribution. It is discovered that
P(x < 3) 5 0.2381.
a) Find the average number of defects per square metre, to the nearest
integer.
b) You randomly pick 1 square metre of this fabric for inspection.
Find the probability of observing at least 3 defects.
8 [Maximum mark: 6]
You invest an amount of $1000 at an interest rate of 6% compounded
semi-annually.
How much money will you have in 20 years?
If you were offered to invest the money at continuous compounding,
how long will it take you to earn the same amount?
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9 [Maximum mark: 5]
Find the equation of the tangent line to the curve defined by
ln(xy) 5 2x
at the point (1, e 2).
10 [Maximum mark: 6]
Solve the differential equation
____
dy e (y 1 sin x)   
___
  y sec x
; y (0) 5 √
  ln 2  .
   5 _______
dx
2 Section B
11 [Maximum mark: 21]
[4 marks]
a) 5x 5 e kx for all real numbers x. Find the value of k.
b) Use the value of k found in a) to find the derivative of f (x) 5 5x.
[3 marks]
c) A random variable X has the following probability density function:
f (x) 5
{
5x 0
0<x<a
otherwise
(i) Find the value of a.
[4 marks]
(ii) Find the expected value of X.
[4 marks]
d) Three values of this random variable are chosen at random. What is
the probability that:
(i) at least one of the values is larger than 0.5?
[3 marks]
(ii) at most two of the values are less than 0.5?
[3 marks]
12 [Maximum mark: 18]
( 
22
Let A be the matrix 24
order 3.
2
2
5
1
)
3
and I be the identity matrix of
5
23  
a) Show that: det(A 2 kI ) 5 2k 3 1 22k 2 6.
[4 marks]
b) With an appropriate choice of k, find the determinant of A.
[3 marks]
c) You are given that the matrix A satisfies the equation
26I 1 22A 2 A3 5 0.
(i) Express the matrix A21 in terms of A.
[5 marks]
(ii) Hence, show that the three planes
20x 2 9y 1 5z 5 2,
2x 1 2z 5 3,
14x 2 6y 1 2z 5 5
intersect at one point. Find the coordinates of that point.
[6 marks]
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Sample Examination Papers
13 [Maximum mark: 21]
Consider the complex number z 5 cos  1 i sin .
a) Use DeMoivre’s theorem to find z 5.
5
[3 marks]
3
4
b) Hence, show that cos 5 5 11 cos  2 10 cos  1 5 sin  cos , and
[6 marks]
sin 5 5 15 sin  cos4  2 10 sin  cos2  1 sin5 .
3
5
1 t c) Hence, show that tan 5 5 ____________
 5t 2 10t   
   where t 5 tan .[6 marks]
1 2 10t 2 1 5t 4
d) Hence, find the solutions to the equation
t 5 2 5t 4 2 10t 3 1 10t 2 1 5t 2 1 5 0,
expressing your answer correct to 3 d.p.
[6 marks]
Paper 2 Sample C
Section A
1 [Maximum mark: 6]
^
Triangle ABC has C 5 42°, BC 5 1.74 cm, and area 1.19 cm2.
a) Find AC.
[2 marks]
b) Find AB.
[4 marks]
2 [Maximum mark: 5]
Find the values of a and b, where a and b are real, given that
(a 1 bi)(2 2 i) 5 5 2 i.
3 [Maximum mark: 6]
2 4  , x ≠ 22.
The function f is defined as f (x) 5 ______
 3x
x12
a) Find an expression for f 21(x).
[5 marks]
b) Write down the domain of f 21.
[1 marks]
4 [Maximum mark: 6]
The function f is defined as f (x) 5 sin x ln x for x [0.5, 3.5].
a) Write down the x-intercepts.
b)The area above the x-axis is A and the total area below the x-axis
is B.
If A 5 kB, find k.
[2 marks]
[4 marks]
5 [Maximum mark: 6]
The weights in grams of bread loaves sold at a supermarket are
normally distributed with mean 200 grams. The weights of 88% of the
loaves are less than 200 grams. Find the standard deviation.
6 [Maximum mark: 6]
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∫
Find e 2x sin x dx.
7 [Maximum mark: 6]
The number of car accidents occurring per day on a highway follows a
Poisson distribution with mean 1.5.
a)Find the probability that more than two accidents will occur on a
given day. [2 marks]
b)Given that at least one accident occurs on another day, find the
probability that more than two accidents occur on that day.
[4 marks]
8 [Maximum mark: 6]
There are 10 seats in a row in a waiting room. There are six people in
the room.
a) In how many different ways can they be seated?
[2 marks]
b)In the group of six people, there are three sisters who must sit next
to each other.
In how many different ways can the group be seated?
[4 marks]
9 [Maximum mark: 6]
Solve the differential equation given that y 5 1 when x 5 21.
dy
    5 4xy (x > 22)
(x 1 2)2 __
dx
10 [Maximum mark: 6]
The radius and height of a cylinder are both equal to x cm. The curved
surface area of the cylinder is increasing at a constant rate of 10 cm2/sec.
When x 5 2, find the rate of change of
a) the radius of the cylinder
[4 marks]
b) the volume of the cylinder.
[2 marks]
Section B
11 [Maximum mark: 12]
A machine is set to produce bags of salt, whose weights are distributed
normally, with a mean of 110 grams and standard deviation of
1.142 grams. If the weight of a bag of salt is less than 108 grams, the
bag is rejected. With these settings, 4% of the bags are rejected.
The settings of the machine are altered and it is found that 7% of the
bags are rejected.
a) (i) If the mean has not changed, find the new standard deviation,
correct to three decimal places.
[4 marks]
The mean is adjusted to operate with this new value of the
standard deviation.
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Sample Examination Papers
(ii)Find the value, correct to two decimal places, at which the
mean should be set so that only 4% of the bags are rejected.
[4 marks]
b)With the new settings from part (a), it is found that 80% of the
bags of salt have a weight which lies between A grams and B grams,
where A and B are symmetric about the mean.
Find the values of A and B, giving your answers correct to two
decimal places.
[4 marks]
12 [Total mark: 22]
Part A
[Maximum mark: 12]
A bag contains a very large number of ribbons. One quarter of the
ribbons are yellow and the rest are blue. Ten ribbons are selected at
random from the bag.
a) Find the expected number of yellow ribbons selected.
[2 marks]
b) Find the probability that exactly six of these ribbons are yellow.
[2 marks]
c) Find the probability that at least two of these ribbons are yellow.
[3 marks]
d) Find the most likely number of yellow ribbons selected. [4 marks]
e)What assumption have you made about the probability of selecting
a yellow ribbon?
[1 mark]
Part B
The continuous random variable X has probability density function
______
  x  2   
, for 0  x  k
1
f (x) 5                                    
  1 x   
  
   
0,   
otherwise
[Maximum mark: 10]
{ 
a) Find the exact value of k.
[5 marks]
b) Find the mode of X.
[2 marks]
c) Calculate P(1  X  2).
[3 marks]
13 [Total mark: 26]
Part A
a)The line L1 passes through the point A(0, 1, 2) and is
perpendicular to the plane x 2 4y 2 3z 5 0. Find a Cartesian
[2 marks]
equation of L1.
b)The line L2 is parallel to L1 and passes through the point
P(3, 28, 211). Find the vector equation of the line L2 . [2 marks]
[Maximum mark: 14]
_ __›
c) (i) The point Q is on the line L1 such that PQ  is perpendicular
to L1 and L2.
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Find the coordinates of Q.
(ii) Hence find the distance between L1 and L2.
Part B
Consider this system of equations.
[10 marks]
[Maximum mark: 12]
x 1 2y 1 kz 5 0
x 1 3y 1 z 5 3
kx 1 8y 1 5z 5 6
a)Find the set of values of k for which this system of equations has a
unique solution.
[6 marks]
b)For each value of k that results in a non-unique solution, find the
solution set.
[6 marks]
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