ZIMSEC A Level Pure Mathematics Paper 2 June 2017

Section A

Section A, Question 1

[3 marks]Algebraic manipulation

Express

\dfrac{3x^{2}+6x+7}{x^{2}+2x+1}

in the form

\dfrac{\mathrm{A}}{(x+\mathrm{B})^{2}}+\mathrm{C}

, stating the values of the constants A, B and C.

Answer this when you sit the paper.

[2 marks]Graph transformations

The graph of

y=\dfrac{4}{(x+1)^{2}}+3

is obtained from the graph of

y=\dfrac{1}{x^{2}}

. Which sequence of transformations does this?

ATranslate 1 unit in the negative $x$ direction, then stretch parallel to the $x$ -axis with scale factor 4, then translate 3 units in the negative $y$ direction.
BTranslate 1 unit in the negative $x$ direction, then stretch parallel to the $y$ -axis with scale factor 4, then translate 3 units in the positive $y$ direction.
CTranslate 1 unit in the positive $x$ direction, then stretch parallel to the $y$ -axis with scale factor 4, then translate 3 units in the positive $y$ direction.
DStretch parallel to the $x$ -axis with scale factor 4, then translate by $\begin{pmatrix}-1\\3\end{pmatrix}$ .

[1 marks]Asymptotes

State the equation of the horizontal asymptote of the curve

y=\dfrac{4}{(x+1)^{2}}+3

Answer this when you sit the paper.

Section A, Question 2

[3 marks]Coordinate geometry: circles

Find the equation of the circle which passes through the points

(-1;\,0)

(1;\,2)

and

(-5;\,4)

, giving your answer in the form

x^{2}+y^{2}+2gx+2fy+c=0

Answer this when you sit the paper.

[2 marks]Coordinate geometry: circles

A circle has equation

x^{2}+y^{2}+4x-6y+3=0

. Find the coordinates of its centre.

Answer this when you sit the paper.

[1 marks]Coordinate geometry: circles

A circle has equation

x^{2}+y^{2}+4x-6y+3=0

. Find its radius, giving an exact answer.

Answer this when you sit the paper.

Section A, Question 3

[2 marks]Series

Given that

\displaystyle\sum_{r=1}^{n}r^{2}=\dfrac16 n(n+1)(2n+1)

, evaluate

\displaystyle\sum_{r=1}^{20}r^{2}

Answer this when you sit the paper.

[2 marks]Proof by mathematical induction

In proving by induction that

\displaystyle\sum_{r=1}^{n}r^{2}=\dfrac16 n(n+1)(2n+1)

, the inductive step forms

\dfrac16 k(k+1)(2k+1)+(k+1)^{2}

. Which expression does this simplify to?

A $\dfrac16(k+1)(k+2)(2k+3)$
B $\dfrac16(k+1)(k+2)(2k+1)$
C $\dfrac16 k(k+2)(2k+3)$
D $\dfrac16(k+1)^{2}(2k+3)$

[2 marks]Proof by mathematical induction

A proof by induction of a formula for

\displaystyle\sum_{r=1}^{n}r^{2}

begins by checking the case

n=1

. What does that step establish?

AThat the sum converges to a finite limit as $n$ becomes arbitrarily large.
BThat the formula holds for the first value of $n$ .
CThat the formula is true for every positive integer $n$ , so no further work is needed.
DThat the formula is the only possible closed form for this particular sum of squares.

Section A, Question 4

[3 marks]Turning points

Find the coordinates of the turning point of the curve

y=e^{x}+4e^{-2x}

Answer this when you sit the paper.

[2 marks]Second derivative test

The curve

y=e^{x}+4e^{-2x}

has a turning point at

x=\ln 2

. Determine the nature of that turning point.

Answer this when you sit the paper.

[2 marks]Implicit differentiation

Find the gradient of the curve

x^{2}-3xy+y^{2}=5

at the point

(1;\,4)

Answer this when you sit the paper.

[3 marks]Normals to a curve

Find the equation of the normal to the curve

x^{2}-3xy+y^{2}=5

at the point

(1;\,4)

, giving your answer in the form

ax+by+c=0

Answer this when you sit the paper.

[2 marks]Differential equations

Find the general solution of the differential equation

x\dfrac{d\theta}{dx}=\cos^{2}\theta

Answer this when you sit the paper.

Section A, Question 5

[2 marks]Matrices: determinants

Find the determinant of the matrix

\mathrm{M}=\begin{pmatrix}1&1&1\\1&1&-1\\-1&-2&3\end{pmatrix}

Answer this when you sit the paper.

[3 marks]Matrices: simultaneous equations

Solve the simultaneous equations

x+y+z=-2

x+y-z=2

-x-2y+3z=3

Answer this when you sit the paper.

[2 marks]Matrix equations

Matrices M and N satisfy

\mathrm{MN}=\mathrm{P}

, where M is a non-singular

3\times3

matrix. Which expression gives N?

A $\mathrm{P}\mathrm{M}$
B $\dfrac{\mathrm{P}}{\mathrm{M}}$
C $\mathrm{M}^{-1}\mathrm{P}$
D $\mathrm{P}\mathrm{M}^{-1}$

[2 marks]Matrix equations

Given that

\mathrm{M}=\begin{pmatrix}1&1&1\\1&1&-1\\-1&-2&3\end{pmatrix}

and

\mathrm{MN}=\begin{pmatrix}1&1&1\\2&0&1\\0&1&-2\end{pmatrix}

, find the element of N in the first row and first column.

Answer this when you sit the paper.

[2 marks]Matrices

Why can the equation

\mathrm{MN}=\mathrm{P}

be solved for N when

\mathrm{M}=\begin{pmatrix}1&1&1\\1&1&-1\\-1&-2&3\end{pmatrix}

ABecause $\det\mathrm{M}\neq0$ , so $\mathrm{M}^{-1}$ exists.
BBecause M is a square matrix, and every square matrix of order 3 has an inverse.
CBecause every entry of M is a whole number, so its inverse has whole number entries.
DBecause M and P have the same number of rows and the same number of columns.

Section A, Question 6

[2 marks]Vectors: points on a plane

The plane P has equation

\mathbf{r}\cdot\begin{pmatrix}2\\-4\\-1\end{pmatrix}=8

and the point A has position vector

3\mathbf{i}-\mathbf{j}+2\mathbf{k}

. Evaluate

\overrightarrow{\mathrm{OA}}\cdot\begin{pmatrix}2\\-4\\-1\end{pmatrix}

and hence state whether A lies in P.

Answer this when you sit the paper.

[2 marks]Vectors

Points A and B have position vectors

3\mathbf{i}-\mathbf{j}+2\mathbf{k}

and

7\mathbf{i}-9\mathbf{j}

respectively. Find

\overrightarrow{\mathrm{BA}}

Answer this when you sit the paper.

[3 marks]Vectors: angle between two vectors

Points A and B have position vectors

3\mathbf{i}-\mathbf{j}+2\mathbf{k}

and

7\mathbf{i}-9\mathbf{j}

respectively, and O is the origin. Calculate the angle

\mathrm{O\hat{B}A}

, giving your answer to the nearest degree.

Answer this when you sit the paper.

[3 marks]Vectors: angle between a line and a plane

The line

l

has equation

\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\2\end{pmatrix}

and the plane P has equation

\mathbf{r}\cdot\begin{pmatrix}2\\-4\\-1\end{pmatrix}=8

. Find the acute angle between

l

and P, to the nearest degree.

Answer this when you sit the paper.

[2 marks]Vectors: distance from a point to a plane

The plane P has equation

\mathbf{r}\cdot\begin{pmatrix}2\\-4\\-1\end{pmatrix}=8

and the point B has position vector

7\mathbf{i}-9\mathbf{j}

. Find the perpendicular distance from B to P, correct to 3 significant figures.

Answer this when you sit the paper.

Section A, Question 7

[3 marks]Complex numbers: equating parts

The complex numbers

W_{1}=1+ix

and

W_{2}=x+iy

, where

x

and

y

are real, satisfy

W_{1}-W_{2}=3i

. Find the value of

x

and the value of

y

Answer this when you sit the paper.

[2 marks]De Moivre's theorem

Use De Moivre's theorem to find the value of

\left(\cos\dfrac14\pi+i\sin\dfrac14\pi\right)^{12}

Answer this when you sit the paper.

[2 marks]Complex numbers: loci

On an Argand diagram, which region satisfies both

\dfrac{\pi}{4}\le\arg z\le\dfrac{\pi}{2}

and

|z-3i|\le3

AThe whole closed disc of radius 3 centred at the point $(0;\,3)$ , including its boundary circle.
BThe whole sector between the half line $y=x$ , $x\ge0$ , and the positive imaginary axis.
CThe part of the disc $|z-3i|\le3$ that lies in the first quadrant, boundary included.
DThe part of the disc $|z-3i|\le3$ inside the sector $\tfrac{\pi}{4}\le\arg z\le\tfrac{\pi}{2}$ .

[2 marks]De Moivre's theorem

Expanding

(\cos\theta+i\sin\theta)^{4}

by the binomial theorem and writing

c=\cos\theta

s=\sin\theta

, which expression is

\sin4\theta

A $c^{4}-6c^{2}s^{2}+s^{4}$
B $4c^{3}s+4cs^{3}$
C $4c^{2}s^{2}-6cs$
D $4c^{3}s-4cs^{3}$

[2 marks]De Moivre's theorem

De Moivre's theorem gives

\cos4\theta=\cos^{4}\theta-6\cos^{2}\theta\sin^{2}\theta+\sin^{4}\theta

. Dividing this by

\cos^{4}\theta

gives the denominator of the standard expression for

\tan4\theta

in terms of

\tan\theta

. Write down that denominator.

Answer this when you sit the paper.

Section B

Section B, Question 8

[2 marks]Projectile motion

A particle is projected horizontally from a point O which is at a height of

45

m vertically above a point P on level ground, and it hits the ground at Q (see diagram). Taking

g=9.81\ \mathrm{ms^{-2}}

, calculate the time taken by the particle to reach Q, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected horizontally from a point O which is

45

m vertically above a point P on level ground, and it lands at Q with

\mathrm{PQ}=15

m (see diagram). Taking

g=9.81\ \mathrm{ms^{-2}}

, find the horizontal speed of projection, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected horizontally from a point O which is

45

m vertically above a point P on level ground, and it lands at Q with

\mathrm{PQ}=15

m (see diagram). Taking

g=9.81\ \mathrm{ms^{-2}}

, find its speed at Q, correct to 3 significant figures.

Answer this when you sit the paper.

Section B, Question 9

[1 marks]Newton's second law

A particle B of mass

2.5

kg is accelerated from rest along a smooth horizontal surface at

4\ \mathrm{ms^{-2}}

. Find the magnitude of the force acting on B.

Answer this when you sit the paper.

[2 marks]Motion under gravity

A particle A is dropped from rest from a point

\left(\dfrac{9g}{2}\right)

metres vertically above a point R on a horizontal surface, where

g

is the acceleration due to gravity. Find the time it takes to reach R.

Answer this when you sit the paper.

[2 marks]Uniform acceleration

A particle B starts from rest and accelerates uniformly at

4\ \mathrm{ms^{-2}}

along a smooth horizontal surface for 3 seconds. Find the distance it covers.

Answer this when you sit the paper.

[1 marks]Motion under gravity

A particle is dropped from a point

\left(\dfrac{9g}{2}\right)

metres above a horizontal surface. Taking

g=9.81\ \mathrm{ms^{-2}}

, find that height in metres, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Displacement-time graphs

Two particles each start from rest under constant acceleration: A falls freely with acceleration

9.81\ \mathrm{ms^{-2}}

and B accelerates at

4\ \mathrm{ms^{-2}}

. Their displacements are plotted against time on the same axes. What do the two graphs look like?

ABoth are parabolas through the origin, with A's the steeper.
BBoth are straight lines through the origin, with A's line the steeper of the two.
CA is a parabola and B is a straight line, and both of them pass through the origin.
DBoth are parabolas through the origin, with B's the steeper of the two throughout.

Section B, Question 10

[2 marks]Friction: limiting equilibrium

A particle rests in limiting equilibrium on a rough plane inclined at

35^{\circ}

to the horizontal. Calculate the coefficient of friction between the particle and the plane, correct to three decimal places.

Answer this when you sit the paper.

[3 marks]Motion on a rough inclined plane

A particle of mass

0.5

kg is released from rest on a rough plane inclined at

65^{\circ}

to the horizontal, where the coefficient of friction is

0.700

. Taking

g=9.81\ \mathrm{ms^{-2}}

, find its acceleration down the line of greatest slope, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Motion on a rough inclined plane

A particle is released from rest

0.8

m up the line of greatest slope of a rough plane inclined at

65^{\circ}

to the horizontal, and slides down with acceleration

5.99\ \mathrm{ms^{-2}}

. Find the time it takes to reach the horizontal surface, correct to 2 significant figures.

Answer this when you sit the paper.

[2 marks]Motion on a rough inclined plane

A particle is released from rest

0.8

m up the line of greatest slope of a rough plane and slides down with acceleration

5.99\ \mathrm{ms^{-2}}

. Find its velocity when it reaches the horizontal surface, correct to 2 significant figures.

Answer this when you sit the paper.

Section C

Section C, Question 11

[2 marks]Geometric distribution

The discrete random variable

X

has a geometric distribution,

X\sim\mathrm{Geo}(0.4)

. Find

\mathrm{P}(X\le7)

, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Geometric distribution

The discrete random variable

X

has a geometric distribution,

X\sim\mathrm{Geo}(0.4)

. Find

\mathrm{P}(X>8\mid X>3)

Answer this when you sit the paper.

Section C, Question 12

[2 marks]Continuous random variables

The continuous random variable

X

has probability density function

f(x)=2e^{-kx}

for

x\ge0

and

f(x)=0

for

x<0

, where

k

is a positive integer. Find the value of

k

Answer this when you sit the paper.

[2 marks]Cumulative distribution functions

The continuous random variable

X

has probability density function

f(x)=2e^{-2x}

for

x\ge0

and

f(x)=0

for

x<0

. Find the cumulative distribution function

F(x)

for

x\ge0

Answer this when you sit the paper.

[2 marks]Median of a continuous distribution

The continuous random variable

X

has cumulative distribution function

F(x)=1-e^{-2x}

for

x\ge0

. Find the exact value of the median of

X

Answer this when you sit the paper.

Section C, Question 13

[2 marks]Median

The marks obtained by 36 students in a Mathematics test were
59 53 74 55 90 57
88 68 59 67 82 62
61 77 74 86 60 83
92 58 60 72 57 96
56 67 73 78 66 79
51 60 54 67 80 63
Find the median mark.

Answer this when you sit the paper.

[3 marks]Interquartile range

The marks obtained by 36 students in a Mathematics test were
59 53 74 55 90 57
88 68 59 67 82 62
61 77 74 86 60 83
92 58 60 72 57 96
56 67 73 78 66 79
51 60 54 67 80 63
Find the interquartile range of the marks.

Answer this when you sit the paper.

[1 marks]Frequency

The marks obtained by 36 students in a Mathematics test were
59 53 74 55 90 57
88 68 59 67 82 62
61 77 74 86 60 83
92 58 60 72 57 96
56 67 73 78 66 79
51 60 54 67 80 63
How many students scored a mark in the class 65 to 69 inclusive?

Answer this when you sit the paper.

Section C, Question 14

[3 marks]Normal distribution

The random variable

X

is normally distributed with mean

\mu

and variance

\sigma^{2}

. Given that

\mathrm{P}(X>65)=0.01

and

\mathrm{P}(X<20)=0.02

, find

\sigma

, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Normal distribution

The random variable

X

is normally distributed with mean

\mu

and standard deviation

\sigma=10.27

. Given that

\mathrm{P}(X>65)=0.01

, find

\mu

, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Normal distribution

For a normally distributed

X

it is given that

\mathrm{P}(X>65)=0.01

and

\mathrm{P}(X<20)=0.02

. Which pair of standardised values does this give?

A $\dfrac{65-\mu}{\sigma}=0.01$ and $\dfrac{20-\mu}{\sigma}=0.02$
B $\dfrac{65-\mu}{\sigma}=2.054$ and $\dfrac{20-\mu}{\sigma}=-2.326$
C $\dfrac{65-\mu}{\sigma}=2.326$ and $\dfrac{20-\mu}{\sigma}=-2.054$
D $\dfrac{65-\mu}{\sigma}=-2.326$ and $\dfrac{20-\mu}{\sigma}=2.054$

Pure Mathematics Paper 2 June 2017

Sit this paper online

The questions

Section A

Section A, Question 1

Section A, Question 2

Section A, Question 3

Section A, Question 4

Section A, Question 5

Section A, Question 6

Section A, Question 7

Section B

Section B, Question 8

Section B, Question 9

Section B, Question 10

Section C

Section C, Question 11

Section C, Question 12

Section C, Question 13

Section C, Question 14

More sittings of this paper

The answers, and why they are the answers

Sit this paper online