ZIMSEC A Level Pure Mathematics Paper 2 November 2017

Section a

Section a, Question 1

[1 marks]Mathematical induction

For the statement

\displaystyle\sum_{r=1}^{n}(r+1)2^{r}=n2^{n+1}

, evaluate the left hand side when

n=1

Answer this when you sit the paper.

[2 marks]Mathematical induction

In a proof by induction of

\displaystyle\sum_{r=1}^{n}(r+1)2^{r}=n2^{n+1}

, the inductive step adds the

(k+1)

th term to the sum of the first

k

terms. Which expression is that

(k+1)

th term?

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

[3 marks]Mathematical induction

Assume that

\displaystyle\sum_{r=1}^{k}(r+1)2^{r}=k2^{k+1}

. Write

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

as a single fully factorised expression.

Answer this when you sit the paper.

Section a, Question 2

[2 marks]Coordinate geometry, circles

The circle defined by the equation

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

passes through the origin. Find the coordinates of its centre.

Answer this when you sit the paper.

[2 marks]Coordinate geometry, circles

The circle defined by the equation

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

passes through the origin. Find its radius, giving an exact answer.

Answer this when you sit the paper.

[2 marks]Coordinate geometry, circles

The circle

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

passes through the origin and crosses the

y

-axis again at a point

\mathbf{P}

. Find the coordinates of

\mathbf{P}

Answer this when you sit the paper.

[3 marks]Coordinate geometry, circles

The circle

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

has centre

(3;2)

and crosses the

y

-axis at

\mathbf{P}(0;4)

. Find the equation of the tangent to the circle at

\mathbf{P}

Answer this when you sit the paper.

Section a, Question 3

[1 marks]Complex numbers, De Moivre's theorem

Find the modulus of the complex number

2+2\sqrt{3}i

Answer this when you sit the paper.

[1 marks]Complex numbers, De Moivre's theorem

Find the argument of the complex number

2+2\sqrt{3}i

, in radians.

Answer this when you sit the paper.

[2 marks]Complex numbers, De Moivre's theorem

Using De Moivre's Theorem, find the value of

\left(2+2\sqrt{3}i\right)^{6}

Answer this when you sit the paper.

[3 marks]Complex numbers, De Moivre's theorem

Express

\dfrac{\sin6\theta}{4\sin\theta}

in terms of

\cos\theta

Answer this when you sit the paper.

[2 marks]Complex numbers, De Moivre's theorem

Why can

\dfrac{\sin6\theta}{4\sin\theta}

be written entirely in terms of

\cos\theta

, with no

\sin\theta

left in it?

ABecause the imaginary part of any power of a complex number is always a polynomial in its real part alone, whatever the angle involved.
BIts expansion carries only odd powers of $\sin\theta$ , so dividing by $\sin\theta$ leaves even powers, each replaceable by $1-\cos^{2}\theta$ .
CBecause $\sin6\theta$ and $4\sin\theta$ cancel exactly, leaving a constant multiple of $\cos\theta$ behind.
DBecause $\sin\theta$ and $\cos\theta$ are equal for every angle to which De Moivre's Theorem may be applied.

Section a, Question 4

[2 marks]Matrices and simultaneous equations

Find the determinant of the matrix

\mathbf{M}=\begin{pmatrix}2&3&4\\-3&2&2\\4&-4&3\end{pmatrix}

Answer this when you sit the paper.

[2 marks]Matrices and simultaneous equations

The matrix

\mathbf{M}=\begin{pmatrix}2&3&4\\-3&2&2\\4&-4&3\end{pmatrix}

has determinant

95

and adjoint

\begin{pmatrix}14&-25&-2\\17&-10&-16\\4&20&13\end{pmatrix}

. Which matrix is

\mathbf{M}^{-1}

A $\dfrac{1}{95}\begin{pmatrix}14&-25&-2\\17&-10&-16\\4&20&13\end{pmatrix}$
B $95\begin{pmatrix}14&-25&-2\\17&-10&-16\\4&20&13\end{pmatrix}$
C $\dfrac{1}{95}\begin{pmatrix}14&17&4\\-25&-10&20\\-2&-16&13\end{pmatrix}$
D $\dfrac{1}{95}\begin{pmatrix}2&3&4\\-3&2&2\\4&-4&3\end{pmatrix}$

[3 marks]Matrices and simultaneous equations

Given that

\mathbf{M}^{-1}=\dfrac{1}{95}\begin{pmatrix}14&-25&-2\\17&-10&-16\\4&20&13\end{pmatrix}

where

\mathbf{M}=\begin{pmatrix}2&3&4\\-3&2&2\\4&-4&3\end{pmatrix}

, solve the simultaneous equations

2x+3y+4z=1

-3x+2y+2z=14

and

4x-4y+3z=22

Answer this when you sit the paper.

[2 marks]Matrices and simultaneous equations

The system

2x+3y+4z=1

-3x+2y+2z=14

4x-4y+3z=22

has coefficient matrix

\mathbf{M}

with

\det\mathbf{M}=95

. What does that determinant tell you about the solution?

AThe determinant is not zero, so $\mathbf{M}^{-1}$ exists and the system has exactly one solution.
BThe determinant is not zero, so the system has infinitely many solutions.
CThe determinant is not zero, so the three planes are parallel and there is no solution.
DThe determinant tells you nothing about the number of solutions; only the constants on the right can do that.

Section a, Question 5

[3 marks]Differentiation and integration

Given that

\ln y=\left(1+8e^{3x}\right)^{\frac{1}{2}}

, find

\dfrac{dy}{dx}

in terms of

x

and

y

Answer this when you sit the paper.

[2 marks]Differentiation and integration

Which is the correct form of the partial fractions for

\dfrac{3}{x^{2}(3-x)}

A $\dfrac{A}{x}+\dfrac{B}{x^{2}}+\dfrac{C}{3-x}$
B $\dfrac{A}{x^{2}}+\dfrac{B}{3-x}$
C $\dfrac{Ax+B}{x^{2}}+\dfrac{Cx+D}{3-x}$
D $\dfrac{A}{x}+\dfrac{B}{x^{2}}+\dfrac{C}{3-x}+\dfrac{D}{(3-x)^{2}}$

[3 marks]Differentiation and integration

Express

\dfrac{3}{x^{2}(3-x)}

in the form

\dfrac{A}{x}+\dfrac{B}{x^{2}}+\dfrac{C}{3-x}

, stating the values of

A

B

and

C

Answer this when you sit the paper.

[3 marks]Differentiation and integration

Find the exact value of

\displaystyle\int_{1}^{2}\frac{3}{x^{2}(3-x)}\,dx

Answer this when you sit the paper.

Section a, Question 6

[1 marks]Trigonometric graphs and equations

State the

y

-intercept of the graph of

f(x)=\cos(x-60^{\circ})

Answer this when you sit the paper.

[2 marks]Trigonometric graphs and equations

State the values of

x

at which the graph of

f(x)=\cos(x-60^{\circ})

crosses the

x

-axis for

0^{\circ}\le x\le360^{\circ}

Answer this when you sit the paper.

[2 marks]Trigonometric graphs and equations

What is the geometrical relationship between the graph of

f(x)=\cos(x-60^{\circ})

and the graph of

f(x)=\cos x

AA reflection in the line $x=60^{\circ}$ .
BA translation of $60^{\circ}$ in the positive $x$ -direction.
CA translation of $60^{\circ}$ in the negative $x$ -direction.
DA stretch parallel to the $x$ -axis with scale factor $60$ .

[2 marks]Trigonometric graphs and equations

Show that the equation

2\sin^{2}(x-60^{\circ})=1+\cos(x-60^{\circ})

may be written as a quadratic equation in

\cos(x-60^{\circ})

, and write down that quadratic equation with all terms on one side.

Answer this when you sit the paper.

[3 marks]Trigonometric graphs and equations

Solve the equation

2\sin^{2}(x-60^{\circ})=1+\cos(x-60^{\circ})

for

0^{\circ}\le x\le360^{\circ}

Answer this when you sit the paper.

Section a, Question 7

[2 marks]Vectors, lines in three dimensions

The point

\mathbf{P}

has position vector

\begin{pmatrix}5\\3\\1\end{pmatrix}

and the line

l

has equation

\mathbf{r}=\begin{pmatrix}1\\0\\-2\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\5\end{pmatrix}

. Does

\mathbf{P}

lie on

l

ANo: the three components give $\lambda=2$ , $-3$ and $\tfrac{3}{5}$ , which disagree.
BYes, because the first component gives $\lambda=2$ , and a parameter value satisfying one component satisfies them all.
CYes, because $\begin{pmatrix}5\\3\\1\end{pmatrix}$ is a scalar multiple of the direction vector $\begin{pmatrix}2\\-1\\5\end{pmatrix}$ .
DNo, because none of the components of the direction vector of the line $l$ is equal to zero.

[3 marks]Vectors, lines in three dimensions

Points

\mathbf{P}

and

\mathbf{Q}

have position vectors

\begin{pmatrix}5\\3\\1\end{pmatrix}

and

\begin{pmatrix}4\\c\\2\end{pmatrix}

. Given that line

\mathbf{PQ}

intersects the line

l

with equation

\mathbf{r}=\begin{pmatrix}1\\0\\-2\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\5\end{pmatrix}

, find the value of

c

Answer this when you sit the paper.

[3 marks]Vectors, lines in three dimensions

The line

l

has direction

\begin{pmatrix}2\\-1\\5\end{pmatrix}

and the line

\mathbf{PQ}

has direction

\begin{pmatrix}-1\\-2\\1\end{pmatrix}

. Calculate the acute angle between the two lines, giving your answer in degrees.

Answer this when you sit the paper.

[3 marks]Vectors, lines in three dimensions

The point

\mathbf{P}

has position vector

\begin{pmatrix}5\\3\\1\end{pmatrix}

and the line

l

has equation

\mathbf{r}=\begin{pmatrix}1\\0\\-2\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\5\end{pmatrix}

. Find the position vector of the point

\mathbf{R}

l

such that

\mathbf{PR}

is perpendicular to

l

Answer this when you sit the paper.

Section b

Section b, Question 8

[2 marks]Statics, equilibrium of forces

A light inextensible string is attached to two points at the same horizontal level. A smooth ring

R

of mass

2

kg slides freely along the string. A horizontal force of magnitude

P

newtons holds the ring in equilibrium, with the two parts of the string inclined at

60^{\circ}

and

30^{\circ}

to the vertical. Taking

g=9,81

^{-2}

, find the tension in the string.

Answer this when you sit the paper.

[2 marks]Statics, equilibrium of forces

A light inextensible string is attached to two points at the same horizontal level. A smooth ring

R

of mass

2

kg slides freely along the string, and the tension throughout the string is

14,36

N. A horizontal force of magnitude

P

newtons holds the ring in equilibrium, with the two parts of the string inclined at

60^{\circ}

and

30^{\circ}

to the vertical. Find the value of

P

Answer this when you sit the paper.

Section b, Question 9

[2 marks]Kinematics, velocity-time graphs

A particle moves in a straight line with a constant velocity of

3

^{-1}

for

3

seconds and then moves with a constant acceleration of

-2

^{-2}

for

8

seconds. Find its velocity at the end of the

11

seconds.

Answer this when you sit the paper.

[3 marks]Kinematics, velocity-time graphs

A particle moves in a straight line with a constant velocity of

3

^{-1}

for

3

seconds and then moves with a constant acceleration of

-2

^{-2}

for

8

seconds. Find the displacement of the particle after the

11

seconds.

Answer this when you sit the paper.

Section b, Question 10

[2 marks]Connected particles, Newton's laws

A particle of mass

12

kg rests on a smooth plane inclined at

45^{\circ}

to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane to a particle of mass

5

kg hanging freely. Taking

g=9,81

^{-2}

, find the acceleration of the system.

Answer this when you sit the paper.

[2 marks]Connected particles, Newton's laws

A particle of mass

12

kg on a smooth plane inclined at

45^{\circ}

is connected by a light inextensible string over a smooth pulley at the top of the plane to a freely hanging particle of mass

5

kg. The system accelerates at

2,011

^{-2}

. Taking

g=9,81

^{-2}

, find the tension in the string.

Answer this when you sit the paper.

[2 marks]Connected particles, Newton's laws

A light inextensible string passes over a smooth pulley fixed at the top of a plane inclined at

45^{\circ}

to the horizontal. One part of the string lies along the plane and the other hangs vertically, and the tension throughout is

59,105

N. Find the magnitude of the force exerted by the string on the pulley.

Answer this when you sit the paper.

Section b, Question 11

[3 marks]Projectile motion

A particle is projected from a point

O

on level ground with speed

20

^{-1}

. It passes through a point

P

that is

10

m from

O

horizontally and

10

m vertically above

O

. Taking

g=9,81

^{-2}

, find the two possible angles of projection, correct to the nearest degree.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected from a point

O

on level ground with speed

20

^{-1}

53,3^{\circ}

to the horizontal. Taking

g=9,81

^{-2}

, find the time taken by the particle to hit the ground.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected from a point

O

on level ground with speed

20

^{-1}

81,6^{\circ}

to the horizontal. State the direction of motion of the particle when it hits the ground.

Answer this when you sit the paper.

Section c

Section c, Question 12

[2 marks]Continuous random variables

A continuous random variable

X

has probability density function

f(x)=\dfrac{2(a-x)}{a^{2}}

for

0\le x\le a

and

f(x)=0

otherwise, where

a

is a constant. Find

\mathrm{E}(X)

in terms of

a

Answer this when you sit the paper.

[3 marks]Continuous random variables

A continuous random variable

X

has probability density function

f(x)=\dfrac{2(a-x)}{a^{2}}

for

0\le x\le a

and

f(x)=0

otherwise. Which equation must the median

m

satisfy?

A $2m^{2}-4am-a^{2}=0$
B $m^{2}-4am+2a^{2}=0$
C $2m^{2}-4am+a^{2}=0$
D $m^{2}-2am+a^{2}=0$

Section c, Question 13

[2 marks]Normal distribution

The random variable

X

is Normally distributed with mean

\mu

and standard deviation

\sigma

. Given that

\mathrm{P}(X>3,6)=0,5

, find the value of

\mu

Answer this when you sit the paper.

[3 marks]Normal distribution

The random variable

X

is Normally distributed with mean

3,6

and standard deviation

\sigma

. Given that

\mathrm{P}(X>2,8)=0,6554

, find the value of

\sigma

Answer this when you sit the paper.

Section c, Question 14

[2 marks]Conditional probability

\mathbf{A}

and

\mathbf{B}

play a series of games, each of which is won by one of them. The probability that

\mathbf{A}

wins the first game is

0,6

. If

\mathbf{A}

wins a game, the probability of winning the next is

0,7

; if

\mathbf{A}

loses a game, the probability of winning the next is

0,4

. Find the probability that

\mathbf{A}

loses the second game.

Answer this when you sit the paper.

[3 marks]Conditional probability

\mathbf{A}

and

\mathbf{B}

play a series of games, each of which is won by one of them. The probability that

\mathbf{A}

wins the first game is

0,6

. If

\mathbf{A}

wins a game, the probability of winning the next is

0,7

; if

\mathbf{A}

loses a game, the probability of winning the next is

0,4

. Find the probability that

\mathbf{A}

wins the first game, given that

\mathbf{A}

loses the second game.

Answer this when you sit the paper.

Section c, Question 15

[1 marks]Stem and leaf diagrams, measures of spread

State one advantage of using a stem and leaf diagram to represent data.

AIt removes outliers automatically before the data is summarised, so the spread is never distorted.
BIt shows the exact mean of the data without any calculation being needed.
CThe original data values are preserved and can still be read off.
DIt works well however many thousands of readings there are, since the leaves simply extend further to the right.

[1 marks]Stem and leaf diagrams, measures of spread

State one disadvantage of using a stem and leaf diagram to represent data.

AIt hides the individual readings, so the original values cannot be recovered.
BIt gives no indication of the shape of the distribution.
CIt cannot be drawn for data that includes negative values.
DEvery value is written out, so it becomes cluttered and hard to read once the data set is large.

[2 marks]Stem and leaf diagrams, measures of spread

The heights, in centimetres, of $30$ students, measured correct to the nearest centimetre, are

$\begin{array}{cccccc}167&174&156&180&162&169\\177&154&165&174&160&184\\169&179&151&163&173&148\\171&168&158&158&167&166\\149&153&171&162&182&162\end{array}$

Find the median height.

Answer this when you sit the paper.

[1 marks]Stem and leaf diagrams, measures of spread

The heights, in centimetres, of $30$ students, measured correct to the nearest centimetre, are

$\begin{array}{cccccc}167&174&156&180&162&169\\177&154&165&174&160&184\\169&179&151&163&173&148\\171&168&158&158&167&166\\149&153&171&162&182&162\end{array}$

Find the lower quartile.

Answer this when you sit the paper.

[1 marks]Stem and leaf diagrams, measures of spread

The heights, in centimetres, of $30$ students, measured correct to the nearest centimetre, are

$\begin{array}{cccccc}167&174&156&180&162&169\\177&154&165&174&160&184\\169&179&151&163&173&148\\171&168&158&158&167&166\\149&153&171&162&182&162\end{array}$

Find the upper quartile.

Answer this when you sit the paper.

[2 marks]Stem and leaf diagrams, measures of spread

The heights, in centimetres, of $30$ students, measured correct to the nearest centimetre, are

$\begin{array}{cccccc}167&174&156&180&162&169\\177&154&165&174&160&184\\169&179&151&163&173&148\\171&168&158&158&167&166\\149&153&171&162&182&162\end{array}$

Find the interquartile range.

Answer this when you sit the paper.

Pure Mathematics Paper 2 November 2017

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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 b, Question 11

Section c

Section c, Question 12

Section c, Question 13

Section c, Question 14

Section c, Question 15

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The answers, and why they are the answers

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