ZIMSEC A Level Pure Mathematics Paper 2 June 2018

Section a

Section a, Question 1

[2 marks]Circular measure

ABCD is a square of side

x

cm, drawn with B and C as the upper vertices and A and D as the lower vertices, and O is the midpoint of AD. Find angle AOB in radians, correct to 3 decimal places.

Answer this when you sit the paper.

[2 marks]Circular measure

ABCD is a square of side

x

cm with B and C the upper vertices, A and D the lower vertices, and O the midpoint of AD. OBC is a sector of a circle centre O and radius OB. Given that angle AOB is

1{,}1071487

radians, find angle BOC in radians, correct to 3 decimal places.

Answer this when you sit the paper.

[2 marks]Circular measure

ABCD is a square of side 5 cm with B and C the upper vertices, A and D the lower vertices, and O the midpoint of AD. OBC is a sector of a circle centre O and radius OB, and angle BOC is

0{,}9272952

radians. Find the area, in cm

^{2}

, of the region between the chord BC and the arc BC, correct to 3 decimal places.

Answer this when you sit the paper.

Section a, Question 2

[2 marks]Logarithms

The equation

\log_{2}(8y)=x+5

is to be written without logarithms. Which of these is it equivalent to?

A $y=2^{x+2}$
B $y=2^{x+5}$
C $y=2^{x-2}$
D $y=8\times 2^{x+5}$

[3 marks]Logarithms and exponentials

Solve the simultaneous equations

\log_{2}(8y)=x+5

and

4^{x}=3y

, and state the value of

y

Answer this when you sit the paper.

[2 marks]Logarithms and exponentials

Solve the simultaneous equations

\log_{2}(8y)=x+5

and

4^{x}=3y

, and state the value of

x

correct to 3 decimal places.

Answer this when you sit the paper.

Section a, Question 3

[3 marks]Implicit differentiation

Find

\dfrac{dy}{dx}

in terms of

x

and

y

for the curve

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

Answer this when you sit the paper.

[2 marks]Turning points

For the curve

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

the gradient is

\dfrac{dy}{dx}=\dfrac{x^{2}-y^{2}}{y^{2}+2xy}

, so at a turning point

y=x

y=-x

. Taking

y=x

, find the value of

x

at that turning point.

Answer this when you sit the paper.

[2 marks]Turning points

For the curve

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

the gradient is

\dfrac{dy}{dx}=\dfrac{x^{2}-y^{2}}{y^{2}+2xy}

, so at a turning point

y=x

y=-x

. Taking

y=-x

, find the exact value of

x

at that turning point.

Answer this when you sit the paper.

Section a, Question 4

[2 marks]Matrices

Find the determinant of the matrix

A=\begin{pmatrix}3&1&4\\2&3&1\\2&1&1\end{pmatrix}

Answer this when you sit the paper.

[3 marks]Matrices

For

A=\begin{pmatrix}3&1&4\\2&3&1\\2&1&1\end{pmatrix}

, which matrix is

A^{-1}

A $\begin{pmatrix}0{,}2&0{,}3&-1{,}1\\0&-0{,}5&0{,}5\\-0{,}4&-0{,}1&0{,}7\end{pmatrix}$
B $\begin{pmatrix}-0{,}2&-0{,}3&1{,}1\\0&0{,}5&-0{,}5\\0{,}4&0{,}1&-0{,}7\end{pmatrix}$
C $\begin{pmatrix}-0{,}2&0&0{,}4\\-0{,}3&0{,}5&0{,}1\\1{,}1&-0{,}5&-0{,}7\end{pmatrix}$
D $\begin{pmatrix}3&1&4\\2&3&1\\2&1&1\end{pmatrix}$

[2 marks]Simultaneous equations

Solve the simultaneous equations

3x+y+4z=15

2x+3y+z=12

and

2x+y+z=10

, and state the value of

x

Answer this when you sit the paper.

[2 marks]Simultaneous equations

Solve the simultaneous equations

3x+y+4z=15

2x+3y+z=12

and

2x+y+z=10

, and state the value of

z

Answer this when you sit the paper.

Section a, Question 5

[2 marks]Algebraic expansion

Expand and simplify

(2r+1)^{3}-(2r-1)^{3}

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[2 marks]Proof by induction

A proof by induction that

U_{n}=2^{6n}+3^{2n-2}

is divisible by 5 begins with the base case

n=1

. Find the value of

U_{1}

Answer this when you sit the paper.

[2 marks]Proof by induction

In a proof by induction for

U_{n}=2^{6n}+3^{2n-2}

, the term

U_{k+1}=2^{6(k+1)}+3^{2(k+1)-2}

is rewritten using

2^{6k}

and

3^{2k-2}

. Which expression is it equal to?

A $64\cdot 2^{6k}+3\cdot 3^{2k-2}$
B $2^{6k}+3^{2k-2}$
C $6\cdot 2^{6k}+2\cdot 3^{2k-2}$
D $64\cdot 2^{6k}+9\cdot 3^{2k-2}$

[3 marks]Proof by induction

Assume

2^{6k}+3^{2k-2}=5m

for an integer

m

, so

2^{6k}=5m-3^{2k-2}

. Substituting into

U_{k+1}=64\cdot 2^{6k}+9\cdot 3^{2k-2}

, which form shows that

U_{k+1}

is divisible by 5?

A $5\left(64m+11\cdot 3^{2k-2}\right)$
B $5\left(64m-3^{2k-2}\right)$
C $5\left(64m-11\cdot 3^{2k-2}\right)$
D $5\left(9m-64\cdot 3^{2k-2}\right)$

Section a, Question 6

[2 marks]Complex numbers

Find

(4+i)^{2}

, giving the answer in the form

a+bi

Answer this when you sit the paper.

[3 marks]Complex numbers

Find the real part of

\left(2+\frac{1}{2}i\right)^{8}

, giving the answer in exact form.

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[2 marks]Polynomials

Both

\left(x+2\sqrt{2}\right)

and

\left(x-2\sqrt{2}\right)

are factors of a polynomial. Write down, in expanded form, the quadratic factor formed by their product.

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[3 marks]Polynomials

It is given that

\left(x+2\sqrt{2}\right)

and

\left(x-2\sqrt{2}\right)

are factors of

f(x)=x^{4}-6x^{3}+ax^{2}+bx-104

. Find the value of

a

Answer this when you sit the paper.

[2 marks]Polynomials

It is given that

\left(x+2\sqrt{2}\right)

and

\left(x-2\sqrt{2}\right)

are factors of

f(x)=x^{4}-6x^{3}+ax^{2}+bx-104

. Find the value of

b

Answer this when you sit the paper.

[3 marks]Polynomials

The polynomial

f(x)=x^{4}-6x^{3}+5x^{2}+48x-104

factorises as

\left(x^{2}-8\right)\left(x^{2}-6x+13\right)

. Find the two non real roots of

f(x)=0

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Section a, Question 7

[3 marks]Vectors and planes

A plane has vector equation

\mathbf{r}\cdot\begin{pmatrix}2\\-3\\1\end{pmatrix}=1

and the point P has coordinates

(-9;17;-2)

. Find the coordinates of the foot of the perpendicular from P to the plane.

Answer this when you sit the paper.

[3 marks]Vectors and planes

The foot of the perpendicular from the point

P(-9;17;-2)

to a plane is

F\left(\frac{9}{7};\frac{11}{7};\frac{22}{7}\right)

. Find the coordinates of the image of P when reflected in that plane.

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[3 marks]Vectors, lines and planes

The plane

\pi

has equation

\mathbf{r}\cdot\begin{pmatrix}2\\-1\\-1\end{pmatrix}=6

and the line

l

has equation

\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z+1}{-2}

. Find the coordinates of the point where

l

meets

\pi

Answer this when you sit the paper.

[3 marks]Vectors, lines and planes

The plane

\pi

has equation

\mathbf{r}\cdot\begin{pmatrix}2\\-1\\-1\end{pmatrix}=6

and the line

l

has equation

\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z+1}{-2}

. Find the acute angle between

\pi

and

l

, correct to the nearest degree.

Answer this when you sit the paper.

[3 marks]Vectors and lines

Find the shortest distance from the origin to the line

\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z+1}{-2}

, giving the exact value.

Answer this when you sit the paper.

[2 marks]Vectors, lines and planes

When the angle between a line and a plane is found from the dot product of the line's direction vector and the plane's normal vector, why does that dot product give the sine of the required angle rather than its cosine?

ABecause a plane's normal vector is always a unit vector, and that converts every cosine into a sine.
BBecause a line and a plane can never be parallel, so the cosine of the angle would be undefined.
CBecause it gives the angle to the normal, and the angle with the plane is its complement.
DBecause a dot product of two direction vectors always returns a sine value and never a cosine value.

Section b

Section b, Question 8

[3 marks]Statics

A particle P is in equilibrium under four forces. One force of magnitude

T

N acts along a ray at

45^{\circ}

above the horizontal to the left of P. A second force of magnitude

T

N acts on the lower left, making

60^{\circ}

with the downward vertical through P. A force of magnitude

Q

N acts horizontally to the right, and a force of 6 N acts vertically downwards. Find the exact value of

T

Answer this when you sit the paper.

[2 marks]Statics

A particle P is in equilibrium under four forces:

T

N at

45^{\circ}

above the horizontal to the left of P,

T

N on the lower left making

60^{\circ}

with the downward vertical,

Q

N horizontally to the right, and 6 N vertically downwards. Given that

T=12\left(\sqrt{2}+1\right)

N, find

Q

correct to 3 significant figures.

Answer this when you sit the paper.

Section b, Question 9

[2 marks]Kinematics

The velocity-time graph of a particle moving in a straight line is a single straight line from the point

(0;2)

to the point

(8;-6)

, where

v

is in ms

^{-1}

and

t

is in seconds. Find the acceleration of the particle, in ms

^{-2}

Answer this when you sit the paper.

[2 marks]Kinematics

A particle moves in a straight line with velocity falling steadily from 2 ms

^{-1}

t=0

-6

^{-1}

t=8

s. Find the average velocity of the particle over the 8 seconds, in ms

^{-1}

Answer this when you sit the paper.

[1 marks]Kinematics

A particle moves in a straight line with velocity falling steadily from 2 ms

^{-1}

t=0

-6

^{-1}

t=8

s, starting from the origin. Which describes its displacement-time graph?

AAn upward parabola through the origin, dipping to $-2$ m at $t=2$ s and then rising to reach 16 m at $t=8$ s.
BA downward parabola through the origin, peaking at 2 m at $t=2$ s and reaching $-16$ m at $t=8$ s.
CA straight line falling steadily from the origin to reach $-16$ m at $t=8$ s, with one gradient throughout.
DA horizontal line resting at zero displacement for the whole of the 8 seconds of the motion.

Section b, Question 10

[2 marks]Friction

A particle of mass 8 kg rests on a rough plane of inclination

\tan^{-1}\left(\frac{3}{4}\right)

to the horizontal, and the coefficient of friction between the particle and the plane is

\frac{1}{5}

. Taking

g=9{,}81

^{-2}

, find the limiting friction force on the particle, in N, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Connected particles

A particle P of mass 8 kg rests on a rough plane of inclination

\tan^{-1}\left(\frac{3}{4}\right)

, with coefficient of friction

\frac{1}{5}

. It is joined by a light inextensible string over a smooth pulley at the top of the plane to a particle Q of mass 2 kg hanging freely, and P slides down the plane. Taking

g=9{,}81

^{-2}

, find the acceleration of the particles, in ms

^{-2}

, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Connected particles

A particle P of mass 8 kg on a rough plane is joined by a light inextensible string over a smooth pulley to a particle Q of mass 2 kg hanging freely. P slides down the plane and the system accelerates at

1{,}49112

^{-2}

. Taking

g=9{,}81

^{-2}

, find the tension in the string, in N, correct to 3 significant figures.

Answer this when you sit the paper.

Section b, Question 11

[2 marks]Projectiles

A particle is projected with an initial speed of 60 ms

^{-1}

at an angle of elevation

\sin^{-1}\left(\frac{3}{5}\right)

. Find the time, in seconds, it takes to reach the point whose horizontal displacement from the point of projection is 144 m.

Answer this when you sit the paper.

[3 marks]Projectiles

A particle is projected from the top of a vertical dam wall, 10 m above water level, with an initial speed of 60 ms

^{-1}

at an angle of elevation

\sin^{-1}\left(\frac{3}{5}\right)

. Taking

g=9{,}81

^{-2}

, find the time, in seconds, the particle takes to reach the water level, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Projectiles

A particle is projected from the top of a vertical dam wall, 10 m above water level, with an initial speed of 60 ms

^{-1}

at an angle of elevation

\sin^{-1}\left(\frac{3}{5}\right)

. Taking

g=9{,}81

^{-2}

, find the maximum height, in metres, attained by the particle above the water level, correct to 3 significant figures.

Answer this when you sit the paper.

Section c

Section c, Question 12

[2 marks]Discrete random variables

A dice is weighted so that the probability of each face coming up is proportional to the face value

x

, where

x=1,2,3,4,5,6

. Write down

P(X=x)

in terms of

x

Answer this when you sit the paper.

[2 marks]Discrete random variables

A random variable

X

takes the values

1,2,3,4,5,6

with

P(X=x)=\dfrac{x}{21}

. Calculate

E(X)

, giving the answer as an exact fraction.

Answer this when you sit the paper.

Section c, Question 13

[2 marks]Binomial probability

A and B played 12 games of chess of which 6 were won by A, 4 were won by B and 2 ended in a draw. Using these results as the probabilities for a single game, they play a tournament of 3 games. Find the probability that exactly two of the games end in a draw, giving the answer as an exact fraction.

Answer this when you sit the paper.

[3 marks]Conditional probability

A and B played 12 games of chess of which 6 were won by A, 4 were won by B and 2 ended in a draw. Using these results as the probabilities for a single game, they play a tournament of 3 games. Find the probability that B wins the tournament given that two of the games ended in a draw.

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Section c, Question 14

[2 marks]Binomial distribution

The probability that an adult chosen at random from a particular group is short sighted is

\frac{1}{5}

. A committee of 7 of the adults in that group is chosen at random. Find the probability that at least one adult on the committee is short sighted, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Normal approximation to the binomial

The probability that an adult from a particular group is short sighted is

\frac{1}{5}

. A bus carried 60 adult passengers from that group. Using a normal approximation with a continuity correction, find the probability that at least 15 of them were short sighted, correct to 3 significant figures.

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Section c, Question 15

[2 marks]Continuous random variables

A continuous random variable

X

has probability density function

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

for

x>0

and

f(x)=0

otherwise, where

c

is a constant. Find the value of

c

Answer this when you sit the paper.

[2 marks]Continuous random variables

A continuous random variable

X

has probability density function

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

for

x>0

and

f(x)=0

otherwise. Calculate the median of

X

, correct to 3 significant figures.

Answer this when you sit the paper.

[1 marks]Continuous random variables

For a function

f

to be a probability density function on

x>0

, which condition fixes the value of the constant multiplying it?

AThe integral of $f(x)$ over its whole range equals 1.
BThe largest value that $f$ reaches anywhere on its range must come out exactly equal to 1.
CThe derivative of $f$ must come out as exactly zero at the median of the distribution.
DThe value that $f$ takes at the left hand end of its range must come out exactly equal to 1.

Section c, Question 16

[2 marks]Measures of location

The sales, in US$, made by a vendor on 30 days were:

11, 37, 20, 15, 16, 23, 20, 26, 10, 21, 28, 13, 36, 18, 15, 32, 31, 16, 28, 26, 45, 46, 23, 58, 15, 54, 43, 37, 10, 32.

Find the median.

Answer this when you sit the paper.

[2 marks]Measures of spread

The sales, in US$, made by a vendor on 30 days were:

11, 37, 20, 15, 16, 23, 20, 26, 10, 21, 28, 13, 36, 18, 15, 32, 31, 16, 28, 26, 45, 46, 23, 58, 15, 54, 43, 37, 10, 32.

Find the interquartile range.

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[1 marks]Skewness

A set of 30 sales figures has minimum 10, lower quartile 16, median 24,5, upper quartile 36 and maximum 58. What does its box and whisker diagram show about the shape of the distribution?

AIt is uniform, because the whiskers and the two halves of the box all come out equal in length to each other.
BIt is negatively skewed: the median sits nearer the upper quartile and the left whisker is much the longer of the two.
CIt is symmetrical, because the median lies exactly halfway between the two quartiles and the whiskers match.
DIt is positively skewed: the median sits nearer the lower quartile and the right whisker is longer.

Pure Mathematics Paper 2 June 2018

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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

Section c, Question 16

More sittings of this paper

The answers, and why they are the answers

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