ZIMSEC A Level Pure Mathematics Paper 2 June 2016

Section a

Section a, Question 1

[2 marks]Integration by substitution

Evaluate

\displaystyle\int_{\pi}^{\frac{3\pi}{2}}\frac{\cos x}{3+\cos^{2}x}\,dx

, using the substitution

u=\sin x

. Give the exact answer as a single natural logarithm.

Answer this when you sit the paper.

[2 marks]Integration by substitution

The substitution

u=\sin x

is used on

\displaystyle\int_{\pi}^{\frac{3\pi}{2}}\frac{\cos x}{3+\cos^{2}x}\,dx

. Which integral in

u

does it become, limits included?

A $\displaystyle\int_{0}^{1}\frac{du}{4-u^{2}}$
B $\displaystyle\int_{-1}^{0}\frac{u\,du}{4-u^{2}}$
C $\displaystyle\int_{0}^{-1}\frac{du}{4-u^{2}}$
D $\displaystyle\int_{0}^{-1}\frac{du}{2+u^{2}}$

[2 marks]Partial fractions

Express

\dfrac{1}{4-u^{2}}

in partial fractions.

A $\dfrac{1}{4}\left(\dfrac{1}{2+u}+\dfrac{1}{2-u}\right)$
B $\dfrac{1}{2+u}-\dfrac{1}{2-u}$
C $\dfrac{1}{4}\left(\dfrac{1}{2-u}-\dfrac{1}{2+u}\right)$
D $\dfrac{1}{2}\left(\dfrac{1}{2+u}-\dfrac{1}{2-u}\right)$

[2 marks]Integration by substitution

In the integral

\displaystyle\int_{\pi}^{\frac{3\pi}{2}}\frac{\cos x}{3+\cos^{2}x}\,dx

the substitution

u=\sin x

is made. State the new lower and upper limits in

u

, in that order.

Answer this when you sit the paper.

Section a, Question 2

[2 marks]Differential equations

The gradient of a curve at a point

(x;y)

is proportional to the product of

\sqrt{x}

and

y^{2}

. Which differential equation says this, with

k

constant?

A $\dfrac{dy}{dx}=kx^{2}\sqrt{y}$
B $\dfrac{dy}{dx}=k\left(\sqrt{x}+y^{2}\right)$
C $\dfrac{dy}{dx}=\dfrac{k\sqrt{x}}{y^{2}}$
D $\dfrac{dy}{dx}=ky^{2}\sqrt{x}$

[3 marks]Differential equations

A curve satisfies

\dfrac{dy}{dx}=ky^{2}\sqrt{x}

and passes through the points

(0;-2)

and

(9;1)

. Find the exact value of the constant

k

Answer this when you sit the paper.

[3 marks]Differential equations

A curve has

\dfrac{dy}{dx}=ky^{2}\sqrt{x}

and passes through

(0;-2)

and

(9;1)

. What is its equation?

A $y=\dfrac{9}{x^{\frac{3}{2}}-18}$
B $y=\dfrac{18}{9-x^{\frac{3}{2}}}$
C $y=\dfrac{18}{x^{\frac{3}{2}}-9}$
D $y=\dfrac{x^{\frac{3}{2}}-9}{18}$

[2 marks]Asymptotes

Write down an equation of an asymptote of the curve

y=\dfrac{18}{x^{\frac{3}{2}}-9}

A $y=x^{\frac{3}{2}}$
B $y=0$
C $x=18$
D $y=18$

Section a, Question 3

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

The roots of

z^{7}-8-8i=0

are written as

r(\cos\theta+i\sin\theta)

. Find the exact value of

r

Answer this when you sit the paper.

[2 marks]Complex numbers, argument

Find the argument of the complex number

8+8i

, in radians.

Answer this when you sit the paper.

[2 marks]Complex numbers, roots of unity

The equation

z^{7}-8-8i=0

has seven roots. Give the smallest positive argument among them, in radians as a multiple of

\pi

Answer this when you sit the paper.

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

Which of these is a root of the equation

z^{7}-8-8i=0

A $8\sqrt{2}\left(\cos\dfrac{\pi}{28}+i\sin\dfrac{\pi}{28}\right)$
B $\sqrt{2}\left(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4}\right)$
C $\sqrt{2}\left(\cos\dfrac{\pi}{28}+i\sin\dfrac{\pi}{28}\right)$
D $2\left(\cos\dfrac{\pi}{7}+i\sin\dfrac{\pi}{7}\right)$

[2 marks]Argand diagrams, loci

On an Argand diagram, what is the locus of the points

z

for which

Arg(z+1)=\dfrac{\pi}{3}

AA half line starting at $(1;0)$ , that point excluded, at $60^{\circ}$ to the positive real axis.
BA circle of radius 1 centred at the origin.
CThe whole straight line through $(-1;0)$ at $60^{\circ}$ to the positive real axis.
DA half line starting at $(-1;0)$ , that point excluded, at $60^{\circ}$ to the positive real axis.

Section a, Question 4

[2 marks]Repeated differentiation

Given

y=x^{2}e^{x}

, find

\dfrac{d^{2}y}{dx^{2}}

A $2e^{x}+2xe^{x}+x^{2}e^{x}$
B $2xe^{x}+x^{2}e^{x}$
C $4e^{x}+4xe^{x}+x^{2}e^{x}$
D $2e^{x}+4xe^{x}+x^{2}e^{x}$

[3 marks]Repeated differentiation

Given

y=x^{2}e^{x}

, find

\dfrac{d^{3}y}{dx^{3}}

in its simplest form.

A $2e^{x}+6xe^{x}+x^{2}e^{x}$
B $6e^{x}+6xe^{x}+x^{2}e^{x}$
C $4e^{x}+6xe^{x}+x^{2}e^{x}$
D $6e^{x}+4xe^{x}+x^{2}e^{x}$

[3 marks]Mathematical induction

For

y=x^{2}e^{x}

the statement

\dfrac{d^{n}y}{dx^{n}}=(n-1)\left(2e^{x}\right)+2xe^{x}+\dfrac{d^{n-1}y}{dx^{n-1}}

is to be proved by induction for

n\geq 2

. What does the base case

n=2

require you to check?

AThat $\dfrac{dy}{dx}=2xe^{x}+x^{2}e^{x}$ .
BThat $\dfrac{d^{2}y}{dx^{2}}=4e^{x}+2xe^{x}+\dfrac{dy}{dx}$ .
CThat $\dfrac{d^{3}y}{dx^{3}}=4e^{x}+2xe^{x}+\dfrac{d^{2}y}{dx^{2}}$ .
DThat $\dfrac{d^{2}y}{dx^{2}}=2e^{x}+2xe^{x}+\dfrac{dy}{dx}$ .

[3 marks]Mathematical induction

In an induction proof it is assumed that

\dfrac{d^{k}y}{dx^{k}}=(k-1)\left(2e^{x}\right)+2xe^{x}+\dfrac{d^{k-1}y}{dx^{k-1}}

. Differentiating both sides with respect to

x

gives which result?

A $\dfrac{d^{k+1}y}{dx^{k+1}}=(k-1)\left(2e^{x}\right)+2xe^{x}+\dfrac{d^{k}y}{dx^{k}}$
B $\dfrac{d^{k+1}y}{dx^{k+1}}=k\left(2e^{x}\right)+2xe^{x}+\dfrac{d^{k}y}{dx^{k}}$
C $\dfrac{d^{k+1}y}{dx^{k+1}}=(k+1)\left(2e^{x}\right)+2xe^{x}+\dfrac{d^{k}y}{dx^{k}}$
D $\dfrac{d^{k+1}y}{dx^{k+1}}=(k-1)\left(2e^{x}\right)+2e^{x}+\dfrac{d^{k}y}{dx^{k}}$

Section a, Question 5

[3 marks]Vectors, lines in three dimensions

The perpendicular bisector of the line joining

A(0;2;2)

and

B(4;1;0)

has Cartesian equation

x-2=\dfrac{2y-3}{4}=z-1

, and it passes through the point

D(1;q;0)

. Find the value of

q

Answer this when you sit the paper.

[3 marks]Vectors, lines in three dimensions

A line has Cartesian equation

x-2=\dfrac{2y-3}{4}=z-1

. What is a direction vector of the line?

A $\begin{pmatrix}1\\2\\1\end{pmatrix}$
B $\begin{pmatrix}4\\-1\\-2\end{pmatrix}$
C $\begin{pmatrix}1\\4\\1\end{pmatrix}$
D $\begin{pmatrix}2\\3\\1\end{pmatrix}$

[3 marks]Vectors, shortest distance

Find the exact shortest distance from the origin to the line

x-2=\dfrac{2y-3}{4}=z-1

Answer this when you sit the paper.

[3 marks]Planes in three dimensions

Find the Cartesian equation of the plane containing the points

A(0;2;2)

B(4;1;0)

and

C(-2;0;3)

Answer this when you sit the paper.

Section a, Question 6

[3 marks]Determinants and singular matrices

The matrix

A=\begin{pmatrix}2&k&4\\3&12&8\\1&8&k\end{pmatrix}

is singular. Find the exact values of

k

Answer this when you sit the paper.

[2 marks]Determinants

Find the determinant of the matrix

\begin{pmatrix}2&5&4\\3&12&8\\1&8&5\end{pmatrix}

Answer this when you sit the paper.

[3 marks]Inverse of a 3 by 3 matrix

Find the inverse of

A=\begin{pmatrix}2&5&4\\3&12&8\\1&8&5\end{pmatrix}

A $\dfrac{1}{5}\begin{pmatrix}-4&7&-8\\-7&6&-4\\12&-11&9\end{pmatrix}$
B $\dfrac{1}{5}\begin{pmatrix}-4&-7&12\\7&6&-11\\-8&-4&9\end{pmatrix}$
C $\begin{pmatrix}-4&7&-8\\-7&6&-4\\12&-11&9\end{pmatrix}$
D $\dfrac{1}{5}\begin{pmatrix}4&-7&8\\7&-6&4\\-12&11&-9\end{pmatrix}$

[3 marks]Simultaneous equations by matrices

Solve the simultaneous equations

2x+5y+4z=1

3x+12y+8z=1

and

x+8y+5z=1

. Give

x

y

and

z

Answer this when you sit the paper.

[3 marks]Matrix transformations

A transformation

P

of the plane maps

A(1;2)

onto

A_{1}(3;2)

and

B(2;5)

onto

B_{1}(7;5)

. Find the matrix representing

P

A $\begin{pmatrix}3&1\\0&1\end{pmatrix}$
B $\begin{pmatrix}1&1\\1&0\end{pmatrix}$
C $\begin{pmatrix}1&1\\0&1\end{pmatrix}$
D $\begin{pmatrix}1&0\\1&1\end{pmatrix}$

[3 marks]Matrix transformations

Describe fully the transformation of the plane represented by the matrix

\begin{pmatrix}1&1\\0&1\end{pmatrix}

AA shear with the $y$ -axis invariant and shear factor 1.
BA shear with the $x$ -axis invariant and shear factor 1.
CAn enlargement of scale factor 1 centred at the origin.
DA reflection in the line $y=x$ .

[2 marks]Area under a transformation

A triangle ABC of area 4 cm

^{2}

is mapped onto triangle

A_{1}B_{1}C_{1}

by the transformation with matrix

\begin{pmatrix}1&1\\0&1\end{pmatrix}

. Find the area of triangle

A_{1}B_{1}C_{1}

Answer this when you sit the paper.

Section b

Section b, Question 7

[1 marks]Projectile motion

A projectile follows the path

y=2x-0,01x^{2}

, where

x

is the horizontal displacement and

y

the vertical displacement from the point of projection. Find the angle of projection, to the nearest degree.

Answer this when you sit the paper.

[2 marks]Projectile motion

A projectile follows the path

y=2x-0,01x^{2}

, where

x

and

y

are the horizontal and vertical displacements in metres from the point of projection. Taking

g=9,81

m s

^{-2}

, find the initial speed of projection in m s

^{-1}

Answer this when you sit the paper.

Section b, Question 8

[2 marks]Resultant of coplanar forces

Three coplanar forces act at a point Y, as shown. The 7 N force is horizontal, the 6 N force is inclined to it at

50^{\circ}

on the upper side, and the 5 N force acts backwards at

30^{\circ}

below the horizontal. Find the magnitude of the resultant, in newtons, correct to 2 significant figures.

Answer this when you sit the paper.

[3 marks]Resultant of coplanar forces

Three coplanar forces act at a point Y, as shown. The 7 N force is horizontal, the 6 N force is inclined to it at

50^{\circ}

on the upper side, and the 5 N force acts backwards at

30^{\circ}

below the horizontal. Find the angle, to the nearest degree, that the resultant makes with the 7 N force.

Answer this when you sit the paper.

Section b, Question 9

[2 marks]Kinematics

A particle starts from rest and accelerates at 2 ms

^{-2}

for 3 seconds. Find the velocity it has reached at the end of those 3 seconds, in m s

^{-1}

Answer this when you sit the paper.

[2 marks]Kinematics

A particle starts from rest and accelerates at 2 ms

^{-2}

for 3 seconds, then maintains the attained velocity for 4 seconds, then decelerates at 5 ms

^{-2}

for 2 seconds. Find the time, in seconds from the start, at which the particle is instantaneously at rest during the last stage.

Answer this when you sit the paper.

[2 marks]Kinematics

A particle starts from rest and accelerates at 2 ms

^{-2}

for 3 seconds, then maintains the attained velocity for 4 seconds, then decelerates at 5 ms

^{-2}

for 2 seconds. Find the total distance travelled in the 9 seconds, in metres.

Answer this when you sit the paper.

[2 marks]Kinematics

A particle starts from rest and accelerates at 2 ms

^{-2}

for 3 seconds, then maintains the attained velocity for 4 seconds, then decelerates at 5 ms

^{-2}

for 2 seconds. Find its displacement from the starting point at the end of the 9 seconds, in metres.

Answer this when you sit the paper.

Section b, Question 10

[3 marks]Connected particles

Two identical small trays, each of mass 0.2 kg, hang from a light inextensible string passing over a fixed smooth pulley, and they balance. A mass of 80 grammes is then placed on one tray, which begins to move downwards. Taking

g=9,81

m s

^{-2}

, find the acceleration of the trays in m s

^{-2}

Answer this when you sit the paper.

[3 marks]Connected particles

^{-2}

. Taking

g=9,81

m s

^{-2}

, find the tension in the string, in newtons.

Answer this when you sit the paper.

[2 marks]Connected particles

A mass of 80 grammes rests on a tray that is accelerating vertically downwards at 1,635 m s

^{-2}

. Taking

g=9,81

m s

^{-2}

, find the force in newtons that the tray exerts on the 80 gramme mass.

Answer this when you sit the paper.

Section c

Section c, Question 11

[2 marks]Probability density functions

A continuous random variable

X

has probability density function

f

with

f(x)=0

outside

1\leq x\leq 5

. On

1\leq x\leq 3

f(x)=k

; on

3\leq x\leq 4

the graph is a straight line rising from

k

2k

; on

4\leq x\leq 5

it is a straight line falling from

2k

to 0. Find the exact value of

k

Answer this when you sit the paper.

[2 marks]Probability density functions

A probability density function has

f(4)=\dfrac{4}{9}

and

f(5)=0

, and its graph is a straight line between

x=4

and

x=5

. What is

f(x)

on that interval?

A $\dfrac{20}{9}-\dfrac{4}{9}x$
B $\dfrac{16}{9}-\dfrac{4}{9}x$
C $\dfrac{4}{9}x-\dfrac{16}{9}$
D $\dfrac{4}{9}-\dfrac{4}{9}x$

Section c, Question 12

[2 marks]Normal distribution

For a standard Normal variable

Z

, find the value of

z

for which

P(Z>z)=0,03

. Give your answer to 3 decimal places.

Answer this when you sit the paper.

[2 marks]Normal distribution

The diameters of washers from a machine are Normally distributed with standard deviation 0.1 mm. There is to be a probability of only 3% that a diameter exceeds 2.0 mm. Find the mean diameter, in millimetres, correct to 3 significant figures.

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Section c, Question 13

[2 marks]Probability

Bag A contains 3 red balls and 2 white balls. Bag B contains 2 red balls and 3 white balls. A bag is selected at random and two balls are drawn from it, one after the other without replacement. Find the probability that both balls drawn are red.

Answer this when you sit the paper.

[3 marks]Conditional probability

Bag A contains 3 red balls and 2 white balls. Bag B contains 2 red balls and 3 white balls. A bag is selected at random and two balls are drawn from it, one after the other without replacement. Given that both balls drawn are red, find the probability that they came from bag A.

Answer this when you sit the paper.

Section c, Question 14

[2 marks]Discrete random variables

Two unbiased tetrahedral dice, each with faces marked 1, 2, 3 and 4, are tossed and

X

is the sum of the two scores. Find

P(X=5)

Answer this when you sit the paper.

[3 marks]Expectation

A player pays $1 to toss two unbiased tetrahedral dice with faces marked 1, 2, 3 and 4. If the sum of the scores is 2, 3 or 4 the player wins nothing; if it is 5, 6 or 7 the player wins $2; if it is greater than 7 the player wins $4. Find the expected gain, in dollars, from each game.

Answer this when you sit the paper.

Section c, Question 15

[2 marks]Measures of location

A stem and leaf diagram records the hectares owned by farmers around a small town, with key 1/3 meaning 13 hectares. Stem 0 has leaves 1, 4, 7; stem 1 has 1, 3, 8, 9; stem 2 has 0, 1, 2, 4, 7, 8; stem 3 has 0, 0, 2, 3, 4, 5, 7, 7; stem 4 has 2, 3, 5, 7. Find the median number of hectares.

Answer this when you sit the paper.

[2 marks]Measures of spread

Answer this when you sit the paper.

[2 marks]Measures of spread

Answer this when you sit the paper.

Pure Mathematics Paper 2 June 2016

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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 b

Section b, Question 7

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

Section c, Question 15

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The answers, and why they are the answers

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