ZIMSEC A Level Additional Mathematics Paper 2 Specimen 2026

Section A

Section A, Question 1

[1 marks]Circular motion: the conical pendulum

A particle is attached to one end of a light inextensible string of length

1.3

m whose other end is fixed at A. The particle moves in a horizontal circle with the string at

35^\circ

to the vertical. Find the radius of the circle, in metres, to 2 decimal places.

Answer this when you sit the paper.

[2 marks]Circular motion: the conical pendulum

A particle of mass

2m

kg on a string of length

L

moves in a horizontal circle with the string at an angle

\theta

to the vertical, at angular speed

\omega

. Resolving vertically gives

T\cos\theta = 2mg

and horizontally gives

T\sin\theta = 2m\omega^2 L\sin\theta

. Which expression for

\omega^2

follows from these two equations?

A $\omega^2 = \dfrac{g}{L\sin\theta}$ , taking the radius as $L$ and the vertical component as $g\sin\theta$ .
B $\omega^2 = \dfrac{g\tan\theta}{L}$ , using the full string length $L$ as the radius of the circle.
C $\omega^2 = \dfrac{gL\cos\theta}{2m}$ , keeping the mass $2m$ in the result rather than cancelling it.
D $\omega^2 = \dfrac{g}{L\cos\theta}$ , since dividing the horizontal equation by the vertical one leaves $\tan\theta$ on the left.

[3 marks]Circular motion: the conical pendulum

A particle of mass

2m

kg hangs from a light inextensible string of length

1.3

m attached to a fixed point A. It moves at constant speed in a horizontal circle with the string at

35^\circ

to the vertical. Taking

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

, calculate its angular speed in

\mathrm{rad\,s^{-1}}

, to 2 significant figures.

Answer this when you sit the paper.

Section A, Question 2

[1 marks]Statics: equilibrium of a rigid lamina

A uniform rectangular lamina ABCD has a mass of

10

kg. Taking

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

, state the weight of the lamina in newtons.

Answer this when you sit the paper.

[2 marks]Statics: equilibrium of a rigid lamina

A uniform rectangular lamina ABCD has

AD = BC = 2a

and

AB = DC = a

. Vertex A rests on a horizontal table and a force

P

acts along the side DC. When moments are taken about A, what is the perpendicular distance from A to the line of action of

P

A $\dfrac{a}{2}$ , because the force acts through the centre of the lamina rather than along an edge.
B $2a$ , because DC is the side parallel to AB and the sides AB and DC are $AD = 2a$ apart.
C $a$ , because $a$ is the length of the side DC along which the force is acting.
D $a\sqrt{5}$ , because that is the length of the diagonal AC of the rectangle.

[3 marks]Statics: equilibrium of a rigid lamina

A uniform rectangular lamina ABCD of mass

10

kg has

AD = BC = 2a

and

AB = DC = a

. It rests in a vertical plane with A on a rough horizontal table and AD inclined at

30^\circ

to the horizontal. A force

P

along DC maintains equilibrium. Taking

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

, find

P

in newtons, correct to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Statics: equilibrium of a rigid lamina

A uniform rectangular lamina of mass

10

kg rests with one vertex A on a horizontal table. A force of

30.22

N acts along a side that makes an angle of

30^\circ

with the vertical, so its vertical component is upward. Taking

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

, find the normal reaction at A in newtons, to 2 decimal places.

Answer this when you sit the paper.

Section A, Question 3

[1 marks]Circular motion in a vertical circle and energy conservation

A small particle is threaded on a smooth circular wire of radius

0.5

m fixed in a vertical plane. It starts from rest at the highest point of the wire. Through what vertical height, in metres, has it fallen when it reaches the lowest point?

Answer this when you sit the paper.

[2 marks]Circular motion in a vertical circle and energy conservation

A particle of mass

1

kg is threaded on a smooth circular wire of radius

0.5

m fixed in a vertical plane with centre O. It is slightly disturbed from rest at the highest point A. Taking

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

, find its speed in

\mathrm{ms^{-1}}

when angle AOP is

90^\circ

, to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Circular motion in a vertical circle and energy conservation

A particle of mass

1

kg is threaded on a smooth circular wire of radius

0.5

m fixed in a vertical plane. It is slightly disturbed from rest at the highest point. Taking

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

, find its speed in

\mathrm{ms^{-1}}

at the lowest point, to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Circular motion in a vertical circle and energy conservation

A particle of mass

1

kg threaded on a smooth circular wire of radius

0.5

m reaches the lowest point of the wire with

v^2 = 2g

. Which expression gives the reaction

R

between the wire and the particle there?

A $R = g - \dfrac{2g}{0.5} = -3g$ , taking the centripetal force as acting away from the centre O.
B $R = \dfrac{2g}{0.5} = 4g$ , since at the lowest point the reaction supplies the centripetal force by itself.
C $R = g + \dfrac{2g}{0.5} = 5g$ , since $R$ must both support the weight and supply the centripetal force.
D $R = g$ , since the particle is momentarily moving horizontally and so is in vertical equilibrium.

Section A, Question 4

[2 marks]Simple harmonic motion of a mass on an elastic spring

A particle of mass

2

kg hangs at rest from a spring of natural length

1

m and modulus of elasticity

\lambda

, fixed at O. At rest the stretch is

0.1

m. Find

\lambda

in terms of

g

Answer this when you sit the paper.

[3 marks]Simple harmonic motion of a mass on an elastic spring

A particle of mass

2

kg hangs from a spring of natural length

1

m and modulus

20g

, fixed at O, and rests with a stretch of

0.1

m. It is then pulled down a further

x

metres. Which expression gives the tension in the spring in that pulled-down position?

A $2g + 20gx$ , since the total extension is now $0.1 + x$ and $T = \dfrac{\lambda(0.1+x)}{1}$ .
B $20gx$ , since only the extra stretch $x$ beyond the resting position contributes to the tension.
C $\dfrac{20g}{0.1 + x}$ , since the tension falls as the total extension of the spring grows.
D $2g - 20gx$ , since the weight of the particle acts against the pull of the stretched spring.

[2 marks]Simple harmonic motion of a mass on an elastic spring

A particle performs simple harmonic motion satisfying

\ddot{x} = -10gx

. Taking

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

, find its angular frequency

\omega

\mathrm{rad\,s^{-1}}

, to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Simple harmonic motion of a mass on an elastic spring

A particle performs simple harmonic motion satisfying

\ddot{x} = -10gx

. Taking

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

, find the period of the motion in seconds, to 2 decimal places.

Answer this when you sit the paper.

Section A, Question 5

[3 marks]Impulse, momentum and the coefficient of restitution

A smooth sphere A of mass

2

kg moving at

4\,\mathrm{ms^{-1}}

on a smooth horizontal plane collides directly with a stationary smooth sphere B of mass

3

kg. The coefficient of restitution is

\dfrac14

. Find the speed of B immediately after the impact, in

\mathrm{ms^{-1}}

Answer this when you sit the paper.

[2 marks]Impulse, momentum and the coefficient of restitution

A smooth sphere A of mass

2

kg moving at

4\,\mathrm{ms^{-1}}

collides directly with a stationary smooth sphere B of mass

3

kg, the coefficient of restitution being

\dfrac14

. After the impact B moves off at

2\,\mathrm{ms^{-1}}

. Find the speed of A immediately after the impact, in

\mathrm{ms^{-1}}

Answer this when you sit the paper.

[2 marks]Impulse, momentum and the coefficient of restitution

A smooth sphere B of mass

3

kg strikes a fixed vertical wall at right angles, moving at

2\,\mathrm{ms^{-1}}

. The coefficient of restitution is

\dfrac14

. Find the speed at which B rebounds, in

\mathrm{ms^{-1}}

Answer this when you sit the paper.

[3 marks]Impulse, momentum and the coefficient of restitution

A smooth sphere B of mass

3

kg strikes a fixed vertical wall at right angles at

2\,\mathrm{ms^{-1}}

and rebounds at

0.5\,\mathrm{ms^{-1}}

. Find the magnitude of the impulse the wall exerts on B, in

\mathrm{Ns}

Answer this when you sit the paper.

[2 marks]Impulse, momentum and the coefficient of restitution

Sphere A of mass

2

kg is moving at

1\,\mathrm{ms^{-1}}

towards a wall. Sphere B of mass

3

kg has rebounded from that wall and is moving at

0.5\,\mathrm{ms^{-1}}

in the opposite direction, straight towards A. They meet and coalesce. What is the speed of the combined body?

A $1.4\,\mathrm{ms^{-1}}$ , in B's direction of motion after the rebound
B $0.1\,\mathrm{ms^{-1}}$ , in A's original direction of motion
C $0.5\,\mathrm{ms^{-1}}$ , in B's direction of motion after the rebound
D $0.7\,\mathrm{ms^{-1}}$ , in A's original direction of motion

Section A, Question 6

[2 marks]Variable acceleration: v dv/dx and motion coming to rest

A particle moving in a straight line meets a retardation that rises at a constant rate with the distance moved. The retardation rises from

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

6\,\mathrm{ms^{-2}}

over a distance of

10

m. By how much, in

\mathrm{ms^{-2}}

, does the retardation rise per metre travelled?

Answer this when you sit the paper.

[3 marks]Variable acceleration: v dv/dx and motion coming to rest

A particle starts with velocity

5\sqrt{6}\,\mathrm{ms^{-1}}

and satisfies

v\dfrac{dv}{dx} = -\dfrac{1}{5}(10+2x)

. Integrating gives

v^2 = 150 - 4x - 0.4x^2

. Find

v^2

when the particle has moved

5

Answer this when you sit the paper.

[3 marks]Variable acceleration: v dv/dx and motion coming to rest

A particle moving in a straight line satisfies

v^2 = 150 - 4x - 0.4x^2

, where

v

is its velocity in

\mathrm{ms^{-1}}

after moving

x

metres. Find the distance in metres it moves before coming to rest.

Answer this when you sit the paper.

[3 marks]Variable acceleration: v dv/dx and motion coming to rest

A particle moving in a straight line satisfies

v^2 = 150 - 4x - 0.4x^2

, and comes to rest after

15

m. Find the time in seconds it takes to come to rest, to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Variable acceleration: v dv/dx and motion coming to rest

A particle moving in a straight line satisfies

v^2 = 150 - 4x - 0.4x^2

. Writing this in the completed-square form

v^2 = C - 0.4(x+5)^2

, find the value of

C

Answer this when you sit the paper.

Section B

Section B, Question 7

[2 marks]Hypothesis testing on a binomial proportion

At a school the pass rate in 'A' level mathematics was

40\%

. After a new teacher was hired,

4

out of

9

students passed. A test is to be run at the

5\%

level for evidence of an improvement. Which pair of hypotheses is correct?

A $H_0: p = 0.4$ against $H_1: p \ne 0.4$ , a two-tailed test since the pass rate could move either way
B $H_0: p > 0.4$ against $H_1: p = 0.4$ , putting the claimed improvement into the null hypothesis
C $H_0: p = 0.44$ against $H_1: p > 0.44$ , using the observed sample proportion as the value under test
D $H_0: p = 0.4$ against $H_1: p > 0.4$ , a one-tailed test since only an improvement is of interest

[3 marks]Hypothesis testing on a binomial proportion

The number of passes

X

among

9

students is modelled by

X \sim B(9, 0.4)

. Find

P(X \ge 4)

, to 3 decimal places.

Answer this when you sit the paper.

[2 marks]Hypothesis testing on a binomial proportion

A one-tailed test at the

5\%

level uses

H_0: p = 0.4

against

H_1: p > 0.4

, with

4

passes out of

9

observed and

P(X \ge 4) = 0.517

under

H_0

. What is the correct conclusion?

AReject $H_0$ , because $0.517$ is larger than the significance level of $0.05$ that was set for the test.
BReject $H_0$ , because the observed proportion $\dfrac49 = 0.444$ is greater than the assumed value of $0.4$ .
CDo not reject $H_0$ : since $0.517 > 0.05$ , there is no evidence at the $5\%$ level of an improvement.
DThe test cannot be carried out, because a sample of $9$ students is too small for a binomial model.

Section B, Question 8

[3 marks]The normal distribution and confidence intervals

The distance travelled by a commuter driver in a day is normally distributed with mean

360

km and standard deviation

60

km. Find the probability that the distance travelled in a day exceeds

370

km, to 3 decimal places.

Answer this when you sit the paper.

[2 marks]The normal distribution and confidence intervals

A confidence interval for a population mean is to be given at the

98\%

level, using a normal distribution. State the value of

z

used, to 3 decimal places.

Answer this when you sit the paper.

[2 marks]The normal distribution and confidence intervals

A random sample of

30

days gives a mean distance of

350

km, where the population standard deviation is known to be

60

km. Which is the

98\%

confidence interval for the mean distance travelled, to 1 decimal place?

A $(321.8,\ 378.2)$ , using the value of $z$ that belongs to a $99\%$ confidence interval
B $(328.5,\ 371.5)$ , using the value of $z$ that belongs to a $95\%$ confidence interval
C $(324.5,\ 375.5)$ , using $z = 2.326$ and the standard error $\dfrac{60}{\sqrt{30}}$
D $(210.4,\ 489.6)$ , using the population standard deviation $60$ in place of the standard error

Section B, Question 9

[1 marks]The Poisson distribution: sums and scaling of rates

Heavy vehicles pass a toll-gate at a mean rate of

2

in any

30

-minute period and light vehicles at a mean rate of

6

in any

30

-minute period, independently and at random. State the mean number of vehicles of all kinds passing in a

30

-minute period.

Answer this when you sit the paper.

[3 marks]The Poisson distribution: sums and scaling of rates

Vehicles pass a toll-gate independently and at random at a combined mean rate of

8

in any

30

-minute period. Find the probability that exactly

4

vehicles pass in a

30

-minute period, to 4 decimal places.

Answer this when you sit the paper.

[3 marks]The Poisson distribution: sums and scaling of rates

Vehicles pass a toll-gate independently and at random at a combined mean rate of

8

in any

30

-minute period. Find the probability that more than

2

vehicles pass in a

5

-minute period, to 4 decimal places.

Answer this when you sit the paper.

Section B, Question 10

[2 marks]The t-test for a population mean from a small sample

t

-test is to be used on a sample of five dogs to test a hypothesis about the mean mass of the population they came from. Which pair of conditions must hold for the test to be valid?

AThe population is normally distributed, and its standard deviation is unknown and estimated from the sample.
BThe population is normally distributed, and its standard deviation is known before the sample is taken.
CThe sample is larger than $30$ , and the population standard deviation is estimated from the sample data.
DThe population has any shape at all, and the sample standard deviation is smaller than the sample mean.

[2 marks]The t-test for a population mean from a small sample

Five dogs were weighed and their masses in kg were

6.2

5.8

7.0

8.7

and

9.1

. Calculate the mean mass in kg.

Answer this when you sit the paper.

[2 marks]The t-test for a population mean from a small sample

Five dogs were weighed and their masses in kg were

6.2

5.8

7.0

8.7

and

9.1

, giving a mean of

7.36

kg. Test the hypothesis

\mu = 6.5

against

\mu > 6.5

by calculating the value of the

t

statistic, to 3 decimal places.

Answer this when you sit the paper.

[3 marks]The t-test for a population mean from a small sample

A one-tailed

t

-test of

H_0: \mu = 6.5

against

H_1: \mu > 6.5

on a sample of five dogs gives

t = 1.302

with

4

degrees of freedom, against a critical value of

2.132

at the

5\%

level. What is the conclusion?

AReject $H_0$ , because the sample mean of $7.36$ kg is larger than the value of $6.5$ kg under test.
BDo not reject $H_0$ : since $1.302 < 2.132$ , there is no evidence the diet increases the dogs' masses.
CReject $H_0$ , because a test statistic of $1.302$ lies more than one standard error away from zero.
DDo not reject $H_0$ , because the critical value should be $1.645$ and the two figures are then too close.

Section B, Question 11

[2 marks]Linear combinations of normal variables and the Poisson approximation

The weight of a female baby is

F \sim N(2200, 60^2)

grams and of a male baby

M \sim N(3000, 50^2)

grams, independently. Find

E(3F - 2M)

in grams.

Answer this when you sit the paper.

[2 marks]Linear combinations of normal variables and the Poisson approximation

The weight of a female baby is

F \sim N(2200, 60^2)

grams and of a male baby

M \sim N(3000, 50^2)

grams, independently. Find

\mathrm{Var}(3F - 2M)

Answer this when you sit the paper.

[2 marks]Linear combinations of normal variables and the Poisson approximation

The variable

D = 3F - 2M

is normally distributed with mean

600

and variance

42400

. Find

P(D < 0)

, to 4 decimal places.

Answer this when you sit the paper.

[3 marks]Linear combinations of normal variables and the Poisson approximation

2\%

of babies develop respiratory problems immediately after birth. Using a suitable approximation, find the probability that more than

2

out of

100

randomly chosen babies develop respiratory problems, to 4 decimal places.

Answer this when you sit the paper.

Section B, Question 12

[1 marks]Product moment correlation and linear regression

Which statement best explains what is meant by the term correlation?

AIt is a measure of the gradient of the line of best fit drawn through a set of plotted points.
BIt is a measure of how far each individual value lies from the mean of its own set of data.
CIt is a measure of the change in one variable that is caused directly by a change in the other.
DIt is a measure of the strength and direction of the linear association between two variables.

[3 marks]Product moment correlation and linear regression

For six pairs of values,

x

being the mass of a baby and

y

the mass of the mother's placenta,

S_{xx} = 0.70

S_{yy} = 0.018283

and

S_{xy} = 0.111

. Calculate the product moment correlation coefficient, to 3 decimal places.

Answer this when you sit the paper.

[3 marks]Product moment correlation and linear regression

For six pairs of values of

x

and

y

\bar{x} = 2.7

\bar{y} = 0.4517

S_{xx} = 0.70

and

S_{xy} = 0.111

. Find the gradient of the regression line of

y

x

, to 4 decimal places.

Answer this when you sit the paper.

[3 marks]Product moment correlation and linear regression

The regression line of the placenta mass

y

on the baby mass

x

, both in kg, is

y = 0.0235 + 0.1586x

, fitted from data with

x

ranging from

2.2

3.2

kg. Find the mass of the placenta, in kg, when the baby has a mass of

2.3

kg, to 3 decimal places.

Answer this when you sit the paper.

Section B, Question 13

[2 marks]Chi-squared goodness of fit test

A company makes

5

deliveries a day for

90

days and each delivery is accepted with probability

0.3

, independently. Under the model

B(5, 0.3)

, find the expected number of days on which exactly

3

deliveries are accepted, to 3 decimal places.

Answer this when you sit the paper.

[3 marks]Chi-squared goodness of fit test

A chi-squared goodness of fit test compares

90

observed daily figures against

B(5, 0.3)

, whose

p

is given rather than estimated. The expected frequencies are

15.126

32.414

27.783

11.907

2.552

and

0.219

. After pooling every class with an expected frequency below

5

into its neighbour, how many degrees of freedom does the test have?

Answer this when you sit the paper.

[3 marks]Chi-squared goodness of fit test

A chi-squared goodness of fit test has four classes with observed frequencies

5

10

15

and

60

against expected frequencies

15.126

32.414

27.783

and

14.677

. Calculate the test statistic, to 1 decimal place.

Answer this when you sit the paper.

[3 marks]Chi-squared goodness of fit test

A chi-squared goodness of fit test of

B(5, 0.3)

against

90

days of delivery data gives

\chi^2 = 168.1

with

3

degrees of freedom, where the critical value at the

5\%

level is

7.815

. What is the conclusion?

AThe model does not fit: $168.1$ far exceeds $7.815$ , so $B(5, 0.3)$ is rejected at the $5\%$ level.
BThe model fits, since a large value of $\chi^2$ shows the observed and expected frequencies agree.
CThe model fits, since the test statistic was worked out from all $90$ days rather than from a sample.
DNo conclusion is possible, since classes had to be pooled before the test statistic could be found.

Additional Mathematics Paper 2 Specimen 2026

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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 B, Question 11

Section B, Question 12

Section B, Question 13

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The answers, and why they are the answers

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