ZIMSEC A Level Pure Mathematics Paper 2 November 2025

Question 101

[3 marks]Modulus functions and inequalities

Solve the inequality

|x^2-2|<-x

Answer this when you sit the paper.

Question 102

[2 marks]Modulus functions and inequalities

In solving

|x^2-2|<-x

by cases, why is only

x=-2

(not

x=1

) accepted from the case

x^2-2\ge0

, and only

x=-1

(not

x=2

) accepted from the case

x^2-2<0

ABecause the original inequality is only ever true for negative values of $x$ , by definition.
BBecause $x=-2$ and $x=-1$ are always the two smallest roots, regardless of which case produced them.
CBecause each case's algebraic solution must also satisfy that case's own defining condition on $x$ , and only one root from each case does so.
DBecause $x=1$ and $x=2$ are not real numbers, so they cannot be solutions to any inequality.

Question 201

[1 marks]Graph transformations

The graph

y=f(x)

has

y

-intercept

(0,16)

. State the

y

-intercept of

y=f(x)-6

Answer this when you sit the paper.

Question 202

[2 marks]Graph transformations

The graph

y=f(x)

has

x

-intercepts

(-2,0)

(2,0)

and

(4,0)

. State the

x

-intercepts of

y=f(x+1)

Answer this when you sit the paper.

Question 203

[1 marks]Graph transformations

The graph

y=f(x)

has

y

-intercept

(0,16)

. State the

y

-intercept of

y=-\tfrac12f(x)

Answer this when you sit the paper.

Question 204

[2 marks]Graph transformations

The graph

y=f(x)

has

x

-intercepts

(-2,0)

(2,0)

and

(4,0)

. State the

x

-intercepts of

y=f(2x)

Answer this when you sit the paper.

Question 301

[3 marks]Differentiation from first principles, tangents

Given

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

, differentiate from first principles to find

f'(x)

Answer this when you sit the paper.

Question 302

[3 marks]Differentiation from first principles, tangents

For

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

, with

f'(x)=\dfrac{-2x+6}{x^3}

, find the equation of the tangent to the curve at

x=2

Answer this when you sit the paper.

Question 401

[2 marks]Complex numbers: division, modulus/argument, loci

The complex number

u=\dfrac{5+4i}{3-2i}

. Express

u

in the form

x+iy

Answer this when you sit the paper.

Question 402

[1 marks]Complex numbers: division, modulus/argument, loci

For

u=\dfrac7{13}+\dfrac{22}{13}i

, find the modulus

|u|

Answer this when you sit the paper.

Question 403

[1 marks]Complex numbers: division, modulus/argument, loci

For

u=\dfrac7{13}+\dfrac{22}{13}i

, find the argument of

u

in degrees.

Answer this when you sit the paper.

Question 404

[2 marks]Complex numbers: division, modulus/argument, loci

The complex number

u\approx0.54+1.69i

. What does the locus

|z-u|=2

represent on an Argand diagram?

AA circle of radius 2, centred at the point representing $u$ .
BA circle of radius 2, centred at the origin, regardless of the value of $u$ .
CA single point, located a distance 2 from $u$ along the positive real axis.
DA straight line perpendicular to the line joining the origin to $u$ , at distance 2 from $u$ .

Question 501

[1 marks]Functions: domain, simultaneous equations, inverse

The function

f:x\mapsto\dfrac{m}{x}+n

. State the domain of

f(x)

Answer this when you sit the paper.

Question 502

[2 marks]Functions: domain, simultaneous equations, inverse

The function

f:x\mapsto\dfrac{m}{x}+n

satisfies

f(-1)=1\tfrac12

and

f(2)=9

. Find the values of

m

and

n

Answer this when you sit the paper.

Question 503

[2 marks]Functions: domain, simultaneous equations, inverse

Given

f(x)=\dfrac5x+\dfrac{13}2

, find an expression for

f^{-1}(x)

Answer this when you sit the paper.

Question 504

[2 marks]Functions: domain, simultaneous equations, inverse

Given

f^{-1}(x)=\dfrac{10}{2x-13}

, evaluate

f^{-1}(4)

Answer this when you sit the paper.

Question 601

[3 marks]Rational and modulus inequalities, simultaneous equations

Solve the inequality

\dfrac2{x-3}<\dfrac3{x-4}

Answer this when you sit the paper.

Question 602

[2 marks]Rational and modulus inequalities, simultaneous equations

The point

(1,5)

satisfies the simultaneous equations

n^2x+my=14

and

ny-mx=8

. Find one valid pair of values for

n

and

m

Answer this when you sit the paper.

Question 603

[3 marks]Rational and modulus inequalities, simultaneous equations

Solve the inequality

|2x-3|<5-|x|

Answer this when you sit the paper.

Question 701

[3 marks]Differential equations: separable, exponential decay

Find the equation of the curve which satisfies

\dfrac{dy}{dx}=-\dfrac{x}{y}

and passes through

(-4,3)

Answer this when you sit the paper.

Question 702

[1 marks]Differential equations: separable, exponential decay

The curve

x^2+y^2=25

satisfies

\dfrac{dy}{dx}=-\dfrac{x}{y}

and passes through

(-4,3)

. Describe this solution curve.

Answer this when you sit the paper.

Question 703

[2 marks]Differential equations: separable, exponential decay

The rate of destruction of a drug by the kidney is proportional to the amount

x

present at time

t

. Form the differential equation relating

x

and

t

Answer this when you sit the paper.

Question 704

[3 marks]Differential equations: separable, exponential decay

A drug decays according to

x=Ae^{-kt}

. At

t=0

x=200

mg. At

t=1

hour,

x=180

mg. Find the quantity of drug remaining after 8 hours.

Answer this when you sit the paper.

Question 801

[3 marks]Series summation, Maclaurin and binomial expansions

The

n

th term of a series is

4^{n-4}+6n

. Find the sum

S_N

of the first

N

terms.

Answer this when you sit the paper.

Question 802

[2 marks]Series summation, Maclaurin and binomial expansions

Find Maclaurin's series for

f(x)=(1-x)^{-1}

up to and including the term in

x^4

Answer this when you sit the paper.

Question 803

[3 marks]Series summation, Maclaurin and binomial expansions

Find the expansion of

(9-x)^{3/2}(1+3x)^{3/2}

up to and including the term in

x^2

Answer this when you sit the paper.

Question 804

[2 marks]Series summation, Maclaurin and binomial expansions

The expansion

(9-x)^{3/2}(1+3x)^{3/2}

requires

|x/9|<1

and

|3x|<1

. Find the set of values of

x

for which the expansion is valid.

Answer this when you sit the paper.

Question 901

[3 marks]Trigonometric identities: harmonic form, double angle

Express

2\cos\theta-\sqrt3\sin\theta

in the form

R\sin(\theta-\alpha)

, with

R>0

and

0^\circ<\alpha<360^\circ

Answer this when you sit the paper.

Question 902

[3 marks]Trigonometric identities: harmonic form, double angle

Solve

2\cos\theta-\sqrt3\sin\theta=1

for

0^\circ\le\theta\le360^\circ

, to the nearest degree.

Answer this when you sit the paper.

Question 903

[1 marks]Trigonometric identities: harmonic form, double angle

State the maximum value of

2\cos\theta-\sqrt3\sin\theta

, for

0^\circ\le\theta\le360^\circ

Answer this when you sit the paper.

Question 904

[1 marks]Trigonometric identities: harmonic form, double angle

State the minimum value of

2\cos\theta-\sqrt3\sin\theta

, for

0^\circ\le\theta\le360^\circ

Answer this when you sit the paper.

Question 905

[1 marks]Trigonometric identities: harmonic form, double angle

Express

2\cos^2\theta-1

in terms of

\cos2\theta

Answer this when you sit the paper.

Question 1001

[3 marks]Numerical integration, integration by parts, Newton-Raphson

Use the trapezium rule with 5 ordinates (

h=0.25

) to find an approximate value of

\displaystyle\int_0^1x^2e^x\,dx

, to 2 d.p.

Answer this when you sit the paper.

Question 1002

[3 marks]Numerical integration, integration by parts, Newton-Raphson

Evaluate the exact value of

\displaystyle\int_0^1x^2e^x\,dx

using integration by parts.

Answer this when you sit the paper.

Question 1003

[2 marks]Numerical integration, integration by parts, Newton-Raphson

The trapezium rule gives

0.76

for

\int_0^1x^2e^x\,dx

, whose exact value is

e-2\approx0.718

. Is this an over-estimation or under-estimation, and why?

AAn over-estimation, since $x^2e^x$ is convex (curves upward) on $[0,1]$ , so the chord segments used by the trapezium rule lie above the curve.
BAn over-estimation, but only because too few ordinates (5) were used; more ordinates would make it an under-estimation instead.
CAn under-estimation, since $x^2e^x$ is concave (curves downward) on $[0,1]$ , so the chord segments lie below the curve.
DNeither: the trapezium rule always gives the exact value for any smooth, continuous function.

Question 1004

[3 marks]Numerical integration, integration by parts, Newton-Raphson

Use the Newton-Raphson method, starting from

x_0=1

, to find a root of

x^3+3x-7=0

, correct to 2 d.p.

Answer this when you sit the paper.

Question 1101

[3 marks]Small errors, partial fractions and integration

An error of 3% is made in measuring the radius of a sphere (volume

=\tfrac43\pi r^3

). Find the percentage error in the volume.

Answer this when you sit the paper.

Question 1102

[3 marks]Small errors, partial fractions and integration

Express

\dfrac{2x+1}{(x-2)(x^2+1)}

in partial fractions.

Answer this when you sit the paper.

Question 1103

[3 marks]Small errors, partial fractions and integration

Using

\dfrac{2x+1}{(x-2)(x^2+1)}=\dfrac1{x-2}-\dfrac{x}{x^2+1}

, evaluate

\displaystyle\int_0^1\dfrac{2x+1}{(x-2)(x^2+1)}\,dx

Answer this when you sit the paper.

Question 1201

[1 marks]Group theory (Cayley table), matrix transformations

In the group

R=(\{O,N,I,T\},r)

, the row and column for

O

reproduce the header row/column exactly. State the identity element of this group.

Answer this when you sit the paper.

Question 1202

[1 marks]Group theory (Cayley table), matrix transformations

In the group

R=(\{O,N,I,T\},r)

, with identity

O

, state the inverse of the element

N

Answer this when you sit the paper.

Question 1203

[2 marks]Group theory (Cayley table), matrix transformations

The Cayley table for

R=(\{O,N,I,T\},r)

shows closure, an identity element

O

, and every element has an inverse. What additional property must be checked to confirm

R

is a group?

AThat every element is its own inverse, which is required for any group.
BAssociativity: that $(a*b)*c=a*(b*c)$ for all elements $a,b,c$ in the set.
CCommutativity: that $a*b=b*a$ for all elements, since this alone is sufficient to define a group.
DThat the set contains an even number of elements, which is required for any group.

Question 1204

[3 marks]Group theory (Cayley table), matrix transformations

Which matrix

M

describes an enlargement by factor 2 followed by a shear, factor 3 in the

x

-axis direction with the

y

-axis invariant?

A $M=\begin{pmatrix}6&2\\0&2\end{pmatrix}$
B $M=\begin{pmatrix}2&3\\0&2\end{pmatrix}$
C $M=\begin{pmatrix}2&0\\6&2\end{pmatrix}$
D $M=\begin{pmatrix}2&6\\0&2\end{pmatrix}$

Question 1205

[1 marks]Group theory (Cayley table), matrix transformations

Using

M=\begin{pmatrix}2&6\\0&2\end{pmatrix}

, find the image

A'

of the point

A(3,2)

Answer this when you sit the paper.

Question 1206

[1 marks]Group theory (Cayley table), matrix transformations

Using

M=\begin{pmatrix}2&6\\0&2\end{pmatrix}

, find the image

B'

of the point

B(1,1)

Answer this when you sit the paper.

Question 1207

[1 marks]Group theory (Cayley table), matrix transformations

Using

M=\begin{pmatrix}2&6\\0&2\end{pmatrix}

, find the image

C'

of the point

C(5,1)

Answer this when you sit the paper.

Question 1208

[2 marks]Group theory (Cayley table), matrix transformations

Triangle

ABC

(area 2 square units) is transformed by

M=\begin{pmatrix}2&6\\0&2\end{pmatrix}

to give triangle

A'B'C'

. Find the area of

A'B'C'

Answer this when you sit the paper.

Pure Mathematics Paper 2 November 2025

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 904

Question 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1207

Question 1208

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