ZIMSEC A Level Pure Mathematics Paper 1 November 2025

Question 101

[1 marks]Variation

A ball's mass

m

varies directly as the cube of its diameter

d

(

m=kd^3

). What power of

d

does

m

vary as?

Answer this when you sit the paper.

Question 102

[3 marks]Variation

A soccer ball's mass varies directly as the cube of its diameter. A ball of diameter 7 cm has mass 0.11 kg. Calculate the mass of a similar ball of diameter 9 cm, to 2 s.f.

Answer this when you sit the paper.

Question 201

[1 marks]Differential equations

Given

\dfrac{dy}{d\theta}+2\sin2\theta=0

, integrating gives

y=\cos2\theta+C

. Using

y=1

when

\theta=\pi/6

, find

C

Answer this when you sit the paper.

Question 202

[3 marks]Differential equations

Given

\dfrac{dy}{d\theta}+2\sin2\theta=0

, where

y=1

when

\theta=\pi/6

, find the equation connecting

y

and

\theta

Answer this when you sit the paper.

Question 301

[2 marks]Complex numbers

For the equation

z^2+3z+4i+2=0

, the sum of the two complex roots equals

-b/a

. Find the sum of the roots.

Answer this when you sit the paper.

Question 302

[3 marks]Complex numbers

Solve

z^2+3z+4i+2=0

, giving one root in the form

x+iy

, correct to 2 d.p.

Answer this when you sit the paper.

Question 401

[2 marks]Small-angle approximations

For small

\theta

(radians,

\theta\to0

), find an approximate value, in terms of

\theta

, for

\dfrac{\sin4\theta+\tan2\theta}{3+\cos2\theta}

, neglecting terms in

\theta^2

and higher.

Answer this when you sit the paper.

Question 402

[3 marks]Small-angle approximations

Given

1^\circ\approx0.018

radians and

(0.018)^2\approx0.000324

, find an approximate value for

\dfrac{\cos2^\circ}{\sin1^\circ}

, to 2 d.p.

Answer this when you sit the paper.

Question 501

[2 marks]Quadratic equations

The equation

x^2+2x+4=P(x^2+4)

has equal roots. Rearranging gives

(1-P)x^2+2x+(4-4P)=0

. Setting the discriminant to zero leads to

(1-P)^2=k

for some constant

k

. Find

k

Answer this when you sit the paper.

Question 502

[3 marks]Quadratic equations

Given that

x^2+2x+4=P(x^2+4)

has equal roots, find the possible values of

P

Answer this when you sit the paper.

Question 601

[3 marks]Matrix transformations

The matrix

\begin{pmatrix}4&-1\\-3&2\end{pmatrix}

maps every point on

y=3x

onto itself. Verify this for the point

(1,3)

: find its image under the matrix.

Answer this when you sit the paper.

Question 602

[2 marks]Matrix transformations

For a general point

(x,3x)

on the line

y=3x

, applying the matrix

\begin{pmatrix}4&-1\\-3&2\end{pmatrix}

gives back exactly

(x,3x)

. What does this prove about the matrix and the line?

AEvery point on the line $y=3x$ is invariant (mapped onto itself) under this transformation.
BThe line $y=3x$ is the only line in the plane invariant under any matrix, so the result holds for every possible transformation matrix, not just this one.
CThe matrix swaps the coordinates of every point it is applied to, for any line.
DThe matrix has no effect on any point in the plane, not just points on $y=3x$ .

Question 701

[3 marks]Exponential equations

Substituting

y=3^x

into

2(3^{2x+1})-20(3^x)=-6

gives a quadratic in

y

. Solve for

y

Answer this when you sit the paper.

Question 702

[3 marks]Exponential equations

Solve the equation

2(3^{2x+1})-20(3^x)=-6

for

x

Answer this when you sit the paper.

Question 801

[2 marks]Proof by induction

To prove

\dfrac12+\dfrac16+\cdots+\dfrac1{n(n-1)}=1-\dfrac1n

by induction for

n>1

, check the base case

n=2

: what is the value of the left-hand side?

Answer this when you sit the paper.

Question 802

[2 marks]Proof by induction

In the induction proof for

\dfrac12+\dfrac16+\cdots+\dfrac1{n(n-1)}=1-\dfrac1n

, going from

n=m

n=m+1

adds one extra term to the left-hand side. What is that extra term?

Answer this when you sit the paper.

Question 803

[2 marks]Proof by induction

Having verified the base case

n=2

and shown that truth at

n=m

implies truth at

n=m+1

, why does this prove the statement for every integer

n>1

ABecause the formula was already given in the question, so no proof of the inductive step is actually needed.
BBy the principle of mathematical induction: truth at the base case plus the inductive step together guarantee truth for every subsequent integer.
CBecause the series converges, and any convergent series formula is automatically true for all $n$ .
DBecause checking just two cases, $n=2$ and $n=m$ , is always enough to prove a statement true for all real numbers, without needing any further justification.

Question 901

[2 marks]Coordinate geometry

The line

2x-14y+6=0

passes through

L(-3,0)

and

M(k,1)

. Find the value of

k

Answer this when you sit the paper.

Question 902

[2 marks]Coordinate geometry

Points

L(-3,0)

and

M(4,1)

. Find the length

LM

, in surd form.

Answer this when you sit the paper.

Question 903

[3 marks]Coordinate geometry

Find the equation of the line through

M(4,1)

parallel to the line

2x+y+5=0

Answer this when you sit the paper.

Question 1001

[2 marks]Completing the square, curve sketching

Express

10x-x^2-27

in the form

-(a-x)^2+b

, where

a

and

b

are integers. State

a

and

b

Answer this when you sit the paper.

Question 1002

[2 marks]Completing the square, curve sketching

Given

10x-x^2-27=-(5-x)^2-2

, why is

10x-x^2-27

always negative for every real

x

ABecause $x^2$ is always positive, so subtracting it from $10x-27$ must always give a negative result.
BBecause the coefficient of $x^2$ in the original expression is negative, which alone guarantees a negative value everywhere.
CBecause $10x-27$ is negative for all $x$ , regardless of the $-x^2$ term.
DBecause $-(5-x)^2\le0$ for every real $x$ , so the whole expression is at most $-2$ , which is always negative.

Question 1003

[2 marks]Completing the square, curve sketching

State the coordinates of the maximum point of the curve

y=10x-x^2-27

Answer this when you sit the paper.

Question 1004

[2 marks]Completing the square, curve sketching

State the coordinates of the point where the curve

y=10x-x^2-27

crosses the

y

-axis.

Answer this when you sit the paper.

Question 1101

[2 marks]Rational functions, transformations

Given

f(x)=\dfrac{2x+5}{x+1}

x\ne-1

, show that

f(x)\equiv2+\dfrac{A}{x+1}

for some constant

A

. Find

A

Answer this when you sit the paper.

Question 1102

[3 marks]Rational functions, transformations

Which sequence of transformations maps the graph of

y=\dfrac1x

onto the graph of

y=\dfrac{2x+5}{x+1}=2+\dfrac3{x+1}

ATranslate 2 units in the positive $x$ -direction, stretch vertically by scale factor 3, then translate 1 unit in the negative $y$ -direction.
BStretch vertically by scale factor 2, translate 3 units in the positive $x$ -direction, then translate 1 unit in the negative $y$ -direction.
CReflect in the $x$ -axis, translate 1 unit in the positive $x$ -direction, then stretch vertically by scale factor 2.
DTranslate 1 unit in the negative $x$ -direction, stretch vertically by scale factor 3, then translate 2 units in the positive $y$ -direction.

Question 1103

[3 marks]Rational functions, transformations

For

y=f(x)=\dfrac{2x+5}{x+1}

with

x\ge-1

, state the vertical and horizontal asymptotes.

Answer this when you sit the paper.

Question 1201

[3 marks]Graphical solution of equations

On the interval

0<x<\pi/2

y_1=\cot x

decreases from

+\infty

0

, while

y_2=2x^2-1

increases from

-1

to about

3.93

. Why does this show the equation

\cot x=2x^2-1

has exactly one root in this interval?

ABoth curves are strictly increasing over the interval, so they can never actually cross at all.
BOne curve is strictly decreasing and the other strictly increasing, their relative order reverses across the interval, so they cross exactly once.
CSince $y_1$ starts above $y_2$ and stays above it throughout the interval, the curves never meet.
DThe two curves have the same value at both endpoints of the interval, which is what proves they cross exactly once regardless of how they behave in between.

Question 1301

[1 marks]Group theory

(G,*)=(\{0,1,2,3\},+\bmod4)

, state the identity element.

Answer this when you sit the paper.

Question 1302

[1 marks]Group theory

(G,*)=(\{0,1,2,3\},+\bmod4)

, state the inverse of 1.

Answer this when you sit the paper.

Question 1303

[1 marks]Group theory

(G,*)=(\{0,1,2,3\},+\bmod4)

, state the inverse of 3.

Answer this when you sit the paper.

Question 1304

[2 marks]Group theory

(G,*)=(\{0,1,2,3\},+\bmod4)

has closure, an identity element (0), and every element has an inverse within

G

. What additional axiom must hold for

(G,*)

to be a group?

ACommutativity, which is not actually required for a group and does not need to be checked here or in any other group.
BThat $G$ has infinitely many elements, since finite sets can never form groups.
CAssociativity: $(a*b)*c=a*(b*c)$ for all $a,b,c$ in $G$ , which is inherited here from ordinary integer addition.
DThat every element equals its own inverse, which is required for any set to form a group.

Question 1305

[1 marks]Group theory

(G,*)=(\{0,1,2,3\},+\bmod4)

a group under addition modulo 4?

Answer this when you sit the paper.

Question 1401

[3 marks]Inverse and composite functions

For

f:x\mapsto e^{3x}

, find

f^{-1}(x)

and state its domain.

Answer this when you sit the paper.

Question 1402

[3 marks]Inverse and composite functions

Given

f:x\mapsto e^{3x}

and

g:x\mapsto\ln(x+2)

x>-2

, find

fg(x)

in terms of

x

Answer this when you sit the paper.

Question 1403

[2 marks]Inverse and composite functions

The graphs of

y=f(x)=e^{3x}

and

y=f^{-1}(x)=\tfrac13\ln x

are sketched on the same axes. What is the geometric relationship between the two curves?

AThey are identical curves, since $f$ and $f^{-1}$ always coincide exactly.
BThey are translations of each other by 1 unit along the $x$ -axis.
CThey are reflections of each other in the line $y=x$ .
DThey are reflections of each other in the $y$ -axis.

Question 1501

[2 marks]Newton-Raphson, root location

Rewriting

e^{2x}+4x-5=0

e^{2x}=5-4x

, why does this equation have exactly one root?

A $y_1=e^{2x}$ is always greater than $y_2=5-4x$ for every value of $x$ , so no root can actually exist.
B $y_1=e^{2x}$ is strictly increasing while $y_2=5-4x$ is strictly decreasing, so an increasing and a decreasing curve can meet at most once, and here they do meet once.
CBoth $y_1=e^{2x}$ and $y_2=5-4x$ are strictly increasing functions of $x$ , so by the general rule for monotonic curves they are guaranteed to intersect at exactly one point somewhere on the real line.
DThe equation is a straight line, so it can only ever have one root by definition.

Question 1502

[1 marks]Newton-Raphson, root location

For

h(x)=e^{2x}+4x-5

, what is the sign of

h(0)

Answer this when you sit the paper.

Question 1503

[1 marks]Newton-Raphson, root location

For

h(x)=e^{2x}+4x-5

, what is the sign of

h(1)

Answer this when you sit the paper.

Question 1504

[2 marks]Newton-Raphson, root location

Using Newton-Raphson with

x_0=0.5

h(x)=e^{2x}+4x-5

, one iteration gives

x_1=x_0-h(x_0)/h'(x_0)

. Find

x_1

, to 4 d.p.

Answer this when you sit the paper.

Question 1505

[3 marks]Newton-Raphson, root location

Use the Newton-Raphson method, starting from

x_0=0.5

, to find the root of

e^{2x}+4x-5=0

, correct to 2 decimal places.

Answer this when you sit the paper.

Question 1601

[2 marks]Stationary points, curve sketching

Find

\dfrac{dy}{dx}

for the curve

y=2x^3+3x^2-12x+1

Answer this when you sit the paper.

Question 1602

[3 marks]Stationary points, curve sketching

Find the stationary points of

y=2x^3+3x^2-12x+1

, using

\dfrac{dy}{dx}=6x^2+6x-12=6(x+2)(x-1)

Answer this when you sit the paper.

Question 1603

[2 marks]Stationary points, curve sketching

At the stationary point

(-2,21)

y=2x^3+3x^2-12x+1

, the second derivative

\dfrac{d^2y}{dx^2}=12x+6

equals

-18

. What kind of stationary point is this?

AA local maximum, since the second derivative is negative there.
BA point of inflection, since the second derivative is negative there.
CA local maximum, but only because $x=-2$ is negative, not because of the second derivative.
DA local minimum, since the second derivative is negative there.

Question 1604

[2 marks]Stationary points, curve sketching

For the curve

y=2x^3+3x^2-12x+1

, state the value of

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

at the stationary point

x=1

, and whether it is a maximum or minimum.

Answer this when you sit the paper.

Question 1605

[1 marks]Stationary points, curve sketching

State the coordinates of the point where the curve

y=2x^3+3x^2-12x+1

crosses the

y

-axis.

Answer this when you sit the paper.

Question 1701

[2 marks]Matrix inverse, simultaneous equations

Find the determinant of

P=\begin{pmatrix}1&1&2\\2&-1&1\\1&-2&3\end{pmatrix}

Answer this when you sit the paper.

Question 1702

[1 marks]Matrix inverse, simultaneous equations

Given

\det P=-12

for

P=\begin{pmatrix}1&1&2\\2&-1&1\\1&-2&3\end{pmatrix}

, and that the top-left cofactor is

-1

, state the value of the top-left entry of

P^{-1}

Answer this when you sit the paper.

Question 1703

[3 marks]Matrix inverse, simultaneous equations

Solve the simultaneous equations

x+y+2z=0

2x-y+z=-3

x-2y+3z=5

for

x

y

and

z

Answer this when you sit the paper.

Question 1704

[2 marks]Matrix inverse, simultaneous equations

For the solution

x=-3

y=-1

z=2

of the simultaneous equations, find

x+y+z

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2025

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 401

Question 402

Question 501

Question 502

Question 601

Question 602

Question 701

Question 702

Question 801

Question 802

Question 803

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1201

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1401

Question 1402

Question 1403

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1601

Question 1602

Question 1603

Question 1604

Question 1605

Question 1701

Question 1702

Question 1703

Question 1704

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