ZIMSEC A Level Pure Mathematics Paper 2 June 2024

Question 101

[3 marks]Direct variation

The extension

y

of an elastic string varies directly as the magnitude of the force

F

extending it. Given

y=0.45

m when

F=6

N, express

F

as a function of

y

Answer this when you sit the paper.

Question 102

[3 marks]Direct variation

Given that

F=\dfrac{40}{3}y

for an elastic string, find

F

, in newtons, when

y=0.2

Answer this when you sit the paper.

Question 201

[2 marks]Groups and Cayley tables

The function

f

is defined by

f:x\to x-1

. Find

f(f(x))

Answer this when you sit the paper.

Question 202

[2 marks]Groups and Cayley tables

Set

S

contains the functions

x

x-1

and

x+1

under function composition. State the identity element of

S

Answer this when you sit the paper.

Question 203

[2 marks]Groups and Cayley tables

In the set

S=\{x, x-1, x+1\}

under function composition, state the inverse of the element

x-1

Answer this when you sit the paper.

Question 204

[2 marks]Groups and Cayley tables

Why is the set

S=\{x, x-1, x+1\}

not a group under function composition?

AIt has no identity element, since none of the three functions leaves the other two of them unchanged.
BNot every element has an inverse inside the set, because $x$ on its own cannot be undone by anything.
CFunction composition is not an associative operation, so the second group axiom fails immediately here.
DIt is not closed, because composing $x-1$ with itself gives $x-2$ , which is not a member of the set.

Question 301

[3 marks]Circular measure: arcs, sectors and radians

OABC

is a rhombus of side

r

with angle

2\theta

O

. Which expression gives its area?

A $r^2\sin2\theta$ , since the area of a rhombus is the product of two adjacent sides and the sine of the angle between them.
B $\dfrac12r^2\sin2\theta$ , since a rhombus is made of two congruent triangles and each triangle needs the factor of one half.
C $2r^2\theta$ , since the angle at $O$ is measured in radians and the two sides meeting there are both of length $r$ .
D $2r^2\sin2\theta$ , since the rhombus is twice the parallelogram built on the two sides that meet at the vertex $O$ .

Question 302

[2 marks]Circular measure: arcs, sectors and radians

A sector of a circle of radius

r

subtends an angle of

2\theta

radians at the centre. Find its area in terms of

r

and

\theta

Answer this when you sit the paper.

Question 303

[3 marks]Circular measure: arcs, sectors and radians

The shaded region has area

r^2(\sin2\theta-\theta)

. Find its exact area, in

\text{cm}^2

, when

r=8

cm and

\theta=\dfrac{\pi}{6}

radians.

Answer this when you sit the paper.

Question 401

[2 marks]Linearising a relationship from experimental data

Two variables are believed to satisfy

y=px^2+qx

. State what should be plotted against

x

so that the graph is a straight line.

Answer this when you sit the paper.

Question 402

[3 marks]Linearising a relationship from experimental data

Experimental values give

\dfrac{y}{x}=1

x=2

and

\dfrac{y}{x}=9

x=6

. For the relation

\dfrac{y}{x}=px+q

, find

p

Answer this when you sit the paper.

Question 403

[3 marks]Linearising a relationship from experimental data

The line

\dfrac{y}{x}=px+q

has gradient

p=2

and passes through the point where

x=1

and

\dfrac{y}{x}=-1

. Find

q

Answer this when you sit the paper.

Question 501

[2 marks]Partial fractions

Factorise

x^3+x^2+2x+2

completely.

Answer this when you sit the paper.

Question 502

[3 marks]Partial fractions

Why must

\dfrac{x^3}{x^3+x^2+2x+2}

be divided out before it is split into partial fractions?

ABecause the cubic on the bottom has three factors, and three separate fractions would need four unknown constants.
BBecause the denominator contains a quadratic factor, and any quadratic factor has to be removed before splitting.
CBecause the numerator is a single term, and partial fractions can only be applied to a numerator built out of two or more separate terms.
DBecause the numerator and denominator have the same degree, so the fraction is improper and no proper split exists yet.

Question 503

[3 marks]Partial fractions

Express

\dfrac{x^3}{(x+1)(x^2+2)}

in partial fractions.

Answer this when you sit the paper.

Question 601

[3 marks]Matrices and transformations of the plane

A matrix

M

maps

A(2;5)

onto

A^1(9;15)

and

B(-1;-4)

onto

B^1(-6;-9)

. Find

M

, giving its four entries in the order top-left, top-right, bottom-left, bottom-right.

Answer this when you sit the paper.

Question 602

[3 marks]Matrices and transformations of the plane

Find the area, in square units, of the triangle with vertices

A^1(9;15)

B^1(-6;-9)

and

C^1(-6;15)

Answer this when you sit the paper.

Question 603

[2 marks]Matrices and transformations of the plane

Find the determinant of

M=\begin{pmatrix}2&1\\5&1\end{pmatrix}

Answer this when you sit the paper.

Question 604

[3 marks]Matrices and transformations of the plane

A triangle is transformed by a

2\times2

matrix

M

. What is the ratio of the image area to the object area?

AThe modulus of the determinant of $M$ , taken as a modulus because a negative determinant only reverses orientation.
BThe determinant of $M$ itself, including its sign, so a negative determinant produces a negative image area.
CThe square of the determinant of $M$ , because area is a two-dimensional quantity and lengths change in two directions.
DThe sum of the four entries of $M$ , since each entry contributes one stretch to the overall change in size.

Question 605

[3 marks]Matrices and transformations of the plane

Triangle

A^1B^1C^1

has area

180

square units and is the image of triangle

ABC

under a matrix with determinant

-3

. Find the area of triangle

ABC

Answer this when you sit the paper.

Question 606

[2 marks]Matrices and transformations of the plane

The matrix

M=\begin{pmatrix}2&1\\5&1\end{pmatrix}

maps a point

C

onto

C^1(-6;15)

. Find the coordinates of

C

Answer this when you sit the paper.

Question 701

[3 marks]Complex numbers: division, modulus/argument, DeMoivre and loci

A complex number

W

satisfies

W(2+3i)=9-6i

. Express

W

in the form

x+iy

Answer this when you sit the paper.

Question 702

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

Find the modulus of the complex number

W=-3i

Answer this when you sit the paper.

Question 703

[3 marks]Complex numbers: division, modulus/argument, DeMoivre and loci

Find the argument of the complex number

W=-3i

, in radians.

Answer this when you sit the paper.

Question 704

[3 marks]Complex numbers: division, modulus/argument, DeMoivre and loci

A complex number lies on the negative imaginary axis of an Argand diagram. What is its principal argument?

A $-\dfrac{\pi}{4}$ , because the negative imaginary axis lies exactly halfway between the two axes that together bound the fourth quadrant.
B $\dfrac{\pi}{2}$ , because the argument of a number on either part of the imaginary axis is a quarter of a full turn.
C $\pi$ , because a number with no real part has been turned through a complete half revolution from the positive axis.
D $-\dfrac{\pi}{2}$ , because it lies a quarter turn clockwise from the positive real axis, and clockwise angles are negative.

Question 705

[3 marks]Complex numbers: division, modulus/argument, DeMoivre and loci

Given

W=-3i

, so that

|W|=3

and

\arg W=-\dfrac{\pi}{2}

, use DeMoivre's theorem to find

W^3

in the form

x+iy

Answer this when you sit the paper.

Question 706

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

On an Argand diagram, the locus

|Z-W|\le3

with

W=-3i

is a disc. State the coordinates of its centre.

Answer this when you sit the paper.

Question 801

[3 marks]Method of differences and summation of series

Express

\dfrac{1}{(r+1)(r+3)}

in partial fractions.

Answer this when you sit the paper.

Question 802

[3 marks]Method of differences and summation of series

Summing

\dfrac{1}{2(r+1)}-\dfrac{1}{2(r+3)}

from

r=1

n

leaves only four terms. Why?

ABecause the two halves of each term are equal in size, so all of them cancel apart from the four written at the ends.
BBecause the sum is a geometric progression, and only the first two and the last two terms of such a progression make any contribution at all.
CBecause each negative part cancels a positive part two rows further down, so only the first two and last two terms survive.
DBecause every term after the fourth one is so small that it may safely be neglected once $n$ has grown large enough.

Question 803

[3 marks]Method of differences and summation of series

Given

\displaystyle\sum_{r=1}^{n}\dfrac{1}{(r+1)(r+3)}=\dfrac{5}{12}-\dfrac{2n+5}{2(n+2)(n+3)}

, state the sum to infinity.

Answer this when you sit the paper.

Question 804

[2 marks]Method of differences and summation of series

For

S(n)=\dfrac{5}{12}-\dfrac{2n+5}{2(n+2)(n+3)}

, evaluate the subtracted fraction

\dfrac{2n+5}{2(n+2)(n+3)}

when

n=25

Answer this when you sit the paper.

Question 805

[3 marks]Method of differences and summation of series

Using

S(n)=\dfrac{5}{12}-\dfrac{2n+5}{2(n+2)(n+3)}

, evaluate

\displaystyle\sum_{r=12}^{25}\dfrac{1}{(r+1)(r+3)}

exactly.

Answer this when you sit the paper.

Question 806

[2 marks]Method of differences and summation of series

To evaluate

\displaystyle\sum_{r=12}^{25}

from a formula

S(n)

for

\displaystyle\sum_{r=1}^{n}

, which expression is used? Write it using

S

Answer this when you sit the paper.

Question 901

[3 marks]Vectors: lines, planes and their intersections

Lines

m

and

l

have direction vectors

\begin{pmatrix}1\\-1\\5\end{pmatrix}

and

\begin{pmatrix}-4\\1\\1\end{pmatrix}

. Calculate their scalar product.

Answer this when you sit the paper.

Question 902

[2 marks]Vectors: lines, planes and their intersections

Plane

\pi_1

has equation

\mathbf{r}\cdot\begin{pmatrix}3\\-2\\-1\end{pmatrix}=0

. Evaluate the left-hand side for the point

P(-3;1;2)

Answer this when you sit the paper.

Question 903

[3 marks]Vectors: lines, planes and their intersections

Line

m

passes through

A(1;2;-1)

and

P

is the point

(-3;1;2)

. Find the vector

\overrightarrow{AP}

Answer this when you sit the paper.

Question 904

[3 marks]Vectors: lines, planes and their intersections

Find a normal to the plane containing the direction

\begin{pmatrix}1\\-1\\5\end{pmatrix}

and the vector

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

, by evaluating their vector product in that order.

Answer this when you sit the paper.

Question 905

[3 marks]Vectors: lines, planes and their intersections

Find the cartesian equation of the plane through

A(1;2;-1)

with normal

\begin{pmatrix}2\\-23\\-5\end{pmatrix}

Answer this when you sit the paper.

Question 906

[2 marks]Vectors: lines, planes and their intersections

Planes

\pi_1

and

\pi_2

turn out to intersect in line

m

itself. Which fact establishes that?

AThe two planes have normals that are parallel, so their intersection has to be one of the lines already named.
B $\pi_2$ was built to contain $m$ , and $A(1;2;-1)$ on $m$ satisfies $\pi_1$ while $m$ 's direction is perpendicular to both normals.
CPoint $P$ lies on $\pi_2$ but not on $\pi_1$ , and any point standing in that position forces the two planes to meet along the line nearest to it.
DLine $l$ is perpendicular to $m$ , so the plane containing $m$ must cut every other plane along the direction of $l$ .

Question 1001

[3 marks]Parametric equations, curves and tangents

Eliminate

\theta

from

x=2+4\cos\theta

and

y=3+4\sin\theta

to give the cartesian equation of the curve.

Answer this when you sit the paper.

Question 1002

[2 marks]Parametric equations, curves and tangents

State the centre and the radius of the circle

(x-2)^2+(y-3)^2=16

Answer this when you sit the paper.

Question 1003

[3 marks]Parametric equations, curves and tangents

For the curve

x=2+4\cos\theta

y=3+4\sin\theta

, find

\dfrac{dy}{dx}

when

\theta=\dfrac{\pi}{4}

Answer this when you sit the paper.

Question 1004

[3 marks]Parametric equations, curves and tangents

Find, in exact form, the point on the curve

x=2+4\cos\theta

y=3+4\sin\theta

at which

\theta=\dfrac{\pi}{4}

Answer this when you sit the paper.

Question 1005

[3 marks]Parametric equations, curves and tangents

Find the equation of the tangent of gradient

-1

at the point

\left(2+2\sqrt2, 3+2\sqrt2\right)

, in the form

y=mx+c

with

c

exact.

Answer this when you sit the paper.

Question 1006

[2 marks]Parametric equations, curves and tangents

Find the coordinates of the point where the line

y=-x+5+4\sqrt2

meets the

x

-axis.

Answer this when you sit the paper.

Question 1101

[2 marks]Functions, inverse functions and proof by induction

Express

f(x)=x^2+2x

in the form

(x+b)^2+c

Answer this when you sit the paper.

Question 1102

[3 marks]Functions, inverse functions and proof by induction

The function

f(x)=x^2+2x

is defined only for

-4\le x\le-1

. State the range of

f

Answer this when you sit the paper.

Question 1103

[3 marks]Functions, inverse functions and proof by induction

For

f(x)=(x+1)^2-1

on the domain

-4\le x\le-1

, why is the inverse

-1-\sqrt{x+1}

rather than

-1+\sqrt{x+1}

ABecause $f$ is a decreasing function on that interval, and the inverse of any decreasing function carries a minus sign.
BBecause the constant inside the bracket is $+1$ , so the sign in front of the root must be the opposite of it.
CBecause $x+1\le0$ throughout $-4\le x\le-1$ , so only the negative root returns values inside the given domain.
DBecause a square root written without a sign in front of it is understood in this subject to mean the negative root.

Question 1104

[3 marks]Functions, inverse functions and proof by induction

Find

f^{-1}(x)

for

f(x)=x^2+2x

defined on

-4\le x\le-1

Answer this when you sit the paper.

Question 1105

[2 marks]Functions, inverse functions and proof by induction

State the domain of

f^{-1}

where

f(x)=x^2+2x

is defined on

-4\le x\le-1

Answer this when you sit the paper.

Question 1106

[3 marks]Functions, inverse functions and proof by induction

In proving

1+4+7+\cdots+(3n-2)=\dfrac{n(3n-1)}{2}

by induction, state the extra term added on going from

n=k

n=k+1

, in terms of

k

Answer this when you sit the paper.

Question 1201

[3 marks]Roots of equations and iterative methods

By sketching

y=x^2-4

and

y=\dfrac1x

on the same axes, state how many roots the equation

x^2-4=\dfrac1x

has.

Answer this when you sit the paper.

Question 1202

[3 marks]Roots of equations and iterative methods

Let

f(x)=x^2-4-\dfrac1x

. Evaluate

f(2)

Answer this when you sit the paper.

Question 1203

[2 marks]Roots of equations and iterative methods

Let

f(x)=x^2-4-\dfrac1x

. Evaluate

f(2.5)

Answer this when you sit the paper.

Question 1204

[3 marks]Roots of equations and iterative methods

For

f(x)=x^2-4-\dfrac1x

, the values

f(2)=-0.5

and

f(2.5)=1.85

have opposite signs. What does that establish?

AThat $f$ has no root between $2$ and $2.5$ , because the two values differ and so cannot both approach zero there.
BThat a root of $f(x)=0$ lies between $2$ and $2.5$ , because a continuous function must cross zero to change sign.
CThat $f$ has a turning point between $2$ and $2.5$ , because the values on either side of one are always opposite in sign.
DThat the root is exactly halfway between them at $x=2.25$ , because the sign change splits the interval evenly in two.

Question 1205

[2 marks]Roots of equations and iterative methods

Rearrange

x^2-4=\dfrac1x

into the form

x=\sqrt{\cdots}

suitable for iteration.

Answer this when you sit the paper.

Question 1206

[3 marks]Roots of equations and iterative methods

Use the iteration

x_{n+1}=\sqrt{\dfrac{1}{x_n}+4}

with

x_1=2.1

to estimate the root of

x^2-4=\dfrac1x

correct to 3 decimal places.

Answer this when you sit the paper.

Pure Mathematics Paper 2 June 2024

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 604

Question 605

Question 606

Question 701

Question 702

Question 703

Question 704

Question 705

Question 706

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1006

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

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