ZIMSEC A Level Pure Mathematics Paper 2 November 2024

Question 101

[2 marks]Partial fractions

Factorise

2x^2-x-1

Answer this when you sit the paper.

Question 102

[2 marks]Partial fractions

Write

\dfrac{2x^2}{2x^2-x-1}

1

plus a proper fraction, before splitting it into partial fractions.

Answer this when you sit the paper.

Question 103

[2 marks]Partial fractions

Express

\dfrac{2x^2}{2x^2-x-1}

in partial fractions.

Answer this when you sit the paper.

Question 201

[3 marks]Integration by parts

Find

\displaystyle\int e^{2x}\cos x\,dx

, omitting the constant of integration.

Answer this when you sit the paper.

Question 202

[3 marks]Integration by parts

Evaluate

\displaystyle\int_0^1e^{2x}\cos x\,dx

, giving the answer correct to 3 significant figures.

Answer this when you sit the paper.

Question 301

[3 marks]Binomial expansion for a rational index

Before expanding

(3-x)^{-\frac12}

by the binomial series, it is written as a constant multiplied by

\left(1-\dfrac{x}{3}\right)^{-\frac12}

. State that constant.

Answer this when you sit the paper.

Question 302

[3 marks]Binomial expansion for a rational index

Expand

\left(1-\dfrac{x}{3}\right)^{-\frac12}

up to and including the term in

x^3

Answer this when you sit the paper.

Question 303

[2 marks]Binomial expansion for a rational index

State the range of values of

x

for which the binomial expansion of

(3-x)^{-\frac12}

is valid.

Answer this when you sit the paper.

Question 401

[3 marks]Cubic equations and inequalities

Solve the equation

x^3-4x^2+x=-6

Answer this when you sit the paper.

Question 402

[2 marks]Cubic equations and inequalities

State the

y

-intercept of the graph of

y=x^3-4x^2+x+6

Answer this when you sit the paper.

Question 403

[3 marks]Cubic equations and inequalities

Given that

x^3-4x^2+x+6=(x+1)(x-2)(x-3)

, solve the inequality

x^3-4x^2+x+6<0

Answer this when you sit the paper.

Question 501

[3 marks]Inverse functions and their graphs

A function is defined by

h:x\longrightarrow\dfrac{2}{x-3}

for

x>3

. Find

h^{-1}(x)

Answer this when you sit the paper.

Question 502

[3 marks]Inverse functions and their graphs

State the domain of

h^{-1}

where

h:x\longrightarrow\dfrac{2}{x-3}

for

x>3

Answer this when you sit the paper.

Question 503

[2 marks]Inverse functions and their graphs

For

h:x\longrightarrow\dfrac{2}{x-3}

with

x>3

, find the value of

hh^{-1}(2)

Answer this when you sit the paper.

Question 504

[2 marks]Inverse functions and their graphs

How are the graphs of

y=h(x)

and

y=h^{-1}(x)

related when they are drawn on the same axes?

AEach is the reflection of the other in the line $y=x$ , so any point $(a,b)$ on one has a partner $(b,a)$ on the other.
BEach is the reflection of the other in the $y$ -axis, since taking an inverse reverses the sign of every input.
COne is the other rotated through a half turn about the origin, so the two curves lie in opposite quadrants.
DOne is the other translated along the $x$ -axis, by a distance equal to the value taken by the horizontal asymptote of the first curve.

Question 505

[2 marks]Inverse functions and their graphs

State the horizontal asymptote of

y=h^{-1}(x)=3+\dfrac{2}{x}

Answer this when you sit the paper.

Question 601

[2 marks]The R-alpha form, trigonometric equations and compound angles

Expressing

2\cos\theta-\sin\theta

in the form

R\cos(\theta+x)

with

R>0

, find

R

in exact form.

Answer this when you sit the paper.

Question 602

[2 marks]The R-alpha form, trigonometric equations and compound angles

Expressing

2\cos\theta-\sin\theta

in the form

R\cos(\theta+x)

with

0^\circ<x<90^\circ

, find

x

to the nearest degree.

Answer this when you sit the paper.

Question 603

[2 marks]The R-alpha form, trigonometric equations and compound angles

In writing

a\cos\theta-b\sin\theta

R\cos(\theta+x)

, how are

R

and

x

obtained?

A $R$ is the sum $a+b$ and $x$ is the angle whose sine is $\dfrac{b}{a+b}$ , taken in the first quadrant so that it comes out acute.
B $R=\sqrt{a^2+b^2}$ and $\tan x=\dfrac{b}{a}$ , from matching the $\cos\theta$ and $\sin\theta$ terms after expanding.
C $R$ is the larger of $a$ and $b$ , and $x$ is the angle whose tangent is the smaller divided by the larger.
D $R=a^2+b^2$ and $\tan x=\dfrac{a}{b}$ , from matching the two terms and then squaring both of the results.

Question 604

[3 marks]The R-alpha form, trigonometric equations and compound angles

Solve

10\cos\theta-5\sin\theta=3

for

0\le\theta\le360^\circ

, giving the answers to the nearest degree.

Answer this when you sit the paper.

Question 605

[3 marks]The R-alpha form, trigonometric equations and compound angles

Find the maximum value of

\dfrac{1}{2\cos\theta-\sin\theta+11}

, correct to 2 significant figures.

Answer this when you sit the paper.

Question 606

[2 marks]The R-alpha form, trigonometric equations and compound angles

Find the minimum value of

\dfrac{1}{2\cos\theta-\sin\theta+11}

, correct to 2 significant figures.

Answer this when you sit the paper.

Question 607

[2 marks]The R-alpha form, trigonometric equations and compound angles

Given that

\sin A=\dfrac35

for

0<A<90^\circ

, find

\tan A

Answer this when you sit the paper.

Question 701

[2 marks]Complex numbers: DeMoivre's theorem, roots and loci

Find the modulus of the complex number

Z=1+i

Answer this when you sit the paper.

Question 702

[2 marks]Complex numbers: DeMoivre's theorem, roots and loci

Find the argument of the complex number

Z=1+i

, in radians.

Answer this when you sit the paper.

Question 703

[3 marks]Complex numbers: DeMoivre's theorem, roots and loci

Each of the five fifth roots of

1+i

has the same modulus. Find it, correct to 3 significant figures.

Answer this when you sit the paper.

Question 704

[3 marks]Complex numbers: DeMoivre's theorem, roots and loci

The fifth roots of

1+i

have arguments

\dfrac{\pi}{4+8k}

divided by nothing; more precisely each is

\dfrac{\frac{\pi}{4}+2k\pi}{5}

for

k=0,1,2,3,4

. State the smallest positive one, in radians.

Answer this when you sit the paper.

Question 705

[3 marks]Complex numbers: DeMoivre's theorem, roots and loci

How many distinct fifth roots does a non-zero complex number have, and how are they arranged on an Argand diagram?

AFive, but all lying on one straight line through the origin, each a fifth of the distance of the one before it.
BTen, arranged in five conjugate pairs, since every root of a complex number is accompanied by its own conjugate.
CFive, spaced $72^\circ$ apart around a circle centred on the origin whose radius is the fifth root of the modulus.
DOne, since a fifth root is an odd root and an odd root of any number is unique, exactly as it is for real numbers.

Question 706

[3 marks]Complex numbers: DeMoivre's theorem, roots and loci

On an Argand diagram the region

0<\arg(Z+1+i)<\dfrac{\pi}{3}

is a wedge. State the coordinates of its vertex.

Answer this when you sit the paper.

Question 801

[3 marks]Arithmetic progressions and proof by induction

In an arithmetic progression with first term

a

and common difference

d

, the eighth term is twice the third term. Express

a

in terms of

d

Answer this when you sit the paper.

Question 802

[3 marks]Arithmetic progressions and proof by induction

An arithmetic progression has

a=3d

and the sum of its first eight terms is

39

. Find the first term.

Answer this when you sit the paper.

Question 803

[2 marks]Arithmetic progressions and proof by induction

An arithmetic progression has

a=3d

and the sum of its first eight terms is

39

. Find the common difference.

Answer this when you sit the paper.

Question 804

[2 marks]Arithmetic progressions and proof by induction

An arithmetic progression has first term

\dfrac94

and common difference

\dfrac34

. Find its ninth term.

Answer this when you sit the paper.

Question 805

[3 marks]Arithmetic progressions and proof by induction

For an arithmetic progression with

a=\dfrac94

and

d=\dfrac34

, the sum of

n

terms takes the form

\dfrac38n(n+k)

. Find

k

Answer this when you sit the paper.

Question 806

[3 marks]Arithmetic progressions and proof by induction

A proof by induction of a formula for

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

begins by checking the formula when

n=1

. What must then be done to complete it?

ASubstitute a large value of $n$ and confirm that both sides agree to within a small margin of rounding error.
BAssume the formula holds for $n=k$ , then use that assumption to deduce that it also holds for $n=k+1$ .
CCheck the formula for a second value such as $n=2$ , and for a third such as $n=3$ , until a clear pattern appears.
DDifferentiate both sides with respect to $n$ and confirm that the two derivatives obtained are equal to each other.

Question 901

[3 marks]Vectors: unit vectors, angles and the equation of a plane

Given

\mathbf{p}=\mathbf{i}+\mathbf{j}+\mathbf{k}

\mathbf{q}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}

and

\mathbf{r}=\mathbf{i}-3\mathbf{j}+2\mathbf{k}

, find

\mathbf{p}+\mathbf{q}+\mathbf{r}

Answer this when you sit the paper.

Question 902

[3 marks]Vectors: unit vectors, angles and the equation of a plane

Find the exact magnitude of the vector

3\mathbf{i}+6\mathbf{k}

Answer this when you sit the paper.

Question 903

[3 marks]Vectors: unit vectors, angles and the equation of a plane

Find a unit vector parallel to

3\mathbf{i}+6\mathbf{k}

Answer this when you sit the paper.

Question 904

[3 marks]Vectors: unit vectors, angles and the equation of a plane

Find the scalar product of

3\mathbf{i}+6\mathbf{k}

and

\mathbf{i}+\mathbf{j}+\mathbf{k}

Answer this when you sit the paper.

Question 905

[2 marks]Vectors: unit vectors, angles and the equation of a plane

Points

P

and

Q

have position vectors

\mathbf{i}+\mathbf{j}+\mathbf{k}

and

\mathbf{i}+2\mathbf{j}+3\mathbf{k}

. Find

\overrightarrow{PQ}

Answer this when you sit the paper.

Question 906

[2 marks]Vectors: unit vectors, angles and the equation of a plane

Points

P(1;1;1)

Q(1;2;3)

and

R(1;-3;2)

all lie on one plane. What is its cartesian equation?

A $x+y+z=3$ , because a plane through three points always has the sum of the coordinates of the first of them.
B $x=1$ , because every one of the three points has $x$ -coordinate 1, so the plane is the one at that fixed value.
C $y=1$ , because the middle coordinate is the one that fixes a plane when three points are given in this form.
D $z=1$ , because the third coordinate is the height, and a plane drawn through any three given points is always a level surface.

Question 1001

[3 marks]Proof by induction and group theory

To start an induction proof of

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

, evaluate the left-hand side when

n=1

Answer this when you sit the paper.

Question 1002

[3 marks]Proof by induction and group theory

In the inductive step of that proof, the expression

(k+2)^2-(2k+3)

arises. Simplify it into a squared bracket.

Answer this when you sit the paper.

Question 1003

[2 marks]Proof by induction and group theory

The set

S=\{a;2;4;6\}

forms a group under addition modulo 8. Write down the value of

a

Answer this when you sit the paper.

Question 1004

[2 marks]Proof by induction and group theory

In the group

\{0;2;4;6\}

under addition modulo 8, evaluate

6+4

Answer this when you sit the paper.

Question 1005

[3 marks]Proof by induction and group theory

For

f(x)=px+q

and

g(x)=mx+n

, find

f(g(x))

, simplified.

Answer this when you sit the paper.

Question 1006

[3 marks]Proof by induction and group theory

For the set of functions

f(x)=px+q

under composition, why is the condition

p\ne0

imposed?

ABecause composition of two such functions would stop being associative as soon as one of them had $p=0$ .
BBecause $q$ would then be forced to be zero as well, and the set would collapse to a single element only.
CBecause the inverse is $f^{-1}(x)=\dfrac{x-q}{p}$ , which does not exist when $p=0$ , so that axiom would fail.
DBecause the identity function is $e(x)=x$ , and allowing $p=0$ would place a second identity element inside the very same set.

Question 1101

[2 marks]Completing the square, quadratic graphs and exponential equations

Express

y=x^2-2x-3

in the form

(x-p)^2-q

Answer this when you sit the paper.

Question 1102

[2 marks]Completing the square, quadratic graphs and exponential equations

Given

x^2-2x-3=(x-p)^2-q

, state the values of

p

and

q

Answer this when you sit the paper.

Question 1103

[3 marks]Completing the square, quadratic graphs and exponential equations

For

y=x^2-2x-3

, state the coordinates of the turning point and of both

x

-intercepts.

Answer this when you sit the paper.

Question 1104

[2 marks]Completing the square, quadratic graphs and exponential equations

State the range of values of

x

for which

y=x^2-2x-3

is negative.

Answer this when you sit the paper.

Question 1105

[3 marks]Completing the square, quadratic graphs and exponential equations

Given

y=3^x

, write the equation

3^{2x}+\dfrac{1}{3^{-x}}=2

as a quadratic equation in

y

equal to zero.

Answer this when you sit the paper.

Question 1106

[2 marks]Completing the square, quadratic graphs and exponential equations

Solve the equation

3^{2x}+\dfrac{1}{3^{-x}}=2

Answer this when you sit the paper.

Question 1107

[2 marks]Completing the square, quadratic graphs and exponential equations

Solving

3^{2x}+\dfrac{1}{3^{-x}}=2

through the substitution

y=3^x

gives

y=-2

y=1

. Why is

y=-2

discarded?

ABecause $3^x$ is positive for every real value of $x$ , so $y$ can never take a negative value at all.
BBecause a quadratic equation with a negative constant term can only ever be satisfied by its positive root.
CBecause $y=-2$ would make $\dfrac{1}{3^{-x}}$ undefined, since a reciprocal cannot be taken of a negative number.
DBecause the substitution $y=3^x$ was only stated to hold for values of $x$ that are greater than or equal to zero.

Question 1201

[3 marks]Determinants, inverse matrices and matrix products

Expand the determinant of

N=\begin{pmatrix}1&1&1\\x+1&1&x\\x&x+1&1\end{pmatrix}

as a simplified expression in

x

Answer this when you sit the paper.

Question 1202

[2 marks]Determinants, inverse matrices and matrix products

Given that the determinant of a matrix simplifies to

x^2-x+1

and equals

7

, with

x>0

, find

x

Answer this when you sit the paper.

Question 1203

[3 marks]Determinants, inverse matrices and matrix products

For

A=\begin{pmatrix}1&1&1\\1&-1&2\\2&1&-1\end{pmatrix}

and

B=\begin{pmatrix}0&2&2\\-2&0&-1\\1&-1&-1\end{pmatrix}

, give the first row of

AB

Answer this when you sit the paper.

Question 1204

[2 marks]Determinants, inverse matrices and matrix products

Find the determinant of

A=\begin{pmatrix}1&1&1\\1&-1&2\\2&1&-1\end{pmatrix}

Answer this when you sit the paper.

Question 1205

[2 marks]Determinants, inverse matrices and matrix products

Find the determinant of

B=\begin{pmatrix}0&2&2\\-2&0&-1\\1&-1&-1\end{pmatrix}

Answer this when you sit the paper.

Question 1206

[2 marks]Determinants, inverse matrices and matrix products

Matrices

A

and

B

have determinants

7

and

-2

. State the determinant of

AB

Answer this when you sit the paper.

Question 1207

[2 marks]Determinants, inverse matrices and matrix products

The inverse of a

3\times3

matrix

A

\dfrac{1}{\det A}

multiplied by which matrix?

AThe matrix $A$ itself with every entry replaced by its own reciprocal, taken one position at a time in order.
BThe transpose of $A$ , formed by writing the rows of the original matrix down as the columns of the new one.
CThe matrix of cofactors of $A$ , used exactly as it stands with no further rearrangement of any one of its entries at all.
DThe adjoint of $A$ , which is the transpose of the matrix of cofactors, so rows and columns are exchanged first.

Pure Mathematics Paper 2 November 2024

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 504

Question 505

Question 601

Question 602

Question 603

Question 604

Question 605

Question 606

Question 607

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 1107

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1207

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The answers, and why they are the answers

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