ZIMSEC A Level Pure Mathematics Paper 1 June 2023

Question 101

[3 marks]Sequences

A sequence

U_n

is defined by

U_n=(-1)^{n+1}+2

. Find the first 3 terms of the sequence.

Answer this when you sit the paper.

Question 102

[1 marks]Sequences

A sequence

U_n

is defined by

U_n=(-1)^{n+1}+2

, whose first terms are

3,\ 1,\ 3,\ 1,\dots

. State the behaviour of the sequence.

AIt oscillates between 3 and 1, with period 2.
BIt increases without limit as $n$ increases, growing steadily larger and larger.
CIt decreases steadily towards 0 as $n$ increases, never quite reaching zero.
DIt converges to the limit 2 as $n$ becomes large, because the terms settle down.

Question 201

[1 marks]Circular measure

A and B are two points on the circumference of a circle centre O and radius

r

. The minor arc AB subtends an angle of

2\theta

radians at O. Write down the area of the minor sector AOB in terms of

r

and

\theta

Answer this when you sit the paper.

Question 202

[2 marks]Circular measure

A and B are two points on the circumference of a circle centre O and radius

r

, and the minor arc AB subtends an angle of

2\theta

radians at O. Which expression gives the area of the minor segment cut off by the chord AB?

A $\tfrac12r^2\theta-r^2\sin2\theta$
B $r^2\theta-r^2\sin\theta$
C $r^2\theta-\tfrac12r^2\sin2\theta$
D $r^2\theta+\tfrac12r^2\sin2\theta$

Question 301

[2 marks]Variation

A variable

p

is inversely proportional to the square of

2q+1

, so that

p=\dfrac{k}{(2q+1)^2}

. Given that

p=\dfrac34

when

q=2

, find the value of

k

Answer this when you sit the paper.

Question 302

[2 marks]Variation

A variable

p

is inversely proportional to the square of

2q+1

, and

p=\dfrac34

when

q=2

. Find the positive value of

q

for which

p=\dfrac13

Answer this when you sit the paper.

Question 401

[3 marks]Exponential inequalities

Solve the inequality

(1.05)^{n-4}<60

, giving the bound on

n

correct to 3 significant figures.

Answer this when you sit the paper.

Question 402

[1 marks]Exponential inequalities

Given that

(1.05)^{n-4}<60

, state the largest integral value of

n

Answer this when you sit the paper.

Question 501

[1 marks]Simultaneous equations

Substituting

y=3-2x

from

2x+y=3

into

xy=1

produces a quadratic in

x

. Write that quadratic in the form

2x^2+bx+c=0

Answer this when you sit the paper.

Question 502

[1 marks]Simultaneous equations

The simultaneous equations

xy=1

and

2x+y=3

reduce to

2x^2-3x+1=0

. Find the two values of

x

Answer this when you sit the paper.

Question 503

[2 marks]Simultaneous equations

Solve the simultaneous equations

xy=1

and

2x+y=3

, giving both pairs

(x,y)

Answer this when you sit the paper.

Question 601

[2 marks]Differentiation

Differentiate

e^{2x}\sin 3x

with respect to

x

Answer this when you sit the paper.

Question 602

[2 marks]Differentiation

Differentiate

\ln(1+x^2)

with respect to

x

Answer this when you sit the paper.

Question 701

[2 marks]Parametric equations

A curve C has parametric equations

x=2t+\dfrac{1}{2t}

and

y=2t-\dfrac{1}{2t}

, where

t

is a parameter. Find

x^2

in terms of

t

Answer this when you sit the paper.

Question 702

[2 marks]Parametric equations

A curve C has parametric equations

x=2t+\dfrac{1}{2t}

and

y=2t-\dfrac{1}{2t}

, where

t

is a parameter. Find

y^2

in terms of

t

Answer this when you sit the paper.

Question 703

[2 marks]Parametric equations

A curve C has parametric equations

x=2t+\dfrac{1}{2t}

and

y=2t-\dfrac{1}{2t}

. By evaluating

x^2-y^2

, find the cartesian equation of C.

Answer this when you sit the paper.

Question 801

[2 marks]Indices

Express

\dfrac{1}{(\sqrt{x})^{\frac43}}

in the form

x^n

Answer this when you sit the paper.

Question 802

[3 marks]Indices

Solve

\dfrac{2^{1,5}}{2^7\times16}=2^x

Answer this when you sit the paper.

Question 901

[2 marks]Polynomials

Divide

a^3-b^3

a-b

Answer this when you sit the paper.

Question 902

[2 marks]Polynomials

Given that

x^4+x^2+x+1\equiv(x^2+A)(x^2-1)+Bx+C

, determine the numerical value of A.

Answer this when you sit the paper.

Question 903

[2 marks]Polynomials

Given that

x^4+x^2+x+1\equiv(x^2+A)(x^2-1)+Bx+C

and that

A=2

, determine the numerical values of B and C.

Answer this when you sit the paper.

Question 1001

[2 marks]Inverse functions

The function

h

is defined by

h(x)=3x+7

for

x\in\mathbb{R}

. Write down

h^{-1}(x)

in terms of

x

Answer this when you sit the paper.

Question 1002

[2 marks]Inverse functions

The function

g

is defined by

g(x)=\dfrac{6}{2x-4}

x\neq2

, for

x\in\mathbb{R}

. Write down

g^{-1}(x)

in terms of

x

Answer this when you sit the paper.

Question 1003

[1 marks]Inverse functions

For

g(x)=\dfrac{6}{2x-4}

x\neq2

, the inverse is

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

. State the value of

x

for which

g^{-1}(x)

is not defined.

Answer this when you sit the paper.

Question 1004

[2 marks]Inverse functions

The graphs of

h(x)=3x+7

and

h^{-1}(x)=\dfrac{x-7}{3}

are drawn on the same axes. What is the relationship between the two graphs?

AEach is the reflection of the other in the $y$ -axis.
BEach is the reflection of the other in the line $y=x$ .
CEach is the reflection of the other in the $x$ -axis, so one of them is upside down.
DThey are parallel straight lines of equal gradient that never meet at any point.

Question 1101

[2 marks]Proof by induction

In proving by induction that

\displaystyle\sum_{r=1}^{n}ap^{r-1}=\frac{a(p^n-1)}{p-1}

, the base case is

n=1

. State the common value that both sides take when

n=1

Answer this when you sit the paper.

Question 1102

[3 marks]Proof by induction

Assume that

\displaystyle\sum_{r=1}^{k}ap^{r-1}=\frac{a(p^k-1)}{p-1}

. Adding the next term

ap^{k}

, write

\dfrac{a(p^k-1)}{p-1}+ap^{k}

as a single fraction in its simplest form.

Answer this when you sit the paper.

Question 1201

[2 marks]Matrices

Given the matrix

A=\begin{pmatrix}1&2&5\\2&-5&3\\4&p&7\end{pmatrix}

, find

\det A

in terms of

p

Answer this when you sit the paper.

Question 1202

[2 marks]Matrices

Given the matrix

A=\begin{pmatrix}1&2&5\\2&-5&3\\4&p&7\end{pmatrix}

, find the value of

p

for which

\det A=54

Answer this when you sit the paper.

Question 1203

[2 marks]Matrices

Find the cofactor of the element in the first row and first column of

\begin{pmatrix}1&2&5\\2&-5&3\\4&-1&7\end{pmatrix}

Answer this when you sit the paper.

Question 1204

[2 marks]Matrices

The matrix

A=\begin{pmatrix}1&2&5\\2&-5&3\\4&-1&7\end{pmatrix}

has determinant 54, and the cofactor of its element in the first row and first column is

-32

. Find the element in the first row and first column of

A^{-1}

, as a fraction in its lowest terms.

Answer this when you sit the paper.

Question 1301

[2 marks]Taylor series

Let

f(x)=\dfrac1x

. Find

f'''(a)

, the third derivative evaluated at

x=a

Answer this when you sit the paper.

Question 1302

[3 marks]Taylor series

The function

f(x)=\dfrac1x

is expanded by Taylor series in ascending powers of

(x-a)

. State the term in

(x-a)^3

Answer this when you sit the paper.

Question 1303

[1 marks]Taylor series

The function

f(x)=\dfrac1x

is expanded by Taylor series in ascending powers of

(x-a)

. State the coefficient of

(x-a)^2

Answer this when you sit the paper.

Question 1304

[2 marks]Taylor series

Using the expansion

\dfrac1x=\dfrac1a-\dfrac{(x-a)}{a^2}+\dfrac{(x-a)^2}{a^3}-\dfrac{(x-a)^3}{a^4}

with

a=1

, evaluate

\dfrac{1}{1,01}

correct to 3 significant figures.

Answer this when you sit the paper.

Question 1401

[1 marks]Trigonometry

Express

4\cos\theta-3\sin\theta

in the form

R\sin(\theta-\alpha)

, where

R>0

. Find the value of

R

Answer this when you sit the paper.

Question 1402

[2 marks]Trigonometry

Express

4\cos\theta-3\sin\theta

in the form

R\sin(\theta-\alpha)

, where

R>0

and

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

. Find

\alpha

to the nearest degree.

Answer this when you sit the paper.

Question 1403

[3 marks]Trigonometry

Given that

4\cos\theta-3\sin\theta=5\sin(\theta-233,13^\circ)

, solve

4\cos\theta-3\sin\theta=3

for

0^\circ\leq\theta\leq360^\circ

, giving your answers correct to 1 decimal place.

Answer this when you sit the paper.

Question 1404

[2 marks]Trigonometry

Given that

4\cos\theta-3\sin\theta=5\sin(\theta-233^\circ)

, find the least value of

\dfrac{1}{4\cos\theta-3\sin\theta+9}

Answer this when you sit the paper.

Question 1501

[2 marks]Complex numbers

Given that

z_1=3+4i

and

z_2=1+i

, find

z_1-z_2

Answer this when you sit the paper.

Question 1502

[2 marks]Complex numbers

Given that

z_1=3+4i

and

z_2=1+i

, find the argument of

z_1-z_2

in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1503

[3 marks]Complex numbers

Solve the equation

z^2-4z+53=0

, expressing the roots in the form

a+bi

, where

a,b\in\mathbb{R}

Answer this when you sit the paper.

Question 1504

[1 marks]Complex numbers

State the sum of the roots of

z^2-4z+53=0

Answer this when you sit the paper.

Question 1505

[1 marks]Complex numbers

State the product of the roots of

z^2-4z+53=0

Answer this when you sit the paper.

Question 1601

[3 marks]Vectors

The points A, B and C have position vectors

2\mathbf{i}-\mathbf{k}

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

and

5\mathbf{i}-3\mathbf{j}+4\mathbf{k}

respectively. Calculate the angle ABC correct to the nearest

0,1^\circ

Answer this when you sit the paper.

Question 1602

[3 marks]Vectors

The point A has position vector

2\mathbf{i}-\mathbf{k}

and the point C has position vector

5\mathbf{i}-3\mathbf{j}+4\mathbf{k}

. N lies on AC so that

\overrightarrow{AN}:\overrightarrow{NC}=1:2

. Find the position vector of N.

Answer this when you sit the paper.

Question 1603

[3 marks]Vectors

The point N has position vector

3\mathbf{i}-\mathbf{j}+\tfrac23\mathbf{k}

and the point C has position vector

5\mathbf{i}-3\mathbf{j}+4\mathbf{k}

. Find the unit vector in the direction of

\overrightarrow{NC}

Answer this when you sit the paper.

Question 1604

[1 marks]Vectors

The point N has position vector

3\mathbf{i}-\mathbf{j}+\tfrac23\mathbf{k}

and the point C has position vector

5\mathbf{i}-3\mathbf{j}+4\mathbf{k}

. Find

\left|\overrightarrow{NC}\right|

correct to 2 decimal places.

Answer this when you sit the paper.

Question 1701

[1 marks]Coordinate geometry

State the gradient of the straight line

2y-x-5=0

Answer this when you sit the paper.

Question 1702

[3 marks]Coordinate geometry

The straight line

\ell

has equation

2y-x-5=0

. Another straight line

m

passes through the point

P(-2,6)

and is perpendicular to

\ell

. Find the equation of

m

, giving the answer in the form

ax+by+c=0

Answer this when you sit the paper.

Question 1703

[3 marks]Coordinate geometry

Find the co-ordinates of the point of intersection of the lines

2y-x-5=0

and

2x+y-2=0

Answer this when you sit the paper.

Question 1704

[2 marks]Coordinate geometry

Find the perpendicular distance from the point

P(-2,6)

to the line

2y-x-5=0

, correct to 2 decimal places.

Answer this when you sit the paper.

Question 1705

[2 marks]Coordinate geometry

P is the point

(-2,6)

, and the points

Q(7;6)

and

R(-7;-1)

lie on the line

2y-x-5=0

, whose perpendicular distance from P is

\dfrac{9}{\sqrt5}

. Find the area of the triangle PQR.

Answer this when you sit the paper.

Question 1801

[1 marks]Newton-Raphson, root location

The graphs of

y=e^{-x}

and

y=\sin x

are drawn on the same axes for

0\leq x\leq\pi

. State the number of roots of the equation

e^{-x}-\sin x=0

in this interval.

Answer this when you sit the paper.

Question 1802

[1 marks]Newton-Raphson, root location

For

f(x)=e^{-x}-\sin x

, write down

f'(x)

Answer this when you sit the paper.

Question 1803

[2 marks]Newton-Raphson, root location

Let

f(x)=e^{-x}-\sin x

. Find

f\left(\dfrac{\pi}{2}\right)

correct to 3 decimal places, and hence say what it shows when taken with

f(0)=1

Answer this when you sit the paper.

Question 1804

[3 marks]Newton-Raphson, root location

The Newton-Raphson method is applied to

f(x)=e^{-x}-\sin x

starting with

x_0=0,8

. Find

x_1

correct to 3 decimal places.

Answer this when you sit the paper.

Question 1805

[3 marks]Newton-Raphson, root location

Starting with

x_0=0,8

, use the Newton-Raphson method twice on

f(x)=e^{-x}-\sin x

to estimate the smallest root of

e^{-x}-\sin x=0

, giving the answer correct to three decimal places.

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2023

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 503

Question 601

Question 602

Question 701

Question 702

Question 703

Question 801

Question 802

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1201

Question 1202

Question 1203

Question 1204

Question 1301

Question 1302

Question 1303

Question 1304

Question 1401

Question 1402

Question 1403

Question 1404

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1601

Question 1602

Question 1603

Question 1604

Question 1701

Question 1702

Question 1703

Question 1704

Question 1705

Question 1801

Question 1802

Question 1803

Question 1804

Question 1805

More sittings of this paper

The answers, and why they are the answers

Sit this paper online