ZIMSEC A Level Pure Mathematics Paper 1 June 2024

Question 101

[2 marks]Sequences

A sequence is defined by

U_{n+1}=3U_n+1

, with

U_1=0

. Write down the first four terms of the sequence.

Answer this when you sit the paper.

Question 102

[1 marks]Sequences

A sequence is defined by

U_{n+1}=3U_n+1

, with

U_1=0

, so its terms are

0

1

4

13

40

, and so on. Which statement best describes the behaviour of the sequence?

AIt converges to the limit $-\tfrac12$
BIt oscillates between two fixed values
CIt is increasing and diverges, the terms growing without bound
DIt is decreasing and converges to zero

Question 103

[2 marks]Sequences

A sequence is defined by

U_{n+1}=3U_n+1

. Express

U_3

in terms of

U_1

Answer this when you sit the paper.

Question 201

[1 marks]Exponentials and logarithms

The variables

x

and

y

are connected by

y=ae^{bx}

, where

a

and

b

are constants. When

\ln y

is plotted against

x

a straight line is obtained. What is the gradient of that line?

A $b$
B $a$
C $\ln a$
D $ab$

Question 202

[2 marks]Exponentials and logarithms

The variables

x

and

y

are connected by

y=ae^{bx}

. When

\ln y

is plotted against

x

, a straight line is obtained passing through

(0;\ln 2)

and

(3;\ln 5)

. Find the exact value of

a

Answer this when you sit the paper.

Question 203

[3 marks]Exponentials and logarithms

The variables

x

and

y

are connected by

y=ae^{bx}

. When

\ln y

is plotted against

x

, a straight line is obtained passing through

(0;\ln 2)

and

(3;\ln 5)

. Find the exact value of

b

Answer this when you sit the paper.

Question 301

[2 marks]Partial fractions

The expression

\dfrac{2x-3}{x^2\left(x^2-4\right)}

is written as

\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x-2}+\dfrac{D}{x+2}

. Find the value of

B

Answer this when you sit the paper.

Question 302

[2 marks]Partial fractions

The expression

\dfrac{2x-3}{x^2\left(x^2-4\right)}

is written as

\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x-2}+\dfrac{D}{x+2}

. Find the value of

C

Answer this when you sit the paper.

Question 303

[3 marks]Partial fractions

Express

\dfrac{2x-3}{x^2\left(x^2-4\right)}

in partial fractions.

Answer this when you sit the paper.

Question 401

[2 marks]Exponential equations

The substitution

u=e^{2x}

turns the equation

\dfrac{e^{2x}+e^{-2x}}{5}=1

into a quadratic in

u

. Write down that quadratic equation.

Answer this when you sit the paper.

Question 402

[2 marks]Exponential equations

Solve the equation

u^2-5u+1=0

, giving both roots in exact surd form.

Answer this when you sit the paper.

Question 403

[3 marks]Exponential equations

Solve the equation

\dfrac{e^{2x}+e^{-2x}}{5}=1

, giving the answer in exact form.

Answer this when you sit the paper.

Question 501

[2 marks]Quadratic inequalities

Expand and simplify the inequality

(x-3)^2>2x+1

into the form

x^2+px+q>0

Answer this when you sit the paper.

Question 502

[2 marks]Quadratic inequalities

Find, in exact surd form, the roots of the equation

x^2-8x+8=0

Answer this when you sit the paper.

Question 503

[3 marks]Quadratic inequalities

Find the set of values of

x

for which

(x-3)^2>2x+1

, giving your answer in exact form.

Answer this when you sit the paper.

Question 601

[2 marks]Circular measure

A chord subtends an angle of

2\theta

radians at the centre of a circle of radius

r

. Write down an expression, in terms of

r

and

\theta

, for the area of the minor segment cut off by the chord.

Answer this when you sit the paper.

Question 602

[3 marks]Circular measure

A chord subtends an angle of

2\theta

radians at the centre of a circle of radius

r

. The area of the minor segment is

\dfrac16

of the area of the major segment. Find, in terms of

\pi

, the exact value of

2\theta-\sin2\theta

Answer this when you sit the paper.

Question 603

[2 marks]Circular measure

For a chord subtending

2\theta

radians at the centre of a circle, the condition that the minor segment is

\dfrac16

of the major segment leads to

2\theta-\sin2\theta=\dfrac{2\pi}{7}

. Which of these is an equivalent form of that result?

A $\sin2\theta=\dfrac{2\pi}{7}-2\theta$
B $\sin2\theta=\dfrac{2}{7}(7\theta-\pi)$
C $\sin2\theta=\dfrac{2}{7}(7\theta+\pi)$
D $\sin2\theta=\dfrac{7}{2}(2\theta-\pi)$

Question 701

[3 marks]Binomial expansion

Expand

(1-2x)^{\frac12}

up to and including the term in

x^2

, simplifying the coefficients.

Answer this when you sit the paper.

Question 702

[1 marks]Binomial expansion

State the values of

x

for which the binomial expansion of

(1-2x)^{\frac12}

is valid.

Answer this when you sit the paper.

Question 703

[2 marks]Binomial expansion

Using the expansion

(1-2x)^{\frac12}\approx1-x-\dfrac12x^2

, find the value it gives when

x=\dfrac19

, as a fraction in its lowest terms.

Answer this when you sit the paper.

Question 704

[2 marks]Binomial expansion

Putting

x=\dfrac19

in the expansion

(1-2x)^{\frac12}\approx1-x-\dfrac12x^2

gives

\dfrac{\sqrt7}{3}\approx\dfrac{143}{162}

. Write down the resulting approximation for

\sqrt7

as a fraction.

Answer this when you sit the paper.

Question 801

[1 marks]Coordinate geometry

The equation of line

l

y+3x-9=0

. Write down the gradient of

l

Answer this when you sit the paper.

Question 802

[3 marks]Coordinate geometry

The equation of line

l

y+3x-9=0

. Find the equation of the line

m

that is perpendicular to

l

and passes through the point A(1;6), giving your answer in the form

ax+by+c=0

Answer this when you sit the paper.

Question 803

[2 marks]Coordinate geometry

The equation of line

l

y+3x-9=0

. Find the equation of the line parallel to

l

that passes through the point B(5;6), giving your answer in the form

ax+by+c=0

Answer this when you sit the paper.

Question 804

[3 marks]Coordinate geometry

Find the perpendicular distance of the point B(5;6) from the line

x-3y+17=0

, giving your answer in exact form.

Answer this when you sit the paper.

Question 901

[2 marks]Completing the square, range of a function

Express

x^2+6x+5

in the form

(x+b)^2+c

, where

b

and

c

are constants.

Answer this when you sit the paper.

Question 902

[1 marks]Completing the square, range of a function

Given that

x^2+6x+5=(x+3)^2-4

, state the turning point of the graph of

y=x^2+6x+5

Answer this when you sit the paper.

Question 903

[2 marks]Completing the square, range of a function

Setting

y=\dfrac{1}{x^2+6x+5}

and rearranging gives the quadratic

yx^2+6yx+(5y-1)=0

x

. Write down the discriminant condition for

x

to be real, simplified to the form

ay^2+by\ge0

Answer this when you sit the paper.

Question 904

[3 marks]Completing the square, range of a function

Find the set of values taken by

y=\dfrac{1}{x^2+6x+5}

for real values of

x

Answer this when you sit the paper.

Question 1001

[3 marks]Complex numbers, polynomials

Given that

f(x)=x^4+x^3+23x^2+25x-50

, evaluate

f(-5i)

, and hence state whether

-5i

is a root of

f(x)=0

Answer this when you sit the paper.

Question 1002

[2 marks]Complex numbers, polynomials

The polynomial

f(x)=x^4+x^3+23x^2+25x-50

has real coefficients, and

-5i

is a root of

f(x)=0

. Which of the following must therefore also be a root?

A $-5$
B $5$
C $-\dfrac{1}{5}i$
D $5i$

Question 1003

[2 marks]Complex numbers, polynomials

The polynomial

f(x)=x^4+x^3+23x^2+25x-50

has

x^2+25

as a factor. Find the other quadratic factor.

Answer this when you sit the paper.

Question 1004

[3 marks]Complex numbers, polynomials

Solve the equation

x^4+x^3+23x^2+25x-50=0

completely, given that

-5i

is one of its roots.

Answer this when you sit the paper.

Question 1101

[2 marks]Functions

The functions

g

and

h

are defined by

g:x\to 4x+1

and

h:x\to\dfrac{1}{x-3}

x\ne3

. Find

gh(x)

, giving your answer as a single fraction.

Answer this when you sit the paper.

Question 1102

[1 marks]Functions

The functions

g

and

h

are defined by

g:x\to 4x+1

and

h:x\to\dfrac{1}{x-3}

x\ne3

, and

gh(x)=\dfrac{x+1}{x-3}

. State the domain of

gh

Answer this when you sit the paper.

Question 1103

[3 marks]Functions

The function

f

is defined by

f:x\to x^2+2x

for

x\in\mathbb{R}

x\ge-1

. Find

f^{-1}(x)

Answer this when you sit the paper.

Question 1104

[2 marks]Functions

The function

g

is defined by

g:x\to 4x+1

for

-2\le x\le3

. State the coordinates of the two end points of the graph of

y=\left|g(x)\right|

Answer this when you sit the paper.

Question 1105

[1 marks]Functions

The function

g

is defined by

g:x\to 4x+1

for

-2\le x\le3

. State the range of

\left|g(x)\right|

Answer this when you sit the paper.

Question 1201

[2 marks]Trigonometric identities and equations

Write

\dfrac{\cos\theta}{1-\cos\theta}-\dfrac{\cos\theta}{1+\cos\theta}

as a single fraction in terms of

\cos\theta

and

\sin\theta

Answer this when you sit the paper.

Question 1202

[2 marks]Trigonometric identities and equations

The expression

\dfrac{2\cos^2\theta}{\sin^2\theta}

is identical to which of the following?

A $\cot^2 2\theta$
B $2\operatorname{cosec}^2\theta$
C $2\cot^2\theta$
D $2\tan^2\theta$

Question 1203

[2 marks]Trigonometric identities and equations

Given the identity

\dfrac{\cos\theta}{1-\cos\theta}-\dfrac{\cos\theta}{1+\cos\theta}\equiv2\cot^2\theta

, the equation

\dfrac{\cos\theta}{1-\cos\theta}-\dfrac{\cos\theta}{1+\cos\theta}=1

reduces to

2\cot^2\theta=1

. Find the exact values of

\tan\theta

Answer this when you sit the paper.

Question 1204

[3 marks]Trigonometric identities and equations

Solve the equation

\dfrac{\cos\theta}{1-\cos\theta}-\dfrac{\cos\theta}{1+\cos\theta}=1

for

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

, giving all answers to the nearest degree.

Answer this when you sit the paper.

Question 1301

[2 marks]Roots of equations, iteration

To show graphically that

x^3+x-4=0

has exactly one real root, the equation is rearranged and two graphs are sketched on the same axes. Which pair of graphs makes the argument work most simply?

A $y=x^3$ and $y=4-x$
B $y=x^3+x$ and $y=x-4$
C $y=x^2$ and $y=4-x$
D $y=x^3-4$ and $y=-x^2$

Question 1302

[2 marks]Roots of equations, iteration

Let

f(x)=x^3+x-4

. Find the values of

f(1)

and

f(1.5)

Answer this when you sit the paper.

Question 1303

[1 marks]Roots of equations, iteration

For

f(x)=x^3+x-4

, the values

f(1)=-2

and

f(1.5)=0.875

have been calculated. What does this pair of values show?

A $f(x)=0$ has two roots between 1 and 1.5
B $f$ changes sign, so a root of $f(x)=0$ lies between 1 and 1.5
C $f$ has a maximum between 1 and 1.5
DThere is no root of $f(x)=0$ between 1 and 1.5

Question 1304

[2 marks]Roots of equations, iteration

Using the iteration

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

with

x_1=1.37

, find

x_2

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1305

[3 marks]Roots of equations, iteration

Using the iteration

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

with

x_1=1.37

, approximate the root of

x^3+x-4=0

correct to 2 decimal places.

Answer this when you sit the paper.

Question 1401

[3 marks]Implicit differentiation

The equation of a curve is

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

. Use implicit differentiation to find

\dfrac{dy}{dx}

in terms of

x

and

y

Answer this when you sit the paper.

Question 1402

[3 marks]Implicit differentiation

The equation of a curve is

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

. Find the value of

y

on the curve where

x=1

and

y>0

Answer this when you sit the paper.

Question 1403

[2 marks]Implicit differentiation

For the curve

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

the gradient is given by

\dfrac{dy}{dx}=\dfrac{3y-4x}{2y-3x}

. Find the gradient of the curve at the point

(1;4)

Answer this when you sit the paper.

Question 1404

[3 marks]Implicit differentiation

The curve

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

has gradient

\dfrac{8}{5}

at the point

(1;4)

. Find the equation of the tangent to the curve at that point, in the form

ax+by+c=0

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2024

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

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 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1401

Question 1402

Question 1403

Question 1404

More sittings of this paper

The answers, and why they are the answers

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