ZIMSEC A Level Pure Mathematics Paper 1 November 2024

Question 101

[2 marks]Indices

Simplify

\left(x^2\right)^{\frac13}\times\left(x^{-\frac12}\right)^3

, giving your answer as a single power of

x

Answer this when you sit the paper.

Question 102

[2 marks]Indices

Simplify

\dfrac{\left(x^2\right)^{\frac13}\times\left(x^{-\frac12}\right)^3}{x^{-\frac16}\times x^{-\frac23}}

Answer this when you sit the paper.

Question 201

[3 marks]Geometric progressions

A geometric progression has first term

a

, where

a\ne0

, and common ratio

r

, where

0<r<1

. Its sum to infinity is double the sum of its first eight terms. Find the value of

r^4

Answer this when you sit the paper.

Question 202

[2 marks]Geometric progressions

A geometric progression has first term

a

and common ratio

r

, where

0<r<1

, and

r^8=\dfrac12

. Given that the 9th term is 20, find the exact value of

a

Answer this when you sit the paper.

Question 301

[3 marks]Variation

Y

varies jointly as the square root of

x

and inversely as the cube of

z

. Given that

Y=16

when

x=9

and

z=\dfrac12

, express

Y

in terms of

x

and

z

Answer this when you sit the paper.

Question 302

[2 marks]Variation

Given that

Y=\dfrac{2\sqrt{x}}{3z^3}

, find the value of

z

when

Y=2.5

and

x=900

Answer this when you sit the paper.

Question 401

[2 marks]Partial fractions

The expression

\dfrac{3-x+6x^2}{(1-x)(2+x)(1+2x)}

is written as

\dfrac{A}{1-x}+\dfrac{B}{2+x}+\dfrac{C}{1+2x}

. Find the value of

A

Answer this when you sit the paper.

Question 402

[3 marks]Partial fractions

Express

\dfrac{3-x+6x^2}{(1-x)(2+x)(1+2x)}

in partial fractions.

Answer this when you sit the paper.

Question 501

[3 marks]Polynomials

The polynomial

h(x)=x^4+3x^3+ax+3

is divisible by

x^2-x+1

. Find the value of

a

Answer this when you sit the paper.

Question 502

[2 marks]Polynomials

The polynomial

h(x)=x^4+3x^3+x+3

is divisible by

x^2-x+1

. Factorise

h(x)

completely.

Answer this when you sit the paper.

Question 601

[3 marks]Modulus inequalities

Squaring both sides of the inequality

3\left|x-2\right|>\left|2x-1\right|

gives a quadratic inequality. Write it in the form

ax^2+bx+c>0

Answer this when you sit the paper.

Question 602

[3 marks]Modulus inequalities

Solve the inequality

3\left|x-2\right|>\left|2x-1\right|

Answer this when you sit the paper.

Question 701

[3 marks]Vectors

The position vectors

\overrightarrow{OM}=2\mathbf{i}+3\mathbf{j}-4\mathbf{k}

and

\overrightarrow{ON}=-\mathbf{i}-2\mathbf{j}+6\mathbf{k}

are given. Find

\cos M\hat{O}N

, correct to 2 significant figures.

Answer this when you sit the paper.

Question 702

[3 marks]Vectors

The position vectors

\overrightarrow{OM}=2\mathbf{i}+3\mathbf{j}-4\mathbf{k}

and

\overrightarrow{ON}=-\mathbf{i}-2\mathbf{j}+6\mathbf{k}

are given. Find the unit vector in the direction of

\overrightarrow{MN}

Answer this when you sit the paper.

Question 801

[3 marks]Differential equations

The gradient of a curve at the point

(x,y)

is given by

\dfrac{dy}{dx}=4xy^2

. Find the general solution of this differential equation.

Answer this when you sit the paper.

Question 802

[3 marks]Differential equations

The gradient of a curve at the point

(x,y)

is given by

\dfrac{dy}{dx}=4xy^2

, and the curve passes through the point where

x=2

and

y=4

. Find the particular solution.

Answer this when you sit the paper.

Question 901

[2 marks]Proof by induction

A proof by induction is to show that

11^n-4^n

is a multiple of 7 for all

n\in\mathbb{Z}^+

. Find the value of

11^n-4^n

when

n=1

, which establishes the base case.

Answer this when you sit the paper.

Question 902

[3 marks]Proof by induction

In a proof by induction that

11^n-4^n

is a multiple of 7, it is assumed that

11^k-4^k=7m

for some integer

m

. Show that

11^{k+1}-4^{k+1}

is a multiple of 7 by writing it in the form

7\times(\text{an integer})

Answer this when you sit the paper.

Question 903

[1 marks]Proof by induction

A proof by induction has shown that a statement is true for

n=1

, and that whenever it is true for

n=k

it is also true for

n=k+1

. What conclusion follows?

ANo conclusion can be drawn without checking $n=2$ separately
BThe statement is true for all positive integers $n$
CThe statement is true only for $n=1$ and $n=2$
DThe statement is true for all real numbers $n$

Question 1001

[3 marks]Implicit differentiation

Given the equation

x^2+y^2-3x^2y+5y+2x=0

, find

\dfrac{dy}{dx}

in terms of

x

and

y

Answer this when you sit the paper.

Question 1002

[2 marks]Implicit differentiation

For the equation

x^2+y^2-3x^2y+5y+2x=0

the gradient is

\dfrac{dy}{dx}=\dfrac{6xy-2x-2}{2y-3x^2+5}

. Evaluate this gradient at the point

(1;3)

Answer this when you sit the paper.

Question 1003

[2 marks]Implicit differentiation

A curve has gradient

\dfrac{7}{4}

at the point

(1;3)

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

ax+by+c=0

Answer this when you sit the paper.

Question 1101

[3 marks]Inverse functions

The function

f(x)=x^2-6x+8

is defined for

0\le x\le3

. Find

f^{-1}(x)

Answer this when you sit the paper.

Question 1102

[2 marks]Inverse functions

The function

f(x)=x^2-6x+8

is defined for

0\le x\le3

. Find the range of

f

Answer this when you sit the paper.

Question 1103

[1 marks]Inverse functions

The function

f(x)=x^2-6x+8

is defined for

0\le x\le3

, and its range is

-1\le f(x)\le8

. State the domain of

f^{-1}(x)

Answer this when you sit the paper.

Question 1104

[2 marks]Inverse functions

The graphs of a function

f

and its inverse

f^{-1}

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

AThey are reflections of each other in the line $y=x$
BThey are reflections of each other in the $x$ axis
CThey are reflections of each other in the $y$ axis
DThey are translations of each other parallel to the $y$ axis

Question 1201

[3 marks]Binomial expansion

Expand

\left(1-\dfrac{x}{3}\right)^{-\frac23}

in ascending powers of

x

up to and including the term in

x^3

, simplifying the coefficients.

Answer this when you sit the paper.

Question 1202

[3 marks]Binomial expansion

Expand

\dfrac{2x+3}{\sqrt[3]{\left(1-\frac{x}{3}\right)^2}}

in ascending powers of

x

up to and including the term in

x^3

, simplifying the coefficients.

Answer this when you sit the paper.

Question 1203

[2 marks]Binomial expansion

State the set of values of

x

for which the binomial expansion of

\left(1-\dfrac{x}{3}\right)^{-\frac23}

is valid.

Answer this when you sit the paper.

Question 1301

[2 marks]Newton-Raphson method

Let

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

. Find the values of

f(1)

and

f(2)

, each correct to 4 decimal places.

Answer this when you sit the paper.

Question 1302

[2 marks]Newton-Raphson method

For

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

, the values

f(1)=-0.8537

and

f(2)=0.1463

have been found. What does this show about the equation

f(x)=0

A $f$ is continuous and changes sign, so a root lies between 1 and 2
B $f$ has a minimum between 1 and 2
CThere is no root of $f(x)=0$ between 1 and 2
DThere are exactly two roots of $f(x)=0$ between 1 and 2

Question 1303

[2 marks]Newton-Raphson method

For

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

, write down an expression for

f'(x)

Answer this when you sit the paper.

Question 1304

[3 marks]Newton-Raphson method

Using

x_0=1.5

as the initial estimate, apply the Newton-Raphson method three times to the equation

\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}-5=0

, and give the root correct to two decimal places.

Answer this when you sit the paper.

Question 1401

[3 marks]Complex numbers

Solve the equation

z^2+4z+7=0

, giving both roots in the form

a+bi

with

b

in exact surd form.

Answer this when you sit the paper.

Question 1402

[2 marks]Complex numbers

Find the modulus of the complex number

z_2=-2-\sqrt3\,i

, giving your answer in exact form.

Answer this when you sit the paper.

Question 1403

[2 marks]Complex numbers

Find the argument of the complex number

z_2=-2-\sqrt3\,i

, giving your answer to the nearest degree.

Answer this when you sit the paper.

Question 1404

[2 marks]Complex numbers

Write the complex number

z_2=-2-\sqrt3\,i

in polar form, using a modulus in exact form and an argument to the nearest degree.

Answer this when you sit the paper.

Question 1405

[2 marks]Complex numbers

The complex numbers

z_1=-2+\sqrt3\,i

and

z_2=-2-\sqrt3\,i

are given. Find

z_1z_2

Answer this when you sit the paper.

Question 1406

[1 marks]Complex numbers

On an Argand diagram, where is the product

z_1z_2=7

of the conjugate pair

z_1=-2+\sqrt3\,i

and

z_2=-2-\sqrt3\,i

plotted?

AAt the point $(-7;0)$ , on the real axis
BAt the point $(-2;7)$
CAt the point $(7;0)$ , on the real axis
DAt the point $(0;7)$ , on the imaginary axis

Question 1501

[3 marks]Circle geometry

A circle has centre C

(-4;5)

and the point A

(0;8)

lies on its circumference. Find the equation of the circle.

Answer this when you sit the paper.

Question 1502

[2 marks]Circle geometry

A circle has centre C

(-4;5)

and radius 5. Show that the point B

(0;2)

lies on its circumference by finding the distance CB.

Answer this when you sit the paper.

Question 1503

[3 marks]Circle geometry

A circle has centre C

(-4;5)

and the point B

(0;2)

lies on its circumference. Find the equation of the tangent to the circle at B, in the form

ax+by+c=0

Answer this when you sit the paper.

Question 1504

[2 marks]Circle geometry

The points B

(0;2)

and D

(-16;8)

are given. Calculate the coordinates of the midpoint of BD.

Answer this when you sit the paper.

Question 1505

[3 marks]Circle geometry

The points B

(0;2)

and D

(-16;8)

are given, and the midpoint of BD is

(-8;5)

. Find the equation of the perpendicular bisector of BD, in the form

ax+by+c=0

Answer this when you sit the paper.

Question 1601

[2 marks]Linear law, logarithmic graphs

An experiment gives values of

x

and

y

believed to satisfy the law

y=ab^{-x}

. Which plot gives a straight line, allowing

a

and

b

to be found from its gradient and intercept?

A $y$ against $\lg x$
B $\lg y$ against $\lg x$
C $y$ against $x^2$
D $\lg y$ against $x$

Question 1602

[3 marks]Linear law, logarithmic graphs

The table shows values of

x

and

y

from an experiment satisfying the law

y=ab^{-x}

, where

x

takes the values 1.0, 1.5, 2.0, 2.5 and 3.0 and

y

takes the values 4.0, 5.7, 8.0, 11.3 and 16.0 in that order. Taking logarithms to base 10, find the gradient of the straight line obtained when

\lg y

is plotted against

x

Answer this when you sit the paper.

Question 1603

[3 marks]Linear law, logarithmic graphs

An experiment satisfies the law

y=ab^{-x}

. Plotting

\lg y

against

x

gives a straight line of gradient 0.301 and intercept 0.301 on the

\lg y

axis. Find the values of

a

and

b

Answer this when you sit the paper.

Question 1604

[2 marks]Linear law, logarithmic graphs

An experiment satisfies the law

y=ab^{-x}

with

a=2

and

b=0.5

. Use the law to find the value of

y

when

x=4.0

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2024

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 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1101

Question 1102

Question 1103

Question 1104

Question 1201

Question 1202

Question 1203

Question 1301

Question 1302

Question 1303

Question 1304

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1601

Question 1602

Question 1603

Question 1604

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