ZIMSEC A-Level Pure Mathematics Paper 1 November 2024 Question Paper

About this question paper

This is the ZIMSEC A-Level Pure Mathematics question paper for Paper 1 from November 2024. Use it for exam practice under timed conditions, then check your work against the marking scheme. The paper code is 6042/1. Time allowed: 3 hours. See the marking scheme.

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Extracted text

# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level

**PURE MATHEMATICS**

**PAPER 1**

**NOVEMBER 2024 SESSION**

Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Scientific calculator [Non-Programmable]

### INSTRUCTIONS TO CANDIDATES

Write your name, centre number and candidate number in the spaces provided on the answer booklet.

### INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 120 .
If a numerical answer cannot be given exactly and the accuracy required is not specified in the question, then in the case of an angle it should be given correct to the nearest degree, and in other cases it should be given correct to 2 significant figures.

Answer all questions.

## Question 1
Simplify \(\frac{\left(x^{2}\right)^{\frac{1}{3}} \times\left(x^{-\frac{1}{2}}\right)^{3}}{x^{-\frac{1}{6}} \times x^{-\frac{2}{3}}}\).

## Question 2
A geometrical progression has the first term \(a\) where \(a \neq 0\), and a common ratio \(r\) where \(0<r<1\). The sum to infinity of the progression is double the sum of the first eight terms.

Given that the \(9^{\text {th }}\) term is 20 , calculate the

### (a)

value of \(r^{4}\).

### (b)

exact value of \(a\).

## Question 3
\(Y\) varies jointly as the square root of \(x\) and inversely as the cube of \(z\). \(Y=16\) when \(x=9\) and \(z=\frac{1}{2}\).

### (a)

Express \(Y\) in terms of \(x\) and \(z\).

### (b)

Find the value of \(z\) when \(Y=2.5\) and \(x=900\).

## Question 4
Express \(\frac{3-x+6 x^{2}}{(1-x)(2+x)(1+2 x)}\) in partial fractions.

## Question 5
Given that the polynomial \(h(x)=x^{4}+3 x^{3}+a x+3\) is divisible by \(x^{2}-x+1\) where \(x \neq 0\).

### (a)

Find the value of \(a\).

### (b)

Hence factorise \(h(x)\) completely.

## Question 6
Solve the inequality \(3|x-2|>|2 x-1|\).

## Question 7
Position vectors \(\overrightarrow{\mathrm{OM}}\) and \(\overrightarrow{\mathrm{ON}}\) are such that \(\overrightarrow{\mathrm{OM}}=2 \boldsymbol{i}+3 \boldsymbol{j}-4 \boldsymbol{k}\) and \(\overrightarrow{\mathrm{ON}}=-i-2 j+6 k\).

Find

### (a)

\(\operatorname{Cos} M \hat{O} N\),

### (b)

unit vector in the direction of \(\overrightarrow{M N}\).

## Question 8
The gradient of a curve at the point \((x, y)\) is given by \(\frac{d y}{d x}=4 x y^{2}\).

Find the

### (a)

general solution of the differential equation,

### (b)

particular solution of \(dy\), given that \(x=2\) and \(y=4\).

## Question 9
Prove by induction that \(11^{n}-4^{n}\) is a multiple of 7 , for all \(n \in Z^{+}\).

## Question 10
Given the equation \(x^{2}+y^{2}-3 x^{2} y+5 y+2 x=0\), find

### (a)

\(\frac{d y}{d x}\),

### (b)

the equation of normal to the curve at the point \((1 ; 3)\).

## Question 11
The function \(f(x)=x^{2}-6 x+8\) is defined for \(0 \leq x \leq 3\).

### (a)

Find \(f^{-1}(x)\).

### (b)

State the domain of \(f^{-1}(x)\).

### (c)

Sketch the graphs of \(f(x)\) and \(f^{-1}(x)\) on the same axes.

## Question 12
### (a)

Expand \(\frac{2 x+3}{\sqrt[3]{\left(1-\frac{x}{3}\right)^{2}}}\) in ascending powers of \(x\) up to and including the term in \(x^{3}\) simplifying the coefficients.

### (b)

State the set of values of \(x\) for which the expansion is valid.

## Question 13
### (a)

Show that the equation \(\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}-5=0\) has a root between 1 and 2 .

### (b)

Using \(x_{0}=1.5\) as the initial estimate, use the Newton - Raphson method three times to estimate root of the equation to two decimal places.

## Question 14
### (a)

Solve the equation \(z^{2}+4 z+7=0\) giving the roots in the form \(z_{1}=a+\sqrt{b i}\) and \(z_{2}=-a-\sqrt{b i}\).

Find

#### (i)

\(z_{1}\) and \(z_{2}\)

#### (ii)

\(z_{2}\) in the polar form.

#### (iii)

\(z_{1} z_{2}\).

### (b)

Represent the complex numbers \(z_{1} ; z_{2}\) and \(z_{1} z_{2}\) on Argand Diagram.

## Question 15
A circle has centre \(C(-4 ; 5)\). The points \(A(0 ; 8)\) and \(D(-16: 8)\) lie on the circumference of the circle. Another point B has coordinates at \((0 ; 2)\).

### (a)

Find the equation of the circle.

### (b)

Show that B lies on the circumference of the circle.

### (c)

Find the equation of the tangent to the circle at \(B\).

### (d)

Calculate the coordinates of the midpoint of the chord BD .

### (e)

Hence or otherwise calculate the equation of the perpendicular bisector of BD .

## Question 16
The table below shows values of \(x\) and \(y\) obtained from an experiment which satisfies the law \(y=a b^{-x}\)

| \(x\) | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| \(y\) | 4.0 | 5.7 | 8.0 | 11.3 | 16.0 |

### (a)

Show graphically that these values satisfy the relationship \(y=a b^{-x}\).

### (b)

Use the graph to find approximate values of \(a\) and \(b\).