ZIMSEC A-Level Pure Mathematics Paper 1 June 2020 Question Paper

About this question paper

This is the ZIMSEC A-Level Pure Mathematics question paper for Paper 1 from June 2020. 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**
**JUNE 2020 SESSION**
3 hours

Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Non-programmable electronic scientific calculator

TIME 3 hours

### INSTRUCTIONS TO CANDIDATES

Write your Name, Centre number and Candidate number in the spaces provided on the answer paper/answer booklet.

Answer all questions.

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.

### 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.

The use of a non-programmable electronic scientific calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

## Question 1
By means of the substitution \(y=x^{\frac{1}{3}}\), or otherwise, find the values of \(x\) for which \(x^{\frac{1}{3}}-3 x^{\frac{-1}{3}}=2\).

## Question 2
### (a)

Find the value of \(k\) for which the line \(k x+(k-2) y+10=0\) is parallel to the line \(3 x+2 y-16=0\).

### (b)

Find the gradient of the line perpendicular to both lines.

## Question 3
### (a)

Express \(6 x^{2}-24 x-25\) in the form \(\mathrm{A}(x+\mathrm{B})^{2}+\mathrm{C}\), giving the numerical values of \(A, B\) and \(C\).

### (b)

Hence, or otherwise, solve exactly the inequality \(6 x^{2}-24 x-25>0\).

## Question 4
Solve the inequality \(|2 x+2|>1-4 x\).

## Question 5
### (a)

Expand \((p-x)^{-2}\) as a series of ascending powers of \(x\) up to the term in \(x^{3}\) where \(p\) is a positive constant.

### (b)

Given that the coefficient of \(x^{2}\) in the expansion is \(\frac{3}{16}\), find the value of \(p\).

### (c)

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

## Question 6
The points \(\mathrm{A}, \mathrm{B}\) and C have position vectors \(a=i-2 j+p k, b=q i+5 j+6 k\) and \(c=5 i+7 j\) respectively relative to the origin.

If \(\overrightarrow{\mathrm{AB}}=-5 i+7 j+3 k\) find the

### (a)

values of \(p\) and \(q\),

### (b)

exact length of AC ,

### (c)

acute angle BAC .

## Question 7
The population of a city is 500000 . The population grows at a rate of \(5 \%\) every year.

Find

### (a)

in terms of \(n\), the population at the end of the \(n\)th year.

### (b)

the population to the nearest thousand after ten years.

### (c)

the year in which the population first exceeds 1000000 .

## Question 8
The function \(f(x)=2-\frac{1}{x}, x>0\).

### (a)

Sketch the graph of \(f(x)\) and state the range.

### (b)

Find \(f^{-1}(x)\), the inverse of \(f(x)\).

### (c)

Calculate the value of \(x\) for which \(f(x)=f^{-1}(x)\).

## Question 9
A circle with centre \((2,-5)\) touches the line \(x+6 y-9=0\). Find the equation of the circle in the forms \((x-a)^{2}+(y-b)^{2}=r^{2}\) where \(a, b\) and \(r\) are constants.

## Question 10
### (a)

Complex numbers \(w\) and \(v\) are such that \(u=1+2 i\) and \(v=-2-i\).

Find \(w=\frac{5 u}{v}\), leaving the answer in the form \(a+i b\), where \(a\) and \(b\) are integers.

### (b)

Hence, or otherwise, find

#### (i)

\(|w|\),

#### (ii)

\(\arg w\).

## Question 11
The polynomial \(2 x^{3}-11 x^{2}+a x+b\) is exactly divisible by \((x-2)\) and leaves a remainder of -36 when divided by \((x+1)\).

### (a)

Find the values of the constants \(a\) and \(b\).

### (b)

Hence factorise the polynomial completely.

## Question 12
\(f(x)=\frac{x^{3}}{x^{2}-5 x+6} x \in \mathrm{R}\).

### (a)

Express \(f(x)\) in the form

$$
A x+B+\frac{C}{x-2}+\frac{D}{x-3}, \text { where } A, B, C \text { and } D \text { are constants. }
$$

### (b)

Hence find \(\int_{4}^{6} f(x) d x\). Leave the answer in exact form.

## Question 13
### (a)

Express \(2 \cos x-5 \sin x\) in the form \(R \cos (x+\theta)\), where \(R>0\) and \(0^{\circ}<\theta<90^{\circ}\).

### (b)

Hence, or otherwise, solve the equation \(2 \cos 2 x-5 \sin 2 x=2.5\) for \(0^{\circ} \leq x \leq 360^{\circ}\).

### (c)

Give values of \(x\) between \(0^{\circ}\) and \(360^{\circ}\) at which the maximum and minimum values of \(2 \cos 2 x-5 \sin 2 x\) occur.

## Question 14
### (a)

Use the trapezium rule with 4 ordinates to evaluate \(\int_{0}^{1.5} x^{3} \sin ^{2} x d x\), giving the answer correct to 3 significant figures.

### (b)

#### (i)

Find the area of the region bounded by the curve \(y^{2}=4 x\) and the line \(\mathrm{y}=x\).

#### (ii)

The region in (i) is rotated through \(360^{\circ}\) about the \(y\)-axis. Find the volume generated giving the answer in terms of \(\pi\).

## Question 15
### (a)

It is given that \(y=\frac{1}{1+\cos x}\).

Find

#### (i)

\(\frac{d^{2} y}{d x^{2}}\) when \(x=0\),

#### (ii)

the Maclaurin's series of \(y\) up to the term in \(x^{2}\).

### (b)

Variables \(x\) and \(y\) are related by the equation \(y=a b^{x}\), where \(a\) and \(b\) are constants. The graph of Iny against \(x\) is a straight line of gradient 0.7 and lny intercept at 2.3 .

Find the values of \(a\) and \(b\).