ZIMSEC A-Level Mathematics Paper 1 November 2012 Question Paper

About this question paper

This is the ZIMSEC A-Level Mathematics question paper for Paper 1 from November 2012. Use it for exam practice under timed conditions, then check your work against the marking scheme. The paper code is 9164/1. 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

**MATHEMATICS**<br>**PAPER 1**

**NOVEMBER 2012 SESSION**

**3 hours**

Additional materials:
Answer paper
Graph paper
List of Formulae

TIME 3 hours

### INSTRUCTIONS TO CANDIDATES

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

There is no restriction on the number of questions which you may attempt.

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 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.
Questions are printed in the order of their mark allocations and candidates are advised to attempt questions sequentially.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.

This question paper consists of 7 printed pages and 1 blank page.
Copyright: Zimbabwe School Examinations Council, N2012.

## Question 1
The diagram shows triangle ABC with \(\mathrm{AB}=m \mathrm{~cm}, \mathrm{BC}=n \mathrm{~cm}, \mathrm{AC}=12 \mathrm{~cm}\) and angle \(\hat{\mathrm{BAC}}=60^{\circ}\).

It is given that the area of triangle ABC is \(\sqrt{675} \mathrm{~cm}^{2}\).

Find the exact values of \(m\) and \(n\).

## Question 2
### (i)

Show that the equation \(6 \sin \left(\pi+x^{2}\right)=3\) has a root between 1.9 and 2.

### (ii)

Starting with \(x_{1}=1.9\) apply the Newton-Raphson method once to find the second estimate for the root, giving your answer correct to 3 significant figures.

## Question 3
### (a)

The function \(y=\mathrm{f}(x)\) is given by \(y=e^{\sin 3 x}\).

Find expressions for

#### (i)

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

#### (ii)

\(\frac{d^{2} y}{d x^{2}}\).

### (b)

Hence write down the Maclaurin series of \(\mathrm{f}(x)\) up to the term in \(x^{2}\).

## Question 4
### (a)

A function is represented parametrically by the equations

\(y=e^{t^{2}}\) and \(x=e^{t^{2}}-1\), for \(1 \leq t \leq 2\).

Use the chain rule to find \(\frac{d y}{d x}\) and show that it is independent of \(t\).

### (b)

Describe briefly stating the exact coordinates of the end points, the shape of the graph of \(y\) against \(x\).

## Question 5
The \(6^{\text {th }}\) term of a geometric progression is \(\frac{1}{81}\) and the \(3^{\text {rd }}\) term is \(-\frac{1}{3}\).

Find

### (a)

the value of the

#### (i)

common ratio,

#### (ii)

first term,

### (b)

the sum of the first \(n\) terms of the progression,

### (c)

the value of the sum to infinity of the progression.

## Question 6
The curve C has equation given as \(y=p+e^{q x}\), where \(p\) and \(q\) are real constants.

It crosses the \(y\)-axis at point \(\mathrm{A}\left(0 ; \frac{3}{2}\right)\) and has gradient -2 at point A.

### (i)

Find the values of \(p\) and \(q\).

### (ii)

State the equation of the asymptote.

### (iii)

Sketch the curve C and its tangent at point A.

![](https://ocr.unopasa.com/p/ocr_assets/10646a131ab00d112e5ffe3d252360646764a497d13449f3f59886d4d4f8c8eb.jpg)

## Question 7
A complex number \(z_{1}\) has modulus 2 and is positioned as shown in the Argand diagram above.

### (i)

State the principal argument of \(z_{1}\) and write \(z_{1}\) in the form \(a+i b\) where \(a\) and \(b\) are exact real numbers.

### (ii)

Find exactly in the form \(a+i b\), the complex number \(w\), given that

$$
w=\frac{(-8 \sqrt{3}) i}{z_{1}}
$$

### (iii)

Show a sketch of \(w\) in an Argand diagram, labelling the modulus and argument values in your diagram.

## Question 8
The function \(\mathrm{f}(x)\) has equation \(\mathrm{f}(x)=2 x^{3}+b x^{2}+x-2\).

Given that \(2 x-1\) is a factor of \(\mathrm{f}(x)\),

### (i)

find the value of \(b\),

### (ii)

factorise \(\mathrm{f}(x)\) completely.

### (iii)

Sketch the curve of \(y=\mathrm{f}(x)\), showing clearly the points of intersection of the curve with the axes without necessarily finding the coordinates of the turning points.

Hence, or otherwise, find the values of \(x\) which satisfy the inequality

$$
\mathrm{f}(x)<0 .
$$

## Question 9
A thin string of length 220 cm is divided (without cutting it up), into \(n\) parts in the ratio \(1: 2: 3: \ldots n\).

### (a)

Show that the length of the longest segment is \(\frac{440}{n+1} \mathrm{~cm}\).

### (b)

The string is made to form a circle so that the ends just touch each other.

#### (i)

Find the radius of this circle.

#### (ii)

Show that the angle subtended at the centre by the largest segment in part (a) is approximately \(\frac{12.57}{n+1}\) radians.

#### (iii)

Hence find the smallest value of \(n\) for which this angle is less than \(\frac{\pi}{12}\) radians.

## Question 10
![](https://ocr.unopasa.com/p/ocr_assets/90c0dc5f16024800561cd02ef5cf14e917badb79bff94fdec89d0503eeb4c56e.jpg)

The diagram shows the graph of the curve \(y=1+\sqrt{3 x-2}\) for \(\frac{2}{3} \leq x \leq 2\).

The region R is enclosed between the curve, the \(x\)-axis and the lines \(x=\frac{2}{3}\) and \(x=2\).

Calculate the

### (i)

area of the region R,

### (ii)

volume generated when the region R is rotated completely about the \(x\)-axis.

## Question 11
The points \(\mathrm{A}, \mathrm{B}\) and C have position vectors \(8 j-4 k ; p i+3 j+4 k\) and \(-2 i+3 j+5 k\) respectively.

### (a)

Calculate the

#### (i)

unit vector parallel to \(\overrightarrow{\mathrm{AC}}\),

#### (ii)

positive value of \(p\) such that BA is perpendicular to BC.

### (b)

Hence or otherwise, find the area of triangle ABC for the value of \(p\) in (a)(ii).

## Question 12
![](https://ocr.unopasa.com/p/ocr_assets/0c2607182ffa2d644ac086a27bce3e3e2a131f1cca248b7dd9ece837c3d27ed9.jpg)

The diagram shows a square …