ZIMSEC A-Level Mathematics Paper 2 June 2012 Question Paper

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

**MATHEMATICS** **9164/2** **PAPER 2** **JUNE 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.

If a numerical value for \(g\) is necessary, take \(g=9.81 \mathrm{~ms}^{-2}\).

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 . Within each section of the paper, questions are printed in the order of their mark allocations and candidates are advised, within each section, 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 8 printed pages.**

Copyright: Zimbabwe School Examinations Council, J2012. ©ZIMSEC J2012

**Section (a): Pure Mathematics**

Question 1

(i)

Verify that the equation

$$ \sin x+\cos x=e^{x}-1 \text { has a root between }-4 \text { and }-3 $$

(ii)

Use the Newton-Raphson method to find this root correct to 3 decimal places.

Question 2

Prove by mathematical induction that \(\sum_{r=1}^{n}(r+3)(2 r+1)=\frac{n}{6}\left(4 n^{2}+27 n+41\right)\)

Question 3

An isosceles triangle with a constant perimeter of \(2 k\) is such that the equal sides are of each length \(x \mathrm{~cm}\).

(a)

Find in terms of \(x\) and \(k\), the expression for

#### (i)

the length of the base of the triangle,

#### (ii)

the perpendicular height of the triangle.

(b)

By first writing the expression for the area, A , of the triangle in terms of \(x\) and \(k\), show that the stationery value of the area occurs when \(x=\frac{2}{3} k\).

Question 4

Mr Khumalo bought a car at \(\$ 4000\). A statistical model pertaining to the depreciation assumes that the value, \(\$ V\), of the car, \(t\) months after it has been bought, decreases at a rate which is proportional to \(V\).

(i)

Using this model, write down a differential equation connecting \(V\) and \(t\) and show that it reduces to \(V=4000 e^{-k t}\), where \(k\) is a positive constant.

(ii)

It is further given that after 3 years, the car is valued at \(\$ 2000\).

Calculate

#### (a)

the value of the car when it is 15 months old, giving your answer to the nearest dollar,

#### (b)

the age of the car when its value is \(\$ 1600\), giving your answer to the nearest month.

Question 5

(a)

The vector equations of the lines \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) are \(\mathbf{r}=(2+\lambda) i+(-3+2 \lambda) j+(1+2 \lambda) \mathbf{k}\) and \(\mathbf{r}=(8+3 \mu) i+(5+2 \mu) j+(13+6 \mu) \mathbf{k}\) respectively.

Find the coordinates of the points of intersection of the lines \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\).

(b)

The plane \(\pi\) is given by \(2 x-z=3\).

Verify that the lines \(L_{1}\) and \(L_{2}\) both lie in the plane \(\pi\).

(c)

Given that the position vector of the point A is \(12 i+5 j+6 k\) and B is the foot of the perpendicular from A to \(\pi\).

#### (i)

Find the vector equation of AB ,

#### (ii)

Hence, find the position of vector of \(B\).

Question 6

(a)

It is given that the complex number \(a\) whose conjugate is \(\bar{a}\). satisfies the equation \(4 a \bar{a}+12 i=8 a+16\).

Find the two possible values of \(a\), giving your answer in the form \(p+\mathrm{i} q\) where \(p\) and \(q\) are real.

(b)

The polynomial \(2 x^{4}+x^{3}+17 x^{2}+9 x-9\) is denoted by \(p(x)\).

#### (i)

Show that \(3 i\) is a root of the equation \(2 x^{4}+x^{3}+17 x^{2}+9 x-9=0\).

#### (ii)

State the other complex root of the equation \(p(x)=0\).

#### (iii)

Hence or otherwise find the other two roots of the equation

$$ p(x)=0 . $$

Question 7

(a)

The transformation matrix \(\left(\begin{array}{rr}-2 & 1 \\ 3 & 2\end{array}\right)\) maps the line \(y\) onto the line \(y^{1}=6 x+8\).

Find the equation of the line \(y\), giving your answer in the form \(a x+b y=c\) where \(a, b\), and \(c\) are integers.

(b)

Given that \(\mathrm{M}=\left(\begin{array}{rrr}3 & 2 & 1 \\ -2 & 2 & 0 \\ 1 & 0 & 0\end{array}\right)\) and \(\mathrm{N}=\left(\begin{array}{rrr}0 & 1 & 3 \\ 1 & 4 & -2 \\ 1 & 0 & 2\end{array}\right)\),

find

#### (i)

MN ,

#### (ii)

\(\quad(\mathrm{MN})^{-1}\).

(c)

Hence find the values of \(x, y\) and \(z\) for which \(\mathrm{MN}\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}3 \\ -6 \\ 3\end{array}\right)\).

**Section (b) : Mechanics**

Question 8

A car travelling along a straight road, passes through a fixed point A with a velocity of \(10 \mathrm{~ms}^{-1}\) and continues with this velocity for \(t_{1}\) seconds. The car then accelerates uniformly over the next \(t_{2}\) seconds till it attains a velocity of \(15 \mathrm{~ms}^{-1}\). The car subsequently decelerates to rest in a further \(t_{3}\) seconds.

(a)

Sketch a well labelled \((t ; v)\) graph to illustrate the motion of the car after passing A .

(b)

Given that the car's acceleration and deceleration are equal, show that \(t_{3}=3 t_{2}\).

Question 9

The diagram shows the trajectory of a ball that is projected from the top of a building 40 m high with initial velocity of \(20 \mathrm{~ms}^{-1}\) at an angle of \(30^{\circ}\) to the horizontal. The ball hits the ground at a point P (see diagram).

Find

#### (i)

the time of flight of the ball,

#### (ii)

the direction in which the ball is moving when it hits point P .

Question 10

Two particles of mass \(m_{1} \mathrm{~kg}\) and \(m_{2} \mathrm{~kg}\) where \(m_{1}>m_{2}\), are connected by a light inelastic string which passes over a smooth fixed pulley.

(a)

Find in terms of \(g, m_{1}\) and \(m_{2}\) the acceleration of the system.

(b)

Show that the tension in the string is given by \(m_{1} g\left(\frac{2 m_{2}}{m_{1}+m_{2}}\right)\).

Question 11

A particle of weight 20 N rests on a rough horizontal surface. A force \(P N\) inclined at an angle \(\theta\) to the horizontal, is applied to the particle until it is on the point of moving. Given that the normal force \(R=16 j\) and the frictional force \(F=-9 i\) act on the particle where \(i\) and \(j\) are unit vectors (see diagram),

calculate

#### (i)

#### (a)

the co-efficient of friction,

#### (b)

the magnitude of the contact force between the particle and the surface.

#### (ii)

#### (a)

\(\theta\),

#### (b)

\(P\).

**Section (c): Statistics**

Question 12

Given that \(W \sim \operatorname{Geo}(p)\) and \(\operatorname{Var}(W)=30\),

find \(\mathrm{E}(W)\).

Question 13

In a certain court, there are only two verdicts on passing judgement, namely "convicted " or "discharged". Of all the cases that have been tried by this court, \(80 \%\) of the verdicts were convictions. Suppose that when the court's verdict is "convicted " or "discharged", the respective probabilities of the accused person being innocent are 0.07 and 0.4 respectively.

By use of a tree diagram, find

(a)

the probability that a person tried by this court is innocent,

(b)

the conditional probability that an innocent person tried by this court is convicted.

Question 14

X is a continuous random variable with probability density function \(\mathrm{f}(x)\) where

$$ f(x)= \begin{cases}2-2 x & 0 \leq x \leq 1 \\ 0 & \text { otherwise }\end{cases} $$

#### (i)

Find the cumulative distribution function \(\mathrm{F}(x)\),

#### (ii)

Show that \(P\left(x>\frac{1}{3}\right)=\frac{4}{9}\),

#### (iii)

Find the value of p such that \(\mathrm{P}(x<\mathrm{p})=\frac{1}{5}\).

Question 15

The performance of a group of 24 students who sat for their final ' O ' - Level examinations in Biology and Integrated Science are shown below.

**Biology**

36, 45, 40, 60, 71 , 66, 53, 42, 35, 54, 35, 43, 72, 37, 39, 34, 49, 43, \(75,58,67,59,36,67\),

**Integrated Science**

88, 89, 30, 34, 48, 49, 59, 65, 67, 78, 41, 70, 54, 66, 39, 49, 37, 59, 45, 63, 52, 75, 38, 38,

(a)

Construct a back to back stem and leaf diagram to represent these two sets of data.

(b)

Using a scale of 2 cm to represent 10 marks, draw a pair of box and whisker plots to represent these two sets of data.

(c)

Comment briefly on the performance of the students in the tests.