ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** **9164/2** **PAPER 2** **JUNE 2008 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 7 printed pages and 1 blank page. Copyright: Zimbabwe School Examinations Council, J2008.
**Section (a): Pure Mathematics**
Question 1
The diagram above shows a region R enclosed by the curve \(y=\frac{a}{x}\), where \(a\) is a positive integer the lines \(y=\frac{1}{2}\), the \(y\)-axis and \(y=2\).
(i)
Find exactly the area of R , giving your answer in a form involving a single logarithm.
(ii)
Hence find the minimum value of \(a\) such that this area exceeds 100 .
Question 2
Use the substitution \(t=e^{x}\) to evaluate \(\int_{0}^{1} \frac{e^{x}}{9-e^{2 x}} d x\), leaving your answer in a form involving logarithms.
Question 3
Prove by induction that
$$ -2+3+8+\ldots+(5 n-7)=\frac{n}{2}(5 n-9) . $$
Question 4
(a)
A bus company will take a minimum of 50 passengers and a maximum of 70 passengers on a trip. If the number of passengers is exactly 50 , the fare is \(\$ 600000\) per passenger. If the number of passengers exceeds 50 the company reduces the fare by \(\$ 10000\) times the excess number of passengers.
Find the number that will maximise the intake.
(b)
A heated metal cube of side \(x \mathrm{~cm}\) is cooled uniformly. Given that the side of the cube decreases at the rate of \(10 \mathrm{~cm} /\) hour , find how fast the volume is changing at the instant when the side is 15 cm long.
Question 5
The diagram above shows a quadrilateral OBCA of which ADBC is a sector of a circle centre C and radius 1 unit. The lines OA and \(\mathrm{OB} x\) are tangents to the sector at A and B respectively, and \(\mathrm{A} \hat{\mathrm{O}} \mathrm{B}=\frac{\pi}{3}\) radians.
(a)
#### (i)
Show that \(\mathrm{A} \hat{\mathrm{C}} \mathrm{B}=\frac{2 \pi}{3}\) radians and find the exact length of line OA .
#### (ii)
Hence find the exact area of the shaded region.
(b)
It is given that the points \(\mathbf{A}, \mathbf{B}, \mathbf{C}\) and \(\mathbf{D}\) lie on an Argand diagram with origin at O .
#### (i)
Write down the complex number representing the point A , giving your answer in the form \(a+i b\).
#### (ii)
The complex number \(Z_{1}\) represents the point \(C\). Find \(Z_{1}\) in the form \(r(\cos \theta+i \sin \theta)\), where \(\theta\) is in terms of \(\pi\).
Hence write down the complex number \(Z_{1}^{2}\), giving your answer in the form \(\mathrm{Re}^{i \alpha}\).
Question 6
Matrices P and Q are given as
$$ P=\left(\begin{array}{llll} k & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right) \text { and } Q=\left(\begin{array}{ccc} 1 & 1 & k \\ -1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right) \text {, } $$
where k is a real number. The matrix M is such that \(\mathrm{M}=\mathrm{PQ}\).
(i)
Find matrix M in terms of k .
(ii)
Given that M is singular and that \(\mathrm{k}>0\), find the exact value of k .
(iii)
Given that \(\mathrm{k}=2\) find the inverse matrix M .
Question 7
The diagram above shows an outline of an open deep freezer OABCDEFG. The lid \(D E F^{\mathrm{i}} G^{\mathrm{l}}\) hinged along side \(E D\) is shown when it is opened through an angle \(\theta\), where \(0^{\circ} \leq \theta<90^{\circ}\). When it is closed the line \(F^{1} G^{1}\) coincides with the line \(F G\). The freezer has a square base of side 2 metres and is 1 metre high. The point O is taken as the origin, and \(x, y\) and \(z\) axes are along \(\mathrm{OA}, \mathrm{OC}\) and OE respectively.
(a)
Find the unit vector in the direction of vector \(\overrightarrow{\mathrm{AD}}\).
(b)
#### (i)
Write down the position vectors of the points D and E and show that the position vector of the point \(\mathrm{F}^{1}\) is \(\mathbf{r}=(2 \cos \theta) \mathbf{i}+(1+2 \sin \theta) \mathbf{k}\).
Hence or otherwise, find the normal vector to plane \(\mathrm{EF}^{1} \mathrm{G}^{1} \mathrm{D}\) in terms of \(\theta\) and write down the vector equation of this plane.
#### (ii)
Deduce the Cartesian equation of the plane \(E F^{\prime} G^{\prime} D\) when \(\theta=60^{\circ}\).
**Section (b): Mechanics**
Question 8
The diagram shows a man pulling a crate of weight 20 N on a rough horizontal floor using a string inclined at \(60^{\circ}\) to the horizontal.
The crate is in limiting equilibrium when the man pulls with a force of magnitude 4 N .
Calculate the exact value of
(i)
the normal reaction between the crate and the floor,
(ii)
the coefficient of friction between the crate and the floor.
Question 9
A car is travelling at \(40 \mathrm{~ms}^{-1}\) when it begins to decelerate uniformly to \(25 \mathrm{~ms}^{-1}\) over 30 seconds. It travels at this speed of \(25 \mathrm{~ms}^{-1}\) for a distance of 500 m . On separate diagrams, draw fully labelled \((t, v)\) and \((t, x)\) graphs for the motion of the car.
Question 10
A stone is thrown from the top of a vertical wall 3 metres high with a velocity of \(\mathrm{V} \mathrm{ms}^{-1}\) at an angle of depression \(\theta^{\circ}\) (see diagram).
The stone hits the ground 0.5 seconds later at a point P which is 1.6 metres from the foot of the wall.
Find V and \(\theta\) by assuming that the air resistance is negligible.
Question 11
A toy car P of mass 4 kg is at rest on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{1}{2} . P\) is connected to another toy car \(Q\) of mass 8 kg by a light inextensible string passing over a smooth pulley X . Q is held at rest on a smooth plane inclined at \(30^{\circ}\) to the horizontal (see diagram).
When Q is released, calculate
(i)
the acceleration of the toy cars and the tension in the string, giving exact answers in terms of \(g\),
(ii)
the magnitude of the force exerted by X on the string, leaving your answer in the form \(k g \cos \alpha\), where \(k\) and \(\alpha\) are rational numbers.
**Section (c): Statistics**
Question 12
The masses in grammes of 24 sweets in a bag are represented by the stem and leaf diagram shown below. The leaves are not ordered.
| 0.7 | \(2,3,9\) | | :--- | :--- | | 0.8 | 0,8 | | 0.9 | \(1,9,1,8,4\) | | 1.0 | \(3,8,6,1\) | | 1.1 | \(3,3,9,3\) | | 1.2 | 1,2 | | 1.3 | 9,3 | | 1.4 | 4,5 |
Key: \(0.7 \mid 2=0.72\)
(a)
#### (i)
Find the median of the distribution,
#### (ii)
Find the mode of the distribution.
(b)
A sweet of mass more than 1.2 g is classified as large. Calculate the mean of large sweets that the bag contains.
Question 13
A fair die is tossed three times. Find the probability that
(i)
exactly one six is obtained,
(ii)
the first score is even, the second is odd and the third is either a one or a two.
Question 14
Legal requirements specify that a pie manufacturer may only produce \(0.3 \%\) of pies below a weight of 80 g . The pie producing machine operates with a standard deviation of 0.8 g . The weekly output is 500000 . Pies with weights in excess of 83 g require additional packaging at a cost of \(\$ 10\) per pie. Assuming the weights are normally distributed, find the firm's weekly cost of extra packaging.
Question 15
A discrete random variable \(X\) takes the values 0,1 and 2 only, with probabilities \(P_{0}, P_{1}\) and \(P_{2}\) respectively. Find the value of \(P_{0}, P_{1}\) and \(P_{2}\) given that \(E(X)=\frac{4}{3}\) and \(\operatorname{Var}(\mathrm{X})=\frac{5}{9}\).