ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS**
PAPER 2
9164/2
3 hours
NOVEMBER 2011 SESSION
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 5 printed pages and 3 blank pages. Copyright: Zimbabwe School Examinations Council, N2011.
**Section (a): Pure Mathematics**
Question 1
Evaluate \(\int_{1}^{e} x^{3} \ln x^{2} d x\), giving your answer in exact form.
Question 2
The diagram below shows a sketch of part of the graph \(y=f(x)\), where \(f(x)=\sin x+\sin 2 x\).
Find the \(x\)-coordinates of the points A and B in terms of \(\pi\).
Question 3
The \(r^{\text {th }}\) term \(U_{r}\) of a series is given by
$$ U_{r}=\left(\frac{1}{3}\right)^{3 r-2}+\left(\frac{1}{3}\right)^{3 r-1} $$
(a)
Express \(\sum_{r=1}^{n} U_{\mathrm{r}}\) in the form \(A\left(1-\frac{B}{27^{n}}\right)\), where \(A\) and \(B\) are constants,
(b)
Find the sum to infinity of the series.
Question 4
Prove by induction that \(\sum_{r=1}^{n} r\left(2^{r-1}\right)=1+(n-1) 2^{n}\) for all positive integral values of \(n\).
Question 5
A certain type of tree grows in such a way that its height, \(h\) metres, \(t\) years after the tree has been planted can be modelled by a differential equation. It is assumed that the rate of increase of the height is directly proportional to the cube root of (9-h).
(a)
Given that \(h=1\), and \(\frac{d h}{d t}=\frac{1}{5}\) when \(t=0\),
#### (i)
form a differential equation in terms of \(h\).
#### (ii)
solve the differential equation and obtain an expression for \(t\) in terms of \(h\).
(b)
Hence find the time taken for the tree to reach a height of 4 m.
Question 6
(a)
Show that the perpendicular distance from the origin to the plane whose equation is \(r .(4 \hat{i}+3 \hat{j}-5 \hat{k})=10\) is \(\sqrt{2}\).
(b)
#### (i)
Find the vector equation of the line \(l_{1}\) which passes through the points \((1 ; 0 ; 2)\) and \((3 ; 1 ; 0)\).
#### (ii)
Find the coordinates of the point of intersection of the line \(l_{1}\) and plane \(\pi\) whose equation is \(r .\left(\begin{array}{c}-1 \\ -1 \\ 1\end{array}\right)=1\).
#### (iii)
The line \(l_{2}\) has equation \(r=\left(\begin{array}{l}2 \\ 1 \\ 0\end{array}\right)+\lambda\left(\begin{array}{l}4 \\ 2 \\ 3\end{array}\right)\). Find the acute angle between the line \(l_{2}\) and the plane \(\pi\).
Question 7
(a)
Find the values of \(x\) for which the matrix \(\left(\begin{array}{lll}x & 1 & 2 \\ 1 & x & 2 \\ 1 & 2 & x\end{array}\right)\) is singular.
(b)
Given that \(M\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}0 \\ 6 \\ 3\end{array}\right)\), where \(M=\left(\begin{array}{ccc}2 & 1 & -1 \\ 1 & -1 & 1 \\ 1 & 2 & 1\end{array}\right)\),
#### (i)
Find the determinant of M .
#### (ii)
Hence find the values of \(x, y\) and \(z\).
Question 8
(a)
Express in exponential form \(\left(\frac{3}{5}+\frac{4 i}{5}\right)^{20}-\left(\frac{3}{5}-\frac{4 i}{5}\right)^{20}\).
(b)
#### (i)
Prove that \(\tan 4 \theta=\frac{4 \tan \theta-4 \tan ^{3} \theta}{1-6 \tan ^{2} \theta+\tan ^{4} \theta}\) based on de Moivres theorem.
#### (ii)
Hence find the first four exact values of \(\theta\) for which \(\tan ^{4} \theta-4 \tan ^{3} \theta-6 \tan ^{2} \theta-4 \tan \theta+1=0\).
**Section (b): Mechanics**
Question 9
Two forces have magnitudes \(P\) and \(Q\) and the angle between them, \(\theta\), is acute. If the resultant of these two forces has magnitude R , show that \(R^{2}=P^{2}+Q^{2}+2 P Q \cos \theta\).
Question 10
A body of mass 5.2 kg is held in equilibrium on a rough plane, by a force, \(F\) newtons acting up the line of greatest slope. The plane is inclined at an angle \(\theta\) to the horizontal where \(\cos \theta=\frac{4}{5}\). When \(F\) is 19.2 newtons, the body is about to slide down the plane.
Find the value of \(\mu\), the coefficient of friction between the body and the plane.
Question 11
Two particles are projected simultaneously from two points A and B on level ground and are 150 m apart. The first particle is projected vertically upwards from A with an initial speed of \(U \mathrm{~m} / \mathrm{s}\), and the second particle is projected from B towards A with an initial velocity \(V \mathrm{~m} / \mathrm{s}\) at an angle of projection \(\propto\). If the particles collide when they are both at their greatest height above the level AB , prove that \(\tan \alpha=\frac{U^{2}}{150 g}\).
Question 12
Two vehicles moving in the same direction pass the point O on a straight road at time \(t=0\). Vehicle A is moving at a constant velocity of \(11 \mathrm{~m} / \mathrm{s}\). Vehicle B has a constant acceleration of \(2 \mathrm{~m} / \mathrm{s}^{2}\) and has a velocity of \(3 \mathrm{~m} / \mathrm{s}\) as it passes O .
(a)
On the same diagram, draw the velocity-time graph for the two vehicles.
(b)
Find
#### (i)
the distance of B from O when its speed is \(21 \mathrm{~ms}^{-1}\),
#### (ii)
the time in seconds when B overtakes A ,
#### (iii)
the distance from O , travelled by A before it was overtaken by B .
**Section (c): Statistics**
Question 13
The continuous random variable X has a probability density function given by
$$ f(x)= \begin{cases}k x, & 0 \leq x \leq 1 \\ k, & 1<x \leq 2 \\ 0, & \text { otherwise }\end{cases} $$
where k is a constant
Find
#### (i)
the value of \(k\),
#### (ii)
the median, \(m\), of X .
Question 14
The meteorological department of a certain country adopts a simple model of the weather in which each day is classified as either fine or rainy. The probability that a fine day is followed by another fine day is 0.8 . The probability that a rainy day is followed by a fine day is 0.4 . The probability that 1 February is fine is 0.75 .
Using a tree diagram or otherwise, find the probability that
(a)
3 February is fine,
(b)
1 February was rainy given that 3 February is fine.
Question 15
(a)
State the conditions under which a normal distribution can be used to approximate a binomial distribution.
(b)
It is estimated that 20\% of people undergoing a medical review are men. If a random sample of 100 people are undergoing a medical review, find the probability that more than 30 are men.
Question 16
An unbiased tetrahedral die has the number 1 written on one face, the number 2 on the other face and the number 3 on the remaining two faces. The die is thrown twice and X is the product of the scores obtained from the two throws.
(a)
Find the probability distribution of X .
(b)
Find
#### (i)
\(\mathrm{E}(\mathrm{X})\),
#### (ii)
\(\quad \operatorname{Var}(\mathrm{X})\).