ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** 9164/2 PAPER 2 JUNE 2013 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 6 printed pages and 2 blank pages. Copyright: Zimbabwe School Examinations Council, J2013.
**Section (a): Pure Mathematics**
Question 1
A farmer intends to build a rectangular pen whose width is \(x \mathrm{~m}\) for his goats. He has 56 m of mesh wire.
(i)
Write down an expression for the area A, of the pen in terms of \(x\).
(ii)
Hence find the maximum area which can be enclosed by the mesh wire, verifying that it is maximum.
Question 2
The function \(g: x \rightarrow x^{2}+5 x, x \geq p\) is one to one.
Find
(i)
the least value of \(p\),
(ii)
the range of \(g\),
(iii)
sketch the graph of this function.
Question 3
Prove by induction that \(9^{2 n}-5^{2 n}\) is divisible by 7 for \(n \in Z^{+}\).
Question 4
(a)
Find \(\int e^{x} \cos x d x\).
(b)
Show that \(\int_{1}^{2} \frac{x^{2}+6 x+7}{(x+2)(x+3)} d x=\ln \left(\frac{75 e}{64}\right)\).
Question 5
The points P, Q, R and S have position vectors \(4 i+j-5 k, 2 i-j-6 k\), \(-2 i+3 j+2 k\) and \(i+j+2 k\) respectively.
(a)
Find the cartesian equation of the line PQ.
(b)
Show that the vector equation of the plane containing the points P, Q and R is \(3 x-5 y+4 z=-13\), and label it \(\pi\).
Hence or otherwise find
#### (i)
the exact value of the area of triangle PQR,
#### (ii)
the perpendicular distance from S to the plane \(\pi\).
Question 6
(a)
Using the substitution \(w=z^{4}\), solve the equation \(z^{8}-z^{4}-6=0\) where \(z\) is a complex number.
(b)
The real part of the complex number \(\frac{z+2}{z-2}\) is zero. Show that the locus of the point representing \(z\) in the Argand plane is a circle centre \((0 ; 0)\) and radius 2.
(c)
Sketch in an argand diagram the set of points representing all complex numbers \(z\) satisfying both the inequalities \(|z-3-i| \leq 4\) and \(\frac{\pi}{3} \leq \arg (z-4-2 i) \leq \frac{\pi}{2}\).
Question 7
(a)
A transformation matrix M is given by \(\mathrm{M}=\left(\begin{array}{cc}4 & -1 \\ 2 & 3\end{array}\right)\).
#### (i)
Show that the transformation matrix has no invariant line using the equation of line \(y=m x\),
#### (ii)
All points on the line \(y-3 x=-2\) are transformed by the matrix M. Find the equation of the image line.
(b)
The matrix P is given by \(\mathrm{P}=\left(\begin{array}{lll}2 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 2 & 1\end{array}\right)\).
#### (i)
Find the inverse of P,
#### (ii)
Hence find the point of intersection of the three planes with equation \(2 x+2 y+z=4\) \(2 x+3 y+z=1\) \(3 x+2 y+z=3\)
**Section (b): Mechanics**
Question 8
A particle of mass 8 kg rests on a rough plane inclined at \(30^{\circ}\) to the horizontal and is about to slip.
Show that the coefficient of friction between the plane and the particle is \(\frac{1}{\sqrt{3}}\).
Question 9
(a)
A ball is projected from point O with velocity \(u \mathrm{~ms}^{-1}\) at an angle \(\theta\) above the horizontal and passes through the point \((x ; y)\).
Show that the equation of the trajectory is given by
$$ y=x \tan \theta-\frac{g x^{2}\left(1+\tan ^{2} \theta\right)}{2 u^{2}} . $$
(b)
A ball is thrown from a height of 0.9 m above the ground at an angle of \(30^{\circ}\) to the horizontal, to clear a wall 10.5 m away horizontally and 4.5 m high. For the ball to clear the wall, show that the velocity of projection must not be less than \(17.1 \mathrm{~ms}^{-1}\).
Question 10
Coplanar forces of magnitude \(3 \mathrm{~N}, 4 \mathrm{~N}\) and 8 N act at a point as shown in the diagram.
(a)
If the resultant force in the \(i\)-direction is \(4(1-\sqrt{3}) \mathrm{N}\), show that \(\theta=30^{\circ}\).
(b)
Calculate the magnitude of the resultant and state its direction.
Question 11
A particle is moving in a straight line. Initially the particle has a velocity of \(8 \mathrm{~ms}^{-1}\) and is subjected to a retardation of \(\frac{3}{2} \mathrm{~ms}^{-2}\) for 6 seconds.
(a)
Draw a velocity-time graph for the particle.
(b)
Find the displacement of the particle from the starting point after the 6 seconds.
(c)
Find the distance travelled by the particle in this time.
**Section (c): Statistics**
Question 12
The table below shows the number of children below the age of 15 known to have suffered from measles in 2009 in a certain village.
| Age (in years) | Number of reported cases | | :--- | :--- | | Under 1 | 14 | | 1-2 | 33 | | 3-4 | 35 | | 5-9 | 39 | | 10-14 | 5 |
(a)
Represent the above information by means of a histogram.
(b)
Calculate the mean age of children who suffered from the disease.
Question 13
During the 2010 World Cup, the probability that there was electricity in a certain city on any particular day was \(\frac{1}{3}\). In the case that there was no electricity, a generator would be switched on. Independently, the probability that John watched a soccer match being screened live was \(\frac{1}{4}\).
(a)
Represent the above information by means of a tree diagram.
(b)
Given that John watched a soccer match, find the probability that there was no electricity.
Question 14
The probability that a learner driver passes his test at a Vehicle Inspection Department is \(\frac{1}{4}\). A learner driver counts the number of attempts \(n\), until he/she passes the driving test.
(a)
State a suitable statistical distribution which can be used to model the above situation.
(b)
Find the mean and variance of the distribution.
(c)
Find the smallest value of \(n\), for which there is a probability of at least 0.7, that the learner will needs only \(n\) or fewer trials to pass the driving test.
Question 15
A continuous random variable \(X\) has a probability density function \(f(x)\) given below
$$ f(x)= \begin{cases}\frac{1}{2}, & 0 \leq x<0.5 \\ \frac{1}{5}(3-x), & 0.5 \leq x \leq 3 \\ 0, & \text { otherwise } .\end{cases} $$
(a)
Sketch the graph of \(f(x)\).
(b)
Find the median.
(c)
Evaluate \(\mathrm{P}(x<1.2)\).