ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS**
PAPER 2 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.
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, N2012.
**Section (a): Pure Mathematics**
Question 1
If \(\mathrm{A}=\left(\begin{array}{ll}2 & a \\ 0 & 1\end{array}\right)\), prove by induction that for every positive integer \(n\),
$$ A^{n}=\left(\begin{array}{cc} 2^{n} & \left(2^{n}-1\right) a \\ 0 & 1 \end{array}\right) $$
Question 2
(i)
Find the exact value of \(\int_{2}^{3} \frac{3}{2 x^{2}-x-1} d x\) giving your answer as a single logarithm.
(ii)
Evaluate \(\int x^{2} e^{-x} d x\).
Question 3
Given that \(\mathrm{M}=\left(\begin{array}{ccc}3 & 1 & -3 \\ 1 & 2 a & 1 \\ 0 & 2 & a\end{array}\right)\),
(i)
find the values of \(a\) for which M is singular.
(ii)
solve the equation
$$ \mathrm{M}\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} -3 \frac{1}{2} \\ 5 \frac{1}{2} \\ 5 \end{array}\right) \text { given that } a=2 $$
Question 4
The diagram shows a rectangular aquarium of length 50 m and width 10 m .
The depth of water in the aquarium at time \(t\) seconds is \(h\) metres. At time \(t=0\), the aquarium is empty and water begins to flow into it at constant rate of \(2 \mathrm{~m}^{3} \mathrm{~s}^{-1}\). At the same time water begins to flow out at a rate proportional to \(h^{\frac{1}{2}}\). When \(h=1, \frac{d h}{d t}=0.002\).
(i)
Show that \(h\) satisfies the differential equation \(\frac{d h}{d t}=\frac{1}{500}(2-\sqrt{h})\).
(ii)
By making the substitution \(x=2-h^{\frac{1}{2}}\), show that the equation in (i) becomes \((2-x)\left(\frac{d x}{d t}\right)=-0.001 x\).
Question 5
The plane \(\pi\) has equation \(\mathbf{r} .\left(\begin{array}{r}2 \\ 0 \\ -1\end{array}\right)=16\) and the lines \(l_{1}\) and \(l_{2}\) have equations \(\mathbf{r}=\left(\begin{array}{l}3 \\ 2 \\ 4\end{array}\right)+\lambda\left(\begin{array}{r}1 \\ 3 \\ -5\end{array}\right)\) and \(\mathbf{r}=\left(\begin{array}{r}-3 \\ 4 \\ 6\end{array}\right)+\mu\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right)\) respectively, where \(\lambda\) and \(\mu\) are parameters.
(i)
Find the position vector of A , the point of intersection of \(l_{1}\) and \(\pi\).
(ii)
Show that \(l_{2}\) and \(\pi\) are parallel.
(iii)
Find the perpendicular distance of the line \(l_{1}\) from the point P with position vector \(\left(\begin{array}{r}-3 \\ -1 \\ 8\end{array}\right)\).
(iv)
Find the exact value of the cosine of the angle between \(l_{1}\) and \(\pi\).
Question 6
(a)
Simplify \(\frac{(1+i)^{4}}{(2-2 i)^{3}}\) giving your answer in the form \(a+b i\).
(b)
#### (i)
Simplify \(\frac{\cos 3 \theta+i \sin 3 \theta}{\cos 2 \theta-i \sin 2 \theta}\).
#### (ii)
Use de Moivre's theorem to express \(\sin 5 \theta\) in terms of \(\sin \theta\).
(c)
#### (i)
Sketch an argand diagram of the locus of \(z\) where
$$ |z-1-i|=|z+2+3 i| $$
#### (ii)
Hence or otherwise state the Cartesian equation of the locus.
**Section (b): Mechanics**
Question 7
Two coplanar concurrent forces of magnitude 4 N and 9 N act on a particle. Given that the angle between the two forces is \(40^{\circ}\), find the magnitude of their resultant.
Question 8
A particle of mass 2 kg is held at rest on a rough horizontal table connected by a light inextensible string which passes over a smooth pulley at the end of the table to a particle of mass 5 kg which hangs freely. The coefficient of friction between the particle and the table is \(\frac{1}{4}\). The particles are released from rest. Find in terms of \(g\), the
(a)
#### (i)
acceleration of the particles,
#### (ii)
tension in the string,
(b)
the force exerted by the string on the pulley.
Question 9
A stone is thrown vertically upwards from a point A, 1 metre above ground level with a speed of \(12 \mathrm{~ms}^{-1}\). On its way down, it hits an obstacle and comes to an instantaneous rest 0.6 metres before hitting the ground.
(a)
Find the
#### (i)
greatest height above its initial position,
#### (ii)
speed with which the stone hits the obstacle.
(b)
Sketch the velocity-time graph for the stone.
Question 10
A ball is projected at an angle of \(30^{\circ}\) above the horizontal with a velocity of \(36 \mathrm{~ms}^{-1}\) from a point which is 1.5 metres above the ground.
Calculate the
(a)
time taken by the ball to reach the ground,
(b)
magnitude of the velocity with which the ball hits the ground.
**Section (c): Statistics**
Question 11
A group of 50 children raised money for charity. The amounts raised by children, to the nearest dollar, were recorded in the table.
| amount (\$) | \(1-5\) | \(6-10\) | \(11-15\) | \(16-20\) | \(21-25\) | \(26-30\) | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | | frequency | 20 | 10 | 5 | 6 | 8 | 1 |
(a)
State the smallest possible amount which may have been raised by one child.
(b)
Find the
#### (i)
mean amount,
#### (ii)
standard deviation.
Question 12
A discrete random variable \(X\), has a probability distribution given in the table.
| \(x\) | 0 | 1 | 2 | | :---: | :---: | :---: | :---: | | \(\mathrm{P}(X=x)\) | \(\mathrm{P}_{0}\) | \(\mathrm{P}_{1}\) | \(\mathrm{P}_{2}\) |
Given that \(\mathrm{E}(X)=1.2\) and \(\operatorname{Var}(X)=0.36\), find the values of \(\mathrm{P}_{0}, \mathrm{P}_{1}\) and \(\mathrm{P}_{2}\).
Question 13
An 'A' Level student wishes to be enrolled at one of the three universities namely Bindura, National University of Science and Technology (NUST) and Midlands State University (MSU). It is equally likely that the student will apply to any one of these three universities. The probability that the student will be admitted at Bindura is \(60 \%\) while the probability that the student will be admitted at NUST and MSU are \(45 \%\) and \(35 \%\) respectively.
Find the probability that
(a)
the student will not be admitted at any one of the universities,
(b)
the student will apply to NUST given that the student will not be admitted into any one of the universities.
Question 14
A continuous variable \(X\), has a probability density function given by
$$ f(x)=\left\{\begin{array}{lc} a x^{2} & 0 \leq x \leq 1 \\ \frac{1}{4}(7-3 x) & 1 \leq x \leq \frac{7}{3} \\ 0 & \text { otherwise } \end{array}\right. $$
where \(a\) is a constant.
(a)
Show that \(a=1\).
(b)
Find the
#### (i)
mean,
#### (ii)
median.