ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** **PAPER 2** **JUNE 2009 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 7 printed pages and 1 blank page.
**Section (a): Pure Mathematics**
Question 1
Given that \(5.427=5.4+0.027+0.00027+0.0000027+\ldots\) Express the number 5.427 in the form \(\frac{a}{b}\), where \(a, b \in \mathrm{Z}\).
Question 2
The parametric equations of a curve are defined by \(x=a \sin ^{2} \theta+\theta\) and
$$ y=a \cos ^{2} \theta-\theta $$
(i)
Show that \(\frac{d y}{d x}=-1\).
(ii)
By eliminating \(\theta\), show that the curve can be expressed in the form \(\mathrm{A} x+\mathrm{B} y-\mathrm{C}=\mathrm{O}\).
Question 3
The diagram shows a \(\triangle \mathrm{ABC}\) in which \(\mathrm{BAC}=\left(\frac{\pi}{2}+\theta\right)\) radians and
$$ \mathrm{A} \hat{\mathrm{C} B}=\left(\theta-\frac{\pi}{2}\right) \text { radians } $$
(i)
Find \(\mathrm{A} \hat{\mathrm{B}} \mathrm{C}\) in radians.
(ii)
Show that \(\frac{\mathrm{BC}}{\mathrm{AC}}=\frac{1}{2 \sin \theta}\).
(iii)
Hence find the value of \(\theta\) for which \(\mathrm{BC}=2 \mathrm{AC}\).
Question 4
Prove by induction that
$$ \frac{\mathrm{d}^{n}\left(x^{m}\right)}{\mathrm{d} x^{n}}=\frac{m!}{(m-n)!} x^{m-n} \text { for } m \geq n \text {. } $$
Question 5
Expand \(\frac{1+a x}{1-b x}\) up to and including the term in \(x^{3}\) and write down the Maclaurin's theorem expansion of \(\mathrm{e}^{x}\), also up to the term in \(x^{3}\). If \(x\) is so small that \(x^{4}\) and higher powers of \(x\) may be neglected, determine the values of \(a\) and \(b\) for which
$$ \frac{1+a x}{1-b x}-\mathrm{e}^{x}=c x^{3} \text { and state the value of the constant } c . $$
Question 6
Given that \(\mathbf{A}=\left(\begin{array}{ccc}-1 & 3 & 1 \\ 2 & 5 & 0 \\ 3 & 1 & -2\end{array}\right)\), find \(\mathbf{A}^{-1}\).
Hence, or otherwise solve the equations
$$ \begin{aligned} & -x+3 y+z=1, \\ & 2 x+5 y=3 \\ & 3 x+y-2 z=3 \end{aligned} $$
Question 7
(a)
The function \(f(z)=z^{4}+\sqrt{3} z^{3}+2 z^{2}+\sqrt{3} z+1\). One root of the equation \(f(z)=0\) is \(e^{5 \pi / 6}\).
#### (i)
Express this root in the form \(a+\mathrm{i} b\), where \(a, b \in I R\).
#### (ii)
Hence or otherwise find the two quadratic factors of \(f(z)\).
#### (iii)
Solve the equation \(f(z)=0\) completely, giving the roots in the form \(e^{i \theta}\).
(b)
On a single sketch, show by shading, the region defined by
$$ |z-1+i| \leq 2 \text { and }-\frac{\pi}{3} \leq \arg (z) \leq \frac{\pi}{3} . $$
Question 8
The lines \(\ell_{1}\) and \(\ell_{2}\) have equations \(\mathbf{r}=\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)+\mu\left(\begin{array}{l}1 \\ 3 \\ 2\end{array}\right)\) and \(\mathbf{r}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)+\lambda\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)\) respectively. The plane \(\pi\) has equation r. \(\left(\begin{array}{r}1 \\ 2 \\ -3\end{array}\right)=2\).
(a)
Find the coordinates of the point A , where \(\ell_{1}\) intersects \(\ell_{2}\).
(b)
Show that \(\ell_{1}\) intersects \(\pi\) at the point B with coordinates \((3 ; 10 ; 7)\) and find the coordinates of \(c\), the point of intersection of \(\ell_{2}\) and the plane \(\pi\).
(c)
Given that the acute angle between \(\ell_{1}\) and \(\ell_{2}\) is \(\theta\),
#### (i)
show that \(\sin \theta=\sqrt{\frac{5}{21}}\),
#### (ii)
hence find the exact area of triangle ABC .
**Section (b): Mechanics**
Question 9
The diagram shows two forces P and Q each of magnitude 80 N inclined at \(60^{\circ}\) to each other and acting on an object. Find the magnitude and directions of a force required to keep the object in equilibrium.
Question 10
A cheetah at point \(A\) sees a stationary antelope grazing at a point \(B\) where \(A B=24 \mathrm{~m}\). The cheetah accelerates at \(3 \mathrm{~ms}^{-2}\) from rest towards the antelope. At the same instant the antelope starts to accelerate away in the direction AB . The antelope is caught at point C on AB . produced where \(\mathrm{AC}=54 \mathrm{~m}\).
Calculate
(i)
the time taken by the cheetah to catch the antelope,
(ii)
the acceleration of the antelope.
Question 11
The diagram shows a mass of 1 kg lying on a rough inclined plane at angle \(\theta\) where \(\theta=\sin ^{-1}\left(\frac{3}{5}\right)\). The mass is connected to one end of a light inextensible string parallel to a line of greatest slope passing over a smooth pulley with the other end connected to a mass of 4 kg hanging freely \(2,5 \mathrm{~m}\) above the floor.
The coefficient of friction between the 1 kg mass and the plane is \(\frac{1}{4}\).
(i)
Show that the 1 kg mass will slide up the plane.
(ii)
Find the velocity with which the 4 kg mass hits the floor.
Question 12
A particle is projected at \(35^{\circ}\) above the horizontal from point O . The point O is 5 m above the level ground. The speed of projection is \(25 \mathrm{~ms}^{-1}\). The particle passes through H , the highest point of its path and hits the ground at point B (see diagram).
Calculate
(i)
the height of H above the ground,
(ii)
the time taken by the particle to travel from H to B ,
(iii)
the horizontal distance between O and B .
**Section (c): Statistics**
Question 13
(a)
An experiment has only two possible outcomes. The first outcome occurs with probability \(p\) and the second outcome with probability \(p^{2}\). Find the value of \(p\) correct to 3 decimal places.
(b)
A survey in a city showed that the probability that a person is in favour of capital punishment is 0.55 and that the person is against it is 0.45 . If two persons are selected at random, find the probability that at least one of them favours capital punishment.
Question 14
(i)
Show that \(f(x)=\frac{1}{28}(6 x-4), 2 \leq x \leq 4\), is a probability density function for the continuous random variable X .
(ii)
Find \(\mathrm{E}(\mathrm{X})\).
Question 15
The table below shows the masses of 200 students measured to the nearest kg .
| Mass (kg) | Number of Students | | :--- | :--- | | 46-50 | 20 | | 51-55 | 60 | | 56-60 | 56 | | 61-65 | 35 | | 66-70 | 19 | | 71-75 | 10 |
(i)
Construct the cumulative frequency distribution table for the masses of these students and draw the cumulative frequency curve.
(ii)
Use your curve to estimate
#### (a)
the interquartile range,
#### (b)
the percentage of students with a mass of 54 kg or less.
Question 16
The heights of flowers in a bed are normally distributed with mean 21.1 cm and standard deviation 4.0 cm .
(a)
Find the probability that a flower has a height greater than 25 cm .
(b)
Eight flowers were picked from the bed at random. Find the probability that fewer than three flowers have heights less than 25 cm , giving your answer to 3 significant figures.