ZIMBABWE SCHOOL EXAMINATIONS COUNCIL General Certificate of Education Advanced Level **MATHEMATICS**
**PAPER 1**
**General Certificate of Education Advanced Level**
**NOVEMBER 2009 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.
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 . Questions are printed in the order of their mark allocations and candidates are advised 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.
Question 1
Find the coefficient of \(x^{-3}\) in the expansion of \(\left(x^{2}-\frac{2}{x}\right)^{9}\).
Question 2
Given that in triangle \(\mathrm{ABC}, \mathrm{AC}=7 \mathrm{~cm}, \mathrm{CB}=3 \mathrm{~cm}\) and \(\mathrm{ABC}=60^{\circ}\), show that \(\cos \mathrm{C}=-\frac{1}{7}\).
Question 3
Express in partial fractions
$$ \frac{5 x^{2}+7 x+9}{(x+2)^{2}(3-x)} $$
Question 4
The function \(f\) is defined by
$$ f: x \rightarrow \frac{1+x^{2}}{x}, x \in \mathrm{IR}, x \neq \dot{k} $$
(i)
State the value of \(k\).
(ii)
Find \(f\left(\frac{1}{x}\right)\) in its simplest form.
(iii)
Find another element in the domain of \(f\) which has the same image as 2 .
Question 5
The complex numbers \(z\) and \(w\) are given by \(z=-3+2 i\) and \(w=5+4 i\). Find
(i)
\(|z|\),
(ii)
\(\arg z\),
(iii)
\(\frac{z}{w}\) in the form \(a+i b\) where \(a\) and \(b\) are exact.
Hence represent \(\frac{z}{w}\) in an Argand diagram.
Question 6
Given that \((3 x-2)\) is a factor of \(6 x^{3}+8 x^{2}+k x+2\), find the value of \(k\).
Hence find all the roots of the equation
$$ 6 x^{3}+8 x^{2}+k x+2=0 $$
giving your answers in exact form.
Question 7
Hence, or otherwise, solve the equation
$$ 3 \cos x-5 \sin x=2 $$
for \(0^{\circ}<x<360^{\circ}\).
Question 8
(a)
Find in the form \(y=m x+c\), the equation of a line of gradient \(m\) that passes through the point ( \(m ; m\) ).
(b)
A curve is represented parametrically by
$$ x=2+\sin \theta, \quad y=6 \cos \theta $$
#### (i)
Find \(\frac{d y}{d x}\) in terms of \(\theta\).
#### (ii)
By expressing \(\sin \theta\) in terms of \(x\) and \(\cos \theta\) in terms of \(y\), use an appropriate identity to show that
$$ y^{2}+36 x^{2}-144 x+108=0 $$
Question 9
(a)
Solve the equation
$$ |2 \sin x-3|=1 $$
for \(0^{\circ} \leq x \leq 360^{\circ}\).
(b)
Solve the equation \(\frac{3}{2} \sin 2 \theta=2 \sin ^{2} \theta\) for \(0^{\circ} \leq \theta \leq 360^{\circ}\), giving your answers correct to the nearest \(0.1^{\circ}\).
Question 10
The radius of a circle depends on time \(t\), according to the formula \(r=t^{3}+1\). Given that its area is represented by A ,
(i)
Find \(\frac{d A}{d t}\), in terms of \(t\), and hence evaluate \(\frac{d A}{d t}\), when \(t=2\).
Express your answer in terms of \(\pi\).
(ii)
Show that the curve \(y=x^{2} \ln x\) has a turning point at \(\left(e^{-\frac{1}{2}},-\frac{1}{2} e^{-1}\right)\) and that this is a minimum point.
Question 11
The position vectors of points \(\mathrm{A}, \mathrm{B}\) and P are given by \(\left(\begin{array}{l}1 \\ 6 \\ 2\end{array}\right) ;\left(\begin{array}{c}-1 \\ 11 \\ -1\end{array}\right) ;\left(\begin{array}{c}2 p+1 \\ -5 p+6 \\ 3 p+2\end{array}\right)\) respectively.
(a)
Show that \(\mathrm{A}, \mathrm{B}\) and P are collinear for all values of p .
(b)
Given that the point C has position vector \(\left(\begin{array}{c}-2 \\ -5 \\ 11\end{array}\right)\), find in terms of \(p\),
#### (i)
an expression for \(\overrightarrow{\mathrm{CP}}\),
#### (ii)
the value of \(\overrightarrow{\mathrm{CP}} \cdot \overrightarrow{\mathrm{AB}}\).
(c)
#### (i)
Given also that CP is perpendicular to AB , find the value of \(p\).
#### (ii)
Hence, or otherwise, obtain the shortest distance from C to the line AB .
Question 12
(a)
Variables \(x\) and \(y\) are related by the equation
$$ y=\frac{3 x-6}{x} $$
Given that \(x\) and \(y\) are functions of \(t\) and that \(\frac{d y}{d t}=0.6\) find the corresponding rate of change of \(x\) with respect to \(t\) when \(y=1\).
(b)
The two variables \(x\) and \(y\) are related by the equation \(y=5-\frac{3}{x}\). Given that the value of \(y\) increases from 4 by a small amount \(\frac{\mathrm{P}}{25}\), determine in terms of \(p\),
#### (i)
the approximate change in \(x\),
#### (ii)
the corresponding percentage change in \(x\).
Question 13
Express \(\frac{6 x^{2}+7 x-2}{(x-1)(x-2)(x+2)}\) into partial fractions.
Question 14
Express \(\frac{2 x^{2}+3 x+5}{(x+1)\left(x^{2}+3\right)}\) and hence show that \(\frac{dy}{dx} = -2/3\) when \(x=0\). Hence find \(\frac{d y}{d x}\) as a sum of three fractions.
Explain why \(\frac{d y}{d x}\) is always negative, except at points where it is not defined.
Question 15
(a)
Show that \(\frac{d}{d x}\left(\frac{x}{\sqrt{4-x^{2}}}\right)=\frac{4}{\left(4-x^{2}\right)^{\frac{3}{2}}}\).
Hence evaluate
$$ \int_{0}^{1} \frac{8}{\left(4-x^{2}\right)^{\frac{3}{2}}} d x $$
(b)
The region enclosed by the curve \(y=\frac{x}{\sqrt{\left(4-x^{2}\right)}}\), the \(x\)-axis, and the lines \(x=-1\) and \(x=1\), is rotated about the \(x\)-axis through four right angles.
Show that the volume generated is given by
$$ 2 \pi(\ln 3-1) $$
Question 16
Express \(12 x-3 x^{2}+15\) in the form \(a+b(x+c)^{2}\).
Hence, or otherwise, sketch the graph of \(y=12 x-3 x^{2}+15\), clearly labelling the coordinates of the turning point.
On the same axes sketch the graph of \(y=e^{-x}\) and state the number of real roots of the equation
$$ e^{-x}+3 x^{2}-12 x-15=0 $$
Taking \(x_{1}=-0.8\), use the Newton - Raphson method to obtain the value of a root of the equation correct to 4 decimal places.
Question 17
(a)
In an arithmetic progression the \(6^{\text {th }}\) term is half the \(4^{\text {th }}\) term and the \(3^{\text {rd }}\) term is 15 .
Find.
#### (i)
the first term and the common difference,
#### (ii)
the number of terms needed to give a sum that is less than 65 .
(b)
Evaluate \(\sum_{r=1}^{10}\left(\frac{3}{4}\right)^{r}\) giving your answer correct to 4 significant figures.