ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
**General Certificate of Education Advanced Level**
**MATHEMATICS**
**PAPER 1 - JUNE 2010 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.
This question paper consists of 5 printed pages and 3 blank pages. Copyright: Zimbabwe School Examinations Council, J2010.
Question 1
Given that \(\cos \theta=-1 / 5\) and \(-180^{\circ}<\theta<-90^{\circ}\), find the exact value of \(\cot \theta\).
Question 2
Differentiate with respect to \(t\);
(i)
\(e^{-2 t} \sin t\),
(ii)
\(\sec ^{2}(3 t-100)\).
Question 3
Solve the inequality \(|3 x+1| \geq 2|x-2|\).
Question 4
Express \(\frac{3 x+8}{(2 x+1)\left(x^{2}+3\right)}\) in partial fractions.
Question 5
A mathematician working with an exponential relation \(y=a b^{x}\) reduced it to linear form and came out with the graph shown in the diagram below.
(i)
State the label on each of the axes.
(ii)
Calculate the value of \(a\) and the value of \(b\).
Question 6
Find the value of \(a\) for which (\(x-2\)) is a factor of \(3 x^{3}+a x^{2}+x-2\).
Show that for this value of \(a\), the cubic equation \(3 x^{3}+a x^{2}+x-2=0\) has only one real root.
Question 7
Write down the equation of a circle with centre \((-3 ; 2)\) and radius \(\sqrt{10}\).
Show that the point \(\mathrm{A}(-2 ;-1)\) lies on the circle, and find the coordinates of B, the other end of the diameter through A.
Question 8
Solve \(\frac{d y}{d x}=x y\) given \(x=0\) when \(y=1\).
Use the series expansion for \(e^{x}\) to write down the first two terms of Maclaurin series for the solution.
Question 9
Given that \(x=\sin ^{2} t\) and \(y=\cos 2 t\), find \(\frac{d y}{d x}\) in its simplest form.
Hence or otherwise describe the shape of the graph of \(y\) against \(x\).
Question 10
The complex number \(z_{1}=1-2 i\) and the complex number \(z_{2}\) is such that
$$ z_{1} z_{2}=-10 i $$
Find \(z_{2}\) in the form \(a+i b\) and sketch it on an Argand diagram.
Question 11
(a)
Find the term independent of \(x\) in the expansion of
$$ \left(x^{2}+\frac{3}{x}\right)^{6}. $$
(b)
Find the series expansion of \(\left(4+x^{2}\right)^{\frac{1}{2}}\) up to and including the term in \(x^{6}\).
Question 12
Use the substitution \(x=a \sin \theta\) to evaluate \(\int_{0}^{a} \sqrt{a^{2}-x^{2}} d x\).
Question 13
The diagram shows a circle centre O and two tangents AP and BP drawn from a point P.
Given that \(\mathrm{AP}=20 \mathrm{~cm}\) and \(\mathrm{AB}=12 \mathrm{~cm}\),
(i)
show that the obtuse angle \(\mathrm{AOB}=2,532\) radians,
(ii)
calculate the radius of the circle,
(iii)
calculate the area of the shaded segment AB.
Question 14
The tangent at point \(P\left(x_{n} ; x_{n}^{2}-2\right)\) to the curve \(y=x^{2}-2\) meets the \(x\)-axis at point \(Q\left(x_{n+1} ; 0\right)\).
If \(x_{n}>0\), show that \(x_{n+1}=\frac{x_{n}^{2}+2}{2 x_{n}}\)
Starting with \(x_{1}=2\), use this iterative formula to find successive approximations \(x_{2}, x_{3}\) for the positive root of the equation \(x^{2}-2=0\).
Hence show that \(x_{4}=\frac{577}{408}\).
Find the error, to 1 significant figure, in using \(\frac{577}{408}\) as an approximation to \(\sqrt{2}\).
Question 15
The function \(f\) is defined as
$$ f: x \rightarrow \frac{2-x}{x+1}, x \neq-1 $$
(a)
#### (i)
Express \(f(x)\) in the form \(a+\frac{b}{x+1}\), where \(a\) and \(b\) are constants.
#### (ii)
Hence, give a sequence of three transformations which take the graph of \(y=\frac{1}{x}\) onto the graph of \(y=f(x)\).
#### (iii)
State the range of \(f\).
(b)
#### (i)
Form the composite function \(f(f(x))\).
#### (ii)
Hence, or otherwise, obtain an expression for \(f^{-1}(x)\).
Question 16
(a)
An established insurance company is planning to launch a new policy package with initial expected sales of 2000 policies by the end of the first year. Policies are expected to increase by 600 per year. Assuming success for such a programme, find the year after starting in which more than 6000 policies are sold.
(b)
Mr Moyo bought a new car for seven hundred million dollars and the car depreciates at \(10 \%\) per year. Find the number of years the car can be used before its resale value is less than five hundred million dollars.
Question 17
The curve \(C\) has equation \(y^{2}-4 y=4 x-12\).
(a)
Find the coordinates of point A at which C meets the \(x\)-axis.
(b)
By completing the square or otherwise, show that the equation of the curve can be written in the form \((y-2)^{2}=4(x-2)\).
(c)
Show that a point Q with coordinates \(\left(2+t^{2} ; 2+2 t\right)\), where \(t\) is any real number, lies on C.
(d)
Find the equation of the tangent at A.
Question 18
(a)
Use the trapezium rule with four equally spaced ordinates, to estimate the value of \(\int_{0}^{\frac{\pi}{6}} \sin ^{4} x d x\), giving your answer to 3 significant figures.
(b)
Show that \(\sin ^{4} x=\frac{3}{8}-\frac{1}{2} \cos 2 x+\frac{1}{8} \cos 4 x\).
Hence, or otherwise,
#### (i)
evaluate exactly \(\int_{0}^{\frac{\pi}{6}} \sin ^{4} x d x\),
#### (ii)
solve the equation \(\cos 4 x-4 \cos 2 x+1=0\) for \(0 \leq x \leq 2 \pi\).