ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS**
**PAPER 1**
9164/1
3 hours
Additional materials: Answer paper Graph paper List of Formulae
JUNE 2011 SESSION
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, 2011.
Question 1
Solve the inequality \(\left|3^{-3 x}-54\right|<27\).
Question 2
Find the expansion of \(\sqrt{\left(4-3 x^{2}\right)}\) up to and including the term in \(x^{4}\).
Question 3
Point \(Q\) lies on the line with equation \(y-3 x=0\). Given that \(Q\) also lies on the perpendicular bisector of the line joining \(M(6 ; 3)\) and \(N(2 ; 1)\), find the coordinates of point \(Q\).
Question 4
Find the value of \(x\) which satisfies the equation
$$ 3\left(2^{2 x+2}\right)-2^{1-2 x}=-5 . $$
Find \(3\left(2^{2 x+2}\right)-2^{1-2 x}=-5\)
Question 5
Using the trapezium rule with 5 trapezia, find the approximate value of the integral \(\int_{0}^{1} x^{2} e^{5} d x\) correct to 4 decimal places.
Hence given that the exact value of the integral is \(e-2\), calculate the percentage error in your approximation.
Question 6
In the diagram above, the lengths of the sides \(BC\), \(CA\) and \(AB\) are denoted by \(a\), \(b\) and \(c\) respectively. \(D\) is the foot of the perpendicular from \(A\) to \(BC\), \(AD=x\) and \(DC=y\).
(i)
Show that \(\cot \hat{\mathrm{C}}=\frac{a}{c} \operatorname{cosec} \hat{\mathrm{~B}}-\cot \hat{\mathrm{B}}\).
(ii)
Hence find, correct to the nearest \(0.1^{\circ}\), the angles A and C given that \(a=14.7 \mathrm{~cm}\), \(c=17.3\) and \(\hat{B} \approx 64.2^{\circ}\).
Question 7
The diagram shows the graph of \(y=f(x)\). Sketch, on separate diagrams showing the images of \(A\), \(B\) and \(C\) the graphs of
(i)
\(y=f(2 x)\),
(ii)
\(y=f(-x)+3\),
(iii)
\(y=2 f(x+1)\).
Question 8
Given that \(y=e^{2 x} \sin 5 x\), show that \(\frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+29 y=0\).
Question 9
It is given that \(z_{3}=2-4 i\) and \(z_{2}=6-2 i\).
(a)
Find \(z_{1}-z_{2}\) and \(z_{1} z_{2}\) in the form \(a+i b\).
(b)
If \(w=\frac{1}{z_{1}}\), obtain the exact values of the modulus and argument of \(w\).
Question 10
A curve has parametric equations given by \(x=t-\ln (2 t+1)\) and \(y=t+\ln (2 t+1)\) where \(t>-0.5\).
(i)
Show that \(\frac{d y}{d x}=\frac{2 t+3}{2 t-1}\).
(ii)
Find the equation of the tangent to the curve at \(t=1\) giving your answer in the form \(y=m x+c\).
Question 11
(a)
Solve the equation
$$ \cos 2 x=\cos x \text { for } 0 \leq x \leq 2 \pi $$
(b)
Given that \(f(x)=\cos 2 x-\cos x, 0 \leq x \leq 2 \pi\), find the \(x\) values of the stationary points.
Question 12
A teacher earns \(\$ x\) in his first year of working. If his annual salary increases by \(10 \%\) of his first year's salary
(i)
show that his total salary after \(n\) years is \(\frac{n x(n+19)}{20}\),
(ii)
Hence calculate the least value of \(n\) when his total salary exceeds 100 times his first salary.
Question 13
In the diagram above, \(PQRS\) is a square. The position vector of \(Q\) relative to an origin, \(O\), is given by \(\overrightarrow{\mathrm{OQ}}=\left(\begin{array}{r}3 \\ -1 \\ 2\end{array}\right)\) and the displacement vectors \(\overrightarrow{\mathrm{PR}}=\left(\begin{array}{r}-3 \\ 3 \\ 12\end{array}\right)\) and \(\overrightarrow{\mathrm{SQ}}=\left(\begin{array}{r}5 \\ -11 \\ 4\end{array}\right)\).
(a)
Find
#### (i)
\(\overline{\mathrm{PQ}}\),
#### (ii)
the position vectors of \(P\), \(R\) and \(S\).
(b)
Calculate the area of the square PQRS.
Question 14
(i)
By using the substitution \(y=\sqrt{x}\) or otherwise, find the exact value of \(\int_{0}^{4} \frac{1}{\sqrt{x}+1} d x\).
(ii)
Using integration by parts, show that
$$ \int_{0}^{1} \frac{\ln (\sqrt{x}+1)}{\sqrt{x}} d x=[2 \sqrt{x} \ln (\sqrt{x}+1)]_{0}^{1}-\int_{0}^{1} \frac{d x}{\sqrt{x}+1} $$
Hence find the exact value of the integral.
Question 15
(a)
Solve the equation \(e^{x}-e^{-x}=0\).
(b)
On the same diagram, sketch the graphs of \(y=e^{x}\), \(y=e^{-x}\) and \(x=1\).
(c)
Calculate
#### (i)
the exact area of the region R bounded by the two curves and the line \(x=1\),
#### (ii)
the exact value of the volume, \(V\), generated when the region R is rotated through \(2 \pi\) radians about the \(x\)-axis.
Question 16
Bread from an oven is placed in a cardboard box of initial temperature of \(20^{\circ} \mathrm{C}\). The rate at which the bread cools down is directly proportional to the difference between its temperature and that of the cardboard box.
(a)
#### (i)
Given that \(\theta^{\circ} \mathrm{C}\) is the temperature of the bread at time \(t\) minutes, write down a differential equation relating \(\theta\) and \(t\).
#### (ii)
Find the general solution of the differential equation, expressing \(\theta\) in terms of \(t\).
(b)
Initially \(\theta=100\) and \(\theta=80\) when \(t=10\),
#### (i)
find the particular solution of the differential equation.
Hence find
#### (ii)
the temperature of the bread at \(t=20\),
#### (iii)
the time to the nearest minute at which the temperature of the bread has dropped by \(40^{\circ} \mathrm{C}\).
(c)
Sketch the graph of \(\theta\) against \(t\).