ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** <br> **PAPER 1**
JUNE 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.
This question paper consists of 5 printed pages and 3 blank pages.
Copyright: Zimbabwe School Examinations Council, 2009.
Question 1
Find the value of \(k\) for which \((x+2)\) is a factor of \(f(x)=x^{3}+2 x^{2}-k x-1\). Hence factorise \(f(x)\) completely.
Question 2
The line \(y=2 x+1\) intersects the curve \(y^{2}-x y=3\) at A and B. Find the coordinates of the mid-point of AB.
Question 3
The function \(f\) is defined by \(f: x \mapsto x^{2}-4 x, x \in \mathbb{R},|x| \leq 1\).
Show by means of a graphical argument or otherwise, that \(f\) is one-one, and find an expression for \(f^{-1}(x)\).
Question 4
Given that \(y=\ln (t+2)\) and \(x=t\) for \(t>-2\), find
(i)
an expression for \(\frac{d y}{d x}\) in terms of \(t\),
(ii)
the values of \(t\) for which \(y\) increases as \(x\) increases.
Question 5
(a)
Solve the inequality \((0.2)^{2 x-3}>10^{6}\).
(b)
Solve the equation \(3 \sqrt{x}-11=4 x^{-1 / 2}\).
Question 6
(a)
Find \(\int x \ln (2 x) d x\).
(b)
Solve the inequality \(|4 x-1| \leq|2 x+7|\).
Question 7
The complex number \(p=3-5 i\) and it is given that \(q=4 i p\).
(a)
State the relationship between
#### (i)
\(|p|\) and \(|q|\).
#### (ii)
\(\arg (p)\) and \(\arg (q)\).
(b)
Given that \(r=p+q\), find \(r\) in the form \(a+b i\) where \(a\) and \(b\) are real numbers.
(c)
The points P, Q and R in an Argand diagram represent the complex numbers \(p\), \(q\) and \(r\) respectively.
#### (i)
State the kind of quadrilateral that OPRQ is, where O is the origin.
#### (ii)
Find the area of OPRQ.
Question 8
Given that \(y=\frac{1}{\sin 3 x}\), show that
(i)
\(\frac{d y}{d x}=-3 \cot 3 x \operatorname{cosec} 3 x\).
(ii)
\(\frac{d^{2} y}{d x^{2}}+9 y=0\).
Question 9
(a)
Triangle ABC is right angled at A and has \(\mathrm{AC}=20\) and angle \(\mathrm{ACB}=\theta\) radians.
Given that the area of region P is 5 times the area of region Q, show that \(6 \theta=5 \tan \theta\).
(b)
Given that the first two terms of a geometric progression are 4 and 6 respectively, calculate the sum of the first 20 terms, giving your answer correct to the nearest integer.
(c)
The first term of an arithmetic progression is 6 and the sum of the first 36 terms is 90. Find the common difference.
Question 10
(a)
Satellite pictures are used to measure the area \(A \mathrm{~km}^{2}\) of a lake after time \(t\) weeks. The results for the first four weeks are given in the table below.
| \(t\) (weeks) | 0 | 1 | 2 | 3 | 4 | | :--- | :--- | :--- | :--- | :--- | :--- | | \(A\left(\mathrm{~km}^{2}\right)\) | 76 | 70 | 62 | 51 | 31 |
Draw a graph of \(A\) against \(t\) and use it to determine whether the rate of decrease of area is constant.
(b)
In a second lake it is found that the rate of decrease of area is inversely proportional to the area at that time. Write down a differential equation relating the area \(A \mathrm{~km}^{2}\) and time \(t\) weeks.
Solve this differential equation given that when \(t=1, A=8\) and when \(t=5\), \(A=6\). Hence find the area when \(t=0\).
Question 11
The points \(A, B\) and \(C\) have position vectors \(a=2 i+j-k, \quad b=3 i+4 j-2 k\) and \(c=5 i-j+2 k\) respectively relative to the origin \(O\).
(a)
Evaluate the scalar product \((a-b) \cdot(c-b)\). Hence calculate the size of angle \(A \hat{B} C\), giving your answer to the nearest \(0.1^{\circ}\).
(b)
Given that ABCD is a parallelogram, determine
#### (i)
the position vector of \(D\),
#### (ii)
the area of ABCD giving your answer in exact form.
Question 12
(a)
Express in their simplest form
#### (i)
\(e^{\ln x+\ln y}\).
#### (ii)
\(\ln e^{2 x}\).
(b)
The number \(N\) of bacteria in a certain culture at time \(t\) hours is given by \(N=600 e^{c t}\) where \(c\) is a constant.
Show that at any given time \(t\), the number of bacteria is increasing at a rate proportional to the number of bacteria present at that time.
The number of bacteria increases from 600 when \(t=0\) to 1800 when \(t=2\).
#### (i)
Show that \(c=\frac{1}{2} \ln 3\).