Download and practise with this ZIMSEC A-Level Mathematics Paper 2 June 2012 question paper. Useful for exam revision, past paper practice, and preparation in Zimbabwe.
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This is the ZIMSEC A-Level Mathematics question paper for Paper 2 from June 2012. Use it for exam practice under timed conditions. The paper code is 9164/2. Time allowed: 3 hours.
# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS**
**9164/2**
**PAPER 2**
**JUNE 2012 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.
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 8 printed pages.**
Copyright: Zimbabwe School Examinations Council, J2012.
©ZIMSEC J2012
**Section (a): Pure Mathematics**
## Question 1
### (i)
Verify that the equation
$$
\sin x+\cos x=e^{x}-1 \text { has a root between }-4 \text { and }-3
$$
### (ii)
Use the Newton-Raphson method to find this root correct to 3 decimal places.
## Question 2
Prove by mathematical induction that \(\sum_{r=1}^{n}(r+3)(2 r+1)=\frac{n}{6}\left(4 n^{2}+27 n+41\right)\)
## Question 3
An isosceles triangle with a constant perimeter of \(2 k\) is such that the equal sides are of each length \(x \mathrm{~cm}\).
### (a)
Find in terms of \(x\) and \(k\), the expression for
#### (i)
the length of the base of the triangle,
#### (ii)
the perpendicular height of the triangle.
### (b)
By first writing the expression for the area, A , of the triangle in terms of \(x\) and \(k\), show that the stationery value of the area occurs when \(x=\frac{2}{3} k\).
## Question 4
Mr Khumalo bought a car at \($ 4000\). A statistical model pertaining to the depreciation assumes that the value, \($ V\), of the car, \(t\) months after it has been bought, decreases at a rate which is proportional to \(V\).
### (i)
Using this model, write down a differential equation connecting \(V\) and \(t\) and show that it reduces to \(V=4000 e^{-k t}\), where \(k\) is a positive constant.
### (ii)
It is further given that after 3 years, the car is valued at \($ 2000\).
Calculate
#### (a)
the value of the car when it is 15 months old, giving your answer to the nearest dollar,
#### (b)
the age of the car when its value is \($ 1600\), giving your answer to the nearest month.
## Question 5
### (a)
The vector equations of the lines \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) are \(\mathbf{r}=(2+\lambda) i+(-3+2 \lambda) j+(1+2 \lambda) \mathbf{k}\) and \(\mathbf{r}=(8+3 \mu) i+(5+2 \mu) j+(13+6 \mu) \mathbf{k}\) respectively.
Find the coordinates of the points of intersection of the lines \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\).
### (b)
The plane \(\pi\) is given by \(2 x-z=3\).
Verify that the lines \(L_{1}\) and \(L_{2}\) both lie in the plane \(\pi\).
### (c)
Given that the position vector of the point A is \(12 i+5 j+6 k\) and B is the foot of the perpendicular from A to \(\pi\).
#### (i)
Find the vector equation of AB ,
#### (ii)
Hence, find the position of vector of \(B\).
## Question 6
### (a)
It is given that the complex number \(a\) whose conjugate is \(\bar{a}\). satisfies the equation \(4 a \bar{a}+12 i=8 a+16\).
Find the two possible values of \(a\), giving your answer in the form \(p+\mathrm{i} q\) where \(p\) and \(q\) are real.
### (b)
The polynomial \(2 x^{4}+x^{3}+17 x^{2}+9 x-9\) is denoted by \(p(x)\).
#### (i)
Show that \(3 i\) is a root of the equation \(2 x^{4}+x^{3}+17 x^{2}+9 x-9=0\).
#### (ii)
State the other complex root of the equation \(p(x)=0\).
#### (iii)
Hence or otherwise find the other two roots of the equation
$$
p(x)=0 .
$$
## Question 7
### (a)
The transformation matrix \(\left(\begin{array}{rr}-2 | 1 3 | 2\end{array}\right)\) maps the line \(y\) onto the line \(y^{1}=6 x+8\).
Find the equation of the line \(y\), giving your answer in the form \(a x+b y=c\) where \(a, b\), and \(c\) are integers.
### (b)
Given that \(\mathrm{M}=\left(\begin{array}{rrr}3 | 2 | 1 -2 | 2 | 0 1 | 0 | 0\end{array}\right)\) and \(\mathrm{N}=\left(\begin{array}{rrr}0 | 1 | 3 1 | 4 | -2 1 | 0 | 2\end{array}\right)\),
find
#### (i)
MN ,
#### (ii)
\(\quad(\mathrm{MN})^{-1}\).
### (c)
Hence find the values of \(x, y\) and \(z\) for which \(\mathrm{MN}\left(\begin{array}{l}x y z\end{array}\right)=\left(\begin{array}{r}3 -6 3\end{array}\right)\).
**Section (b) : Mechanics**
## Question 8
A car travelling along a straight road, passes through a fixed point A with a velocity of \(10 \mathrm{~ms}^{-1}\) and continues with this velocity for \(t_{1}\) seconds. The car then accelerates uniformly over the next \(t_{2}\) seconds till it attains a velocity of \(15 \mathrm{~ms}^{-1}\). The car subsequently decelerates to rest in a further \(t_{3}\) seconds.
### (a)
Sketch a well labelled \((t ; v)\) graph to illustrate the motion of the car after passing A .
### (b)
Given that the car's acceleration and deceleration are equal, show that \(t_{3}=3 t_{2}\).
## Question 9
The diagram shows the trajectory of a ball that is projected from the top of a building 40 m high with initial velocity of \(20 \mathrm{~ms}^{-1}\) at an angle of \(30^{\circ}\) to …