ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS ..... 9164/2** **PAPER 2** **NOVEMBER 2010 SESSION** **3 hours**
**Section (A): Pure Mathematics**
Question 1
If \(A+B=45^{\circ}\), show that
$$ \tan A+\tan B+\tan A \tan B=1 $$
Hence find the exact value of \(\tan 22 \frac{1}{2}^{\circ}\) in its simplest form.
Question 2
Prove by mathematical induction that \(7^{n}-6n-1\) is divisible by 36 for \(n \in \mathbb{N}, n>1\).
Question 3
(a)
Using the Maclaurin series for \(e^{x}\) and \(\sin x\), write down the series for \(e^{-2x}\) and \(\sin 3x\), up to the \(x^{3}\) terms, simplifying the terms.
(b)
Using your results in part (a) find the series expansion of \(f(x)=e^{-2x}\sin 3x\), up to the term in \(x^{3}\).
(c)
By differentiating \(f(x)\) once, check your results in (b) up to the term in \(x\).
Question 4
(a)
The point A and B are given by the coordinates \((1;2;4)\) and \((3;-2;1)\) respectively.
Find
#### (i)
\(\overrightarrow{AB}\),
#### (ii)
\(|\overrightarrow{AB}|\),
#### (iii)
the vector equation of the straight line that passes through the points A and B.
(b)
Given that the plane \(p\) has equation \(r \cdot (2i+j-3k)=28\) and the equation of plane \(q\) is \(r \cdot (4i-7j+k)=31\), find
#### (i)
a vector equation of the line of intersection of the two planes,
#### (ii)
the angle between plane \(p\) and plane \(q\) giving your answer correct to the nearest \(0.1^{\circ}\).
Question 5
(a)
A sequence of matrices is given by \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}^{n}\) where \(n \in \mathbb{N}\).
Evaluate the first three terms of the sequences.
State the behaviour of the sequence as \(n\) increases.
(b)
Given that \(A=\begin{pmatrix}1 & x & y\end{pmatrix}\), show that \(|A|=2\) is the equation of a straight line in \(x\) and \(y\).
(c)
If \(T=\begin{pmatrix}1 & 0 \\ 2 & 1\end{pmatrix}\) and \(M=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\), describe completely the single transformations represented by each of the matrices M and T.
Find the coordinates of the point whose image under \(TM^{-1}\) is \((3;1)\).
Question 6
(a)
The complex number \(z=a+bi\) where \(a\) and \(b\) are positive real numbers.
#### (i)
Given that \(w=iz\), write down \(w\) in terms of \(a\) and \(b\) and explain the geometrical relationship between \(z\) and \(w\).
#### (ii)
Another complex number \(v=\frac{1}{2}z+w\).
Represent clearly on the same Argand diagram the complex numbers \(z\), \(w\) and \(v\).
Find \(v\) if the complex number \(z=3+2i\).
(b)
Use De Moivre's theorem to find the 4 roots of unity giving your answers in exponential form.
Question 7
(a)
Prove that \(\frac{\cos^{4}\theta-\sin^{4}\theta}{\sin\theta\cos^{3}\theta+\sin^{3}\theta\cos\theta}=\frac{\cos^{2}\theta-\sin^{2}\theta}{\sin\theta\cos\theta}\)
Hence show that \(\frac{\cos^{4}\theta-\sin^{4}\theta}{\sin\theta\cos^{3}\theta+\sin^{3}\theta\cos\theta}=2\cot 2\theta\).
(b)
The diagram shows a rectangular piece of cardboard which is fixed against a vertical wall. The heights, in cm, of A, B and C above the ground are 41, 62 and 104 respectively.
#### (i)
Given that \(AB=3BC\) and that AB and BC are inclined at \(\theta^{\circ}\) and \(\alpha^{\circ}\) respectively to the horizontal, show that
1. \(\theta+\alpha=90\),
2. \(6\sin\theta=\sin\alpha\).
#### (ii)
Solve for \(\theta\) and hence calculate the length of AB.
#### (iii)
Find the height of D above the horizontal.
**Section (B): Mechanics**
Question 8
A stone is projected from point A with a speed of \(20 ms^{-1}\), at an angle of \(30^{\circ}\) to the horizontal, towards a bird 30 m away. The bird is standing on a pole \(2.5 m\) vertically above the level ground, (see diagram).
Assuming that the path of the stone is not impeded, determine whether the stone will hit the bird and if not by how far vertically above or below the bird does the stone miss it.
Question 9
A block of weight 20 N is placed on a rough horizontal plane inclined at an angle \(\beta=4/5\). The block is acted on by a horizontal force of P N and is about to slip downwards, (see diagram).
Given that the coefficient of friction between the block and plane is \(\frac{1}{4}\), show that the horizontal force, P N, needed to prevent the block from slipping is 16.25 N.
Question 10
Masses of 2 kg and 5 kg are connected by a light inextensible string and hang vertically on either side of a smooth fixed pulley. The system is released from rest with both portions of string vertical and both particles at a height of 2.5 metres above the ground (see diagram).
(a)
Find the acceleration of the particles in terms of \(g\).
(b)
Find the velocity of the 5 kg mass as it hits the ground.
(c)
If the 5 kg mass hits the ground and does not rebound, find the greatest height reached by the 2 kg mass if it does not hit the pulley.
Question 11
A particle starts from rest and moves in a straight line. Its acceleration \(a ms^{-2}\) is given by:
$$ a = \begin{cases} 3 & 0 \leq t \leq 2 \\ -3 & 2 < t \leq 8 \end{cases} $$
**Section (C): Statistics**
Question 12
In a large batch of bolts, the probability that a randomly chosen bolt is defective is \(1/6\). A random sample of 42 bolts is taken, find the probability that more than two bolts are defective.
Question 13
The probability density function for a discrete random variable \(X\), is given as follows:
$$ \begin{aligned} & P(X=x)=\frac{x-1}{16} \text{ for } x=2,3,4,5 \\ & P(X=x)=\frac{9-x}{16} \text{ for } x=6,7,8 \end{aligned} $$
(a)
Draw up a table showing the probability distribution of X.
(b)
Calculate \(E(X)\).
State the nature of the distribution.
Question 14
A continuous random variable X has a probability density function,
$$ f(x)=\begin{cases} kx, & 0 \leq x \leq 3 \\ 3k(4-x), & 3 \leq x \leq 4 \\ 0, & \text{otherwise} \end{cases} $$
where \(k\) is a constant.
(a)
Find the value of \(k\) and sketch the graph of \(f(x)\).
(b)
Find the probability that \(x>2\).
Question 15
Three players A, B and C, in that order, throw a fair cubical die. The first to throw a 6 wins. The game is continued indefinitely until one of the players wins.
(a)
Find the probability that A wins
#### (i)
on his first throw,
#### (ii)
on his second throw,
#### (iii)
the game.
(b)
Given that the probability that B wins is \(30/91\), find the probability that C wins.