ZIMSEC A-Level Mathematics Paper 4 November 2011 Question Paper

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL General Certificate of Education Advanced Level

**MATHEMATICS**

PAPER 4 NOVEMBER 2011 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 6 printed pages and 2 blank pages.

**Section (a): Statistics**

Question 1

A roulette wheel contains 38 numbers of which 18 are red, 18 are black and 2 are green. When the roulette wheel is spun, it is equally likely to land on any of the 38 numbers. In two plays at the wheel, find the probability that

(a)

the ball lands on red both times,

(b)

the ball lands on green the first time and on black the second time.

Question 2

The continuous random variable X has a probability density function given by

$$ f(x)= \begin{cases}k x, & 0 \leq x \leq 1 \\ k, & 1<x \leq 2 \\ 0, & \text { otherwise }\end{cases} $$

where k is a constant.

Find

(i)

the value of $k$,

(ii)

the median, $m$, of X .

Question 3

In a chemical industry workmen had a 20% chance of suffering from an occupational disease.

Find the number of workmen who could have been selected at random before the probability that at least one of them contracted the disease, became greater than 0.9 .

Question 4

After some rain the depth of moisture, $X$ metres, in Arda Gardens can be taken as a continuous random variable with a probability density function

$$ f(x)=\left\{\begin{array}{l} \frac{12 x}{5}(b-x), \\ 0, \text { otherwise } \end{array}, 0 \leq x \leq 1\right. $$

(a)

Find the value of $b$.

(b)

Calculate the probability that the depth of moisture exceeds 0.9 .

Question 5

The meteorological department of a certain country adopts a simple model of the weather in which each day is classified as either fine or rainy. The probability that a fine day is followed by another fine day is 0.8 . The probability that a rainy day is followed by a fine day is 0.4 . The probability that 1 February is fine is 0.75 .

Using a tree diagram or otherwise, find the probability that

(a)

the $3^{\text {rd }}$ February is fine,

(b)

the $1^{\text {st }}$ February was rainy given that $3^{\text {rd }}$ February is fine.

Question 6

(a)

State the conditions under which a normal distribution be used to approximate a binomial distribution.

(b)

It is estimated that 20% of people undergoing medical review are men. If a random sample of 100 people are undergoing a medical review, find the probability that more than 30 are men.

Question 7

An unbiased tetrahedral die has the number 1 written on one face, the number 2 on the other face and the number 3 on the remaining two faces. The die is thrown twice and X is the product of the scores obtained from the two throws.

(a)

Find the probability distribution of X .

(b)

Find $\mathrm{E}(\mathrm{X})$ and $\operatorname{Var}(\mathrm{X})$.

Question 8

The number of patients admitted at a medical centre each day is found to have a Poisson distribution with mean 2.

(a)

Evaluate the probability that on a particular day there will be no admissions.

(b)

At the beginning of one day, the hospital has five beds available. Calculate the probability that this will be an insufficient number for the day.

(c)

Calculate the probability that there will be exactly three admissions altogether on two consecutive days.

(d)

150 patients are attended to, at the centre on a particular day and the probability that a patient will be admitted is 0.02 .

Using a suitable approximation, find the probability that exactly 4 patients are admitted.

Question 9

Boxes marked B contain big fruits and boxes marked S contain small fruits. The masses of the boxes are continuous random variables having independent normal distributions with means and standard deviations given in the table below.

| Size of fruit | Mean mass of a box $(\mathrm{kg})$ | Standard deviation $(\mathrm{kg})$ | | :---: | :---: | :---: | | Big | 10 | 2 | | Small | 12 | 3 |

(a)

Find the probability that the mass of

#### (i)

a box marked S is less than 10 kg ,

#### (ii)

4 big fruit boxes and 5 small fruit boxes is greater than 90 kg .

(b)

Find the value $m$ such that $\mathrm{P}\left(\mathrm{B}_{1}+\mathrm{B}_{2}<m\right)=\frac{1}{4}$ where $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ are independent observations.

Question 10

A sports director wants to know whether the interest distribution of form one students in sporting disciplines is different from the form two interest distribution. The form two interest distribution is given in the Table 1.

Table 1

| Sporting discipline | Percentage | | :--- | :---: | | Cricket | 21.1 | | Hockey | 27.0 | | Rugby | 33.9 | | Soccer | 18.0 |

A random sample of 200 form ones was taken and gave the results in Table 2.

Table 2

| Sporting discipline | Frequency | | :--- | :---: | | Cricket | 42 | | Hockey | 62 | | Rugby | 64 | | Soccer | 32 |

Show, at 5% level of significance, whether the data provide sufficient evidence to conclude that the current form one interests are different from form two interest distribution.

Question 11

At Kurerana High School, 8 students studying chemistry prepared for a test and the time, T, in hours, each student spent studying was recorded. The test was marked out of 50 and the results of the scored mark, M, for each student were as follows:

| Study time (T) | 4 | 3 | 4 | 5 | 4 | 7 | 7 | 8 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Scored mark (M) | 37 | 32 | 35 | 40 | 40 | 44 | 42 | 48 |

(a)

Plot a scatter diagram showing study time, T , against the mark, M .

(b)

Calculate the equation of the regression line $\mathrm{M}=a+b \mathrm{~T}$ where $a$ and $b$ are constants to be determined.

(c)

Draw the regression line on the graph and use it to estimate the study in hours and minutes for a student who scored 41 marks.

(d)

Find the product moment correlation coefficient and comment on the relationship between study times and test marks.

**Section (b): Mechanics**

Question 12

Two forces have magnitudes $P$ and $Q$ and the angle between them, $\theta$, is acute. If the resultant of these two forces has magnitude $R$, show that $R^{2}=P^{2}+Q^{2}+2 P Q \cos \theta$.

Question 13

A body of mass 5.2 kg is held in equilibrium on a rough plane, by a force P N acting up the line of greatest slope. The plane is inclined at an angle $\theta$ to the horizontal where $\cos \theta=\frac{4}{5}$. When $P$ is 19.2 N , the body is about to slide down the plane.

Find the value of $\mu$, the coefficient of friction between the body and the plane.

Question 14

Two particles are projected simultaneously from two points A and B on level ground which are 150 m apart. The first particle is projected vertically upwards from A with an initial speed of $U \mathrm{~m} / \mathrm{s}$, and the second particle is projected from B towards A with an initial velocity $V \mathrm{~m} / \mathrm{s}$ at an angle of projection $\propto$. If the particles collide when they are both at their greatest height above the level AB, prove that $\tan \alpha=\frac{U^{2}}{150 g}$.

Question 15

Two vehicles moving in the same direction pass the point O on a straight road at time $t=0$. Vehicle A is moving at a constant velocity of $11 \mathrm{~ms}^{-1}$. Vehicle B has a constant acceleration of $2 \mathrm{~ms}^{-2}$ and it has a velocity of $3 \mathrm{~ms}^{-1}$ as it passes O .

(a)

On the same diagram, draw the velocity-time graph for the two vehicles.

(b)