# Principal Mathematics

P 425/1

PRINCIPAL MATHEMATICS

TIME: 3 hours

INTERNAL MOCK EXAMINATION 2016

PAPER 2

INSTRUCTIONS:

• Attempt all the questions in section A and ONLY five questions in section B.
• Any extra questions attempted will not be marked.
• All the necessary working must be clearly shown.
• Indicate clearly in a grid the questions you have attempted.

Section A

1. Below are the weights of books

8 / 9 / 6 / 12 / 11 / 12 / 9 / 12 / 7 / 7 / 6 / 12 / 12 / 13

Use the data above to compute the mean, range and standard deviation of the data above

(5 marks)

1. Using six ordinates of the trapezium rule estimate the value of

to four significant figures, hence an appropriate percentage error in the evaluation above.

(5 marks)

1. A random variable B has mean 1.8 and variance 0.99. If B follows a binomial distribution,Draw a frequency distribution table for this Random variable and hence find
1. The mode of this distribution
2. The probability (5 marks)
1. A random variable has a distribution of the form,

Find c and the expected value of the distribution.(5marks)

1. Given the numbers A and B with A = 7.35 B = 9.214 measured to the nearest decimal places indicted,

a.Determine the absolute error in .

b.Find the limits within which the quotient lies correct to 3 decimalplaces. (5marks)

Section B

1. In a certain farm maize is grown in bags of mean weight 40kg and standard deviation of 2kg. Given that the weight is normally distributed, find
1. The probability that the weight of any bag taken at random will be between 41.0kg and 42.5kg. (3marks)
2. The percentage of bags whose weight exceeds 43kg.(3marks)
3. Calculate 99.9% confidence limits of the weight of the bags.(6marks)
1. Below are marks for ten students who were tested in Mathematics and English.

Student / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
English / 40 / 90 / 54 / 32 / 80 / 65 / 55 / 48 / 55 / 30
Maths / 68 / 40 / 47 / 64 / 55 / 41 / 62 / 76 / 74 / 80
1. Draw a scatter diagram to represent the above data and hence comment on your graph.
2. Draw the line of best fit and use it to estimate the mark a student who scored 60 in mathematics would get in English.
3. Calculate the rank correlation coefficient between mathematics and English. Hence comment on the result.
1. In a football match of Russia versus England, the Russian supporters formed an eighth of the total spectators in the field. The probability that a spectator in the field was engaged in fighting during that match was ½. If the probability that a person who was not Russian fought was 20/21, What is the probability that
1. A Russian did not fight
2. A person caught fighting was Russian(4 marks)
1. John plays two games of squash. The probability that he wins his first game is 0.3. If he wins his first game, the probability that he wins his second game is 0.6. If he loses his first game, the probability that he wins his second game is 0.15. Given that he wins his second game, find the probability that he won his first game (4 marks)
2. Dan has a pack of 15 cards. 10 cards have a picture of a robot on them and 5 cards have a picture of an aeroplane on them. Emma has a pack of cards. 7 cards have a picture of a robot on them and x − 3 cards have a picture of an aeroplane on them. One card is taken at random from Dan’s pack and one card is taken at random from Emma’s pack. The probability that both cards have pictures of robots on them is 7/18. Write down an equation in terms of x and hence find the value of x. (4 marks)
1. Show that the equation

has a real root between and (3marks)

1. Using linear interpolation, find the first approximation for this root. (3marks)
2. Using the Newton-Raphson formula, find the value of this root correct to 3 significant figures. (6marks)

1. / Solution
x / f / fx / fx2
6 / 2 / 12 / 72
7 / 2 / 14 / 98
8 / 1 / 8 / 64
9 / 2 / 18 / 162
11 / 1 / 11 / 121
12 / 5 / 60 / 720
13 / 1 / 13 / 169
/ 136 / 1406

2. /
0 / Y0 / 7.389056
0.2 / Y1 / 6.049647
0.4 / Y2 / 4.953032
0.6 / Y3 / 4.0552
0.8 / Y4 / 3.320117
1 / Y5 / 2.718282
/ 10.10734 / 18.378

3. /

4. / x / /
0 / 1C / 0
1 / 0.8C / 0.8C
2 / 0.64C / 1.28C
3 / 0.512C / 1.536C
4 / 0.4096C / 1.6384C
5 / 0.32768C / 1.6384C
6 / 0.262144C / 1.572864C
/
For a Discrete PDF,

The Expectation of x E(X)=

5.
6.
7. / Solution
b(12 marks)

For the mid-point to the left,

For the mid-point to the right,

Student / English x / Maths y / / /
10 / 30 / 80 / 1 / 10 / 81
4 / 32 / 64 / 2 / 6 / 16
1 / 40 / 68 / 3 / 7 / 16
8 / 48 / 76 / 4 / 9 / 25
3 / 54 / 47 / 5 / 3 / 4
7 / 55 / 62 / 6.5 / 5 / 2.25
9 / 55 / 74 / 6.5 / 8 / 2.25
6 / 65 / 41 / 8 / 2 / 36
5 / 80 / 55 / 9 / 4 / 25
2 / 90 / 40 / 10 / 1 / 81
549 / 607 / 288.5
8. / Let the probability that a spectator is Russian be P(R)= 1/8
Let the probability that a spectator is not Russian be P(R)= 7/8
Let the probability that a spectator fought be P(F)=12/13
Let the probability that a spectator fought be P(F)=1/13
If

b.ii.

c.

d.
Probability that two cards with robots are picked by both Emma and Dan

9. / x / -3 / a / -2
f(x) / -1 / 0 / 3

1. N.R.M

xo / -2.75
x1 / -2.8939
x2 / -2.8795
x3 / -2.8794
x4 / -2.8794
x5 / -2.8794
10