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SENIOR FIVE REVISION QUESTIONS-APPLIED MATHEMATICS P425/2 AUG.2012

1. A car takes 5 minutes to cover a distance of 3km between two stations A and B. Starting from

rest at A , it accelerates at a constant rate to a speed of 40kmh-1 and maintains this speed until it

is brought uniformly to rest at B. If the car takes three times as long to decelerate as it does to

accelerate, find the time taken by the car to accelerate.

2. A ball is thrown vertically upward with speed of 14ms-1. Two seconds later a second ball is

dropped from the same point. Find where the two balls meet?

3. A random variable X has the following probability distribution.

P(X = 1) = 0.1, P(X = 2) = 0.2, P(X = 3) = 0.3 and P(X = 4) = 0.4

(a) Show that the distribution above is a discrete probability distribution.

(b) Determine the expected value of X.

4. A particle of mass 5kg is pulled along a rough horizontal surface by a string which is inclined at 60o to the horizontal. If the acceleration of the particle is [pic]gms-2 and the coefficient of friction between the particle and the plane is [pic], find the tension in the string.

(READ ABOUT FRICTION BEFORE YOU ATTEMPT THIS QUESTION)

5. A farmer keeps two breeds A, B of chicken, 70% of the egg production is from birds of breed

A. Of the eggs laid by the A hens, 30% are large, 50% medium and the remainder small: For

the B hen the corresponding values are 40%, 30% and 30%. Egg color (brown or white) is

manifested independently of size in each breed, 30% of A eggs and 40% of B eggs are brown.

Find:

(a) The probability that an egg laid by an A hen is large and brown.

(b) The probability that an egg is large and brown.

6. (a) John wishes to send a message to Mary. The probabilities that he uses e-mail, letter or

personal contact are 0.4, 0.1 and 0.5 respectively. He uses only one method; the

probabilities of Mary receiving the message if John uses e-mail, letter or personal

contact are 0.6, 0.8 and 1 respectively.

(i) Find the probability that Mary receives the message.

(ii) Given that Mary receives the message find the probability that she received it

via e-mail

(b) If A and B are two events and P(A) = 0.6, P(B) = 0.3 and P(A∪B) = 0.8, find

(i) P(A∩B)

(ii) P(Ᾱ∩[pic])

(C ) Two events A and B are such that P (A) = 2/5, P(A n B1) = 3/10, P(A1/B) = 3/5.

Find (i) P(A n B (ii) P(A/B)

7. The weights of Buffalos in a certain game park were according to the following frequency

table.

|Weight (kg) |Frequency (f) |

|400 – 449 |4 |

|450 – 499 |7 |

|500 – 549 |6 |

|550 – 599 |13 |

|600 – 649 |9 |

|650 – 699 |1 |

(a) Estimate the mean and standard deviation of the weights of the Buffalos.

(b) Draw a cumulative frequency curve and use it to estimate the semi –interquartile

range.

8. Two particles P and Q are moving along a straight path. When Q is ahead of P by [pic] the speed of Q is [pic]and that of P is [pic].Given that P and Q have a constant retardation of [pic] and [pic] respectively, find the distance Q has travelled when it is first overtaken by P.

9. A particle moves so that after t seconds its displacement S is given by

S = (3t2 + 1)i+ (t4 – 5t)k. Find

i) its speed after 3 seconds

ii) its acceleration after 3 second

(Apply [pic] )

10. (a) The resultant of forces F1 = 3i + (a – c)j, F2 = (2a + 3c)i + 5j, F3 = 4i + 6j acting on a

particle is 10i + 12j

Find

i) values of a and c

ii) magnitude of force F2

(b) The figure below shows four forces acting on a particle

5N 1N

30o 45o

3N

5N

i) Find the resultant of the forces and its direction.

ii) The magnitude and direction of the 5th force required to keep the particle in equilibrium.

11. 100 students of a certain school sat an examination that was marked out of 135. Below

are the marks and number of students.

|Marks |45 – 55 |55 - |65 - |75- |

|P( x = x) |3/16 |½ |¼ |1/16 |

Find the mean and the variance of x

b) Two events A and B are neither independent nor mutually exclusive. Given that P(B) = 1/3

P(A) = ½ and P(A∩B’ ) = 1/3 , find

i) P( A’ U B’ ) (ii) P( A’/B’ )

(c) Given that A and Bare independent events such that P( A' ) = 3/5 and

P(AuB)= 4/5, find P(B) and P(A' u B').

(d ) A Mathematics student measured the times taken in seconds for a trolley to run down slopes of varying gradients and obtained the following results;

35.2 , 34.5 , 33.5 , 29.3 , 30.9 , 31.8

Calculate;

i) the mean times

ii) the standard deviation

13. (a) A particle travelling in a straight line with constant acceleration covers distances S1 and S2 in the third and fourth seconds of its motion respectively.

Show that its initial speed U is given by; U = ½ ( 7 S1 – 5 S2 )

(b) Forces of magnitude 2N, 3N, 4N, 4N and 5N act along the sides AB, BC, CD, DA and AC

respectively of the square ABCD of side a, the direction of the forces being given by the

order of the letters. Determine the resultant force.

(c) Forces 8N, 10N and 12N act along the sides AB, BC and CA of an

equilateral triangle. Find the magnitude and the direction of the resultant

force with AB.

14. The table below gives the cumulative distribution of the heights of 400 children in a certain school

|Height (cm) |Cumulative frequencies |

|< 100 |0 |

|< 110 |27 |

|< 120 |85 |

|< 130 |215 |

|< 140 |320 |

|< 150 |370 |

|< 160 |395 |

|< 170 |400 |

(a) Draw the cumulative frequency curve and use it to determine the estimated;

(i) median

(ii) interquartile range

(iii) the 10 to 90 percentage range

(b) 8 students took an examination in Mathematics and Physics and their grades were as follows;

|Maths |A |A |D |O |D |B |F |

|Physics |C |C |E |E |C |A |F |

Calculate the rank correlation coefficient between the grades and comment on your results

15. A particle of mass m kg is projected vertically upwards with speed um/s

and when it reaches its greatest height, a second particle of mass 2mkg is

projected vertically upwards with a speed of 2u m/s from the same point as

the first. Find the time taken between when the second particle is projected and its collision with the first particle.

16. (a) The probability that a married man watches a football match on television is 0.4 and the probability that a married woman watches the same match is 0.5.The probability that the man watches the match, given that his wife does, is 0.7. Find:

i) The probability that a married couple watches the football match

ii) The probability that a wife watches the match given that her husband does.

b) There are 3 black and 2 white balls in each of the two bags. A ball is taken from the first

bag and put in the second bag, then a ball is taken from the second into the first.

What is the probability that there are now the same number of black and white balls in each bag as there were to begin with?

17. In order to determine whether or not there is any correlation between secondary school pupils’ performances in mathematics and English two standard tests (one in mathematics and the other in English) were given to 12 pupils. The following marks were recorded:

|Pupils |A |

|Under 10 |28 |

|10 and under 20 |54 |

|20 and under 30 |81 |

|30 and under 40 |57 |

|40 and under 50 |23 |

|50 and under 60 |7 |

|60 and over |0 |

a) Calculate the mean and the standard deviation of the time taken.

b) Plot an Ogive and use it to estimate;

i) the median time

ii) number of faults repaired between 27 and 42 minutes.

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