7/29/2019 Unit 2 Module 3 Practice

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UNIT 2: PURE MATHEMATICS

Module 3 Practice

1. Let

1 2 3

. x y z

 y z z x x y

 

(a) Use a row operation to show that ( ) x y z is a factor of  .  

(b) Hence, or otherwise, express as a product of linear factors.

2. Find the general solution of the differential equation

2 2

d 2 1,

d  4

 y y

 x x x x

where 0 2. x  

3. The matrix M is given by

1

3 1 .

4 2 1

a a

a

M  

(a) Find, in terms of a, the determinant of M.

(b) Hence find the values of a for which 1M does not exist.

(c) Determine whether the simultaneous equations

6 6 3

3 6 0

4 2

 x y z k 

 x y z

 x y z k 

 

where k is a non-zero constant, have a unique solution, no solution or an infinite number of solutions,

 justifying your answer.

4. A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers and 

2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least 1 wicketkeeper.

(a) Find the number of different ways in which the team can be chosen.

Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries

and 2 identical notebooks.

(b) Find the number of different arrangements of the presents if they are all displayed in a row.

(c) 10 of these 11 presents are chosen and arranged in a row. Find the number of different arrangements

that are possible.

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6. Laura frequently flies to business meetings and often finds that her flights are delayed. A fight may be

delayed due to technical problems, weather problems or congestion problems, with probabilities 0.2, 0.15

and 0.1 respectively. The tree diagram below shows this information.

(a) Write down the values of the probabilities a, b and c shown in the tree diagram.

One of Laura’s flights is selected at random.

(b) Find the probability that Laura’s flight is not delayed and hence write down the probability that it

is delayed.

(c) Find the probability that Laura’s flight is delayed due to just one of the three problems.

(d) Given that Laura’s flight is delayed, find the probability that the delay is due to just one of the three

 problems.

(e) Given that Laura’s flight has no technical problems, find the probability that it is delayed.

5. A reservoir supplies a large city. At time t days the level of the water above a fixed mark is  x metres, where

 x and t are related by2

2

d d 2 2 30cos3 35sin 3 .

d d 

 x x x t t 

t t   

When 0, 2t x and the water level is rising at a rate of 14 metres per day.

(a) Find  x in terms of t .

(b) Show that, after a long time, the difference between the highest and lowest water levels is approximately

10 metres.