3.1 Practice PDF
3.1 Practice PDF
3.1 Practice PDF
MARCH 2014
Time: 1hr 30 min
GENERAL INSTRUCTIONS
Answer ALL Questions
Unless otherwise stated in the question, any numerical answer that is not exact MUST be written
correct to three significant figures.
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 60.
You are reminded of the need for clear presentation in your answers.
1.
Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a
row.
(a)
[2]
The seven cards are now shuffled and 2 cards are selected at random, without replacement.
(b)
2.
3.
Find the probability that at least one of these 2 cards has D printed on it?
[3]
Maria chooses toast for her breakfast with probability 0.85. If she does not choose toast then she has
a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8. If she has a
bread roll, then the probability that she will have jam on it is 0.4.
(a)
[2]
(b)
Given that Maria did not have jam for breakfast, find the probability that she had toast.
[3]
P=
k 1 .
1 1 k and Q =
3
5 3 2
7
3 2
(a)
[2]
(b)
Given that det (PQ) = 16, find the two possible values of k.
[4]
2
4.
d2 y
dy
5 + 4 y = 8 x 10 10cos 2 x.
2
dx
dx
5.
(a)
Show that =
y 2 x + sin 2 x is a particular integral of the given differential equation.
[3]
(b)
[4]
dy sec 2 x
+
y=
tan x.
dx
tan x
Hence, solve this differential equation given that y = 3 when x = .
4
6.
[2]
[6]
0 1
2 1 .
2 3
(a)
1
Given that A = 5
10
(b)
2 4
6 4 , find the value of p.
p
9
A 1= 1 ( A 2 6 A + 11I ) .
6
r
1
1
=
6
2
2 2
5 2 , find the value of r and the value of s.
s
2
(c)
Given that A
(d)
[2]
[2]
[2]
x
z=
k
x + 2y + z =
5
2 x + 2 y + 3z =
7
giving your answers in terms of k.
[3]
3
7.
There are 10 spaniels, 14 retrievers and 6 poodles at a dog show. 7 dogs are selected to go through to
the final.
(a)
How many selections of 7 different dogs can be made if there must be at least 1 spaniel, at least
2 retrievers and at least 3 poodles?
[4]
2 spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line.
8.
(b)
How many different arrangements of these 7 dogs are there if the spaniels stand together and
the retrievers stand together?
[3]
(c)
How many different arrangements of these 7 dogs are there if no poodle is next to another
poodle?
[3]
A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of
children living in these houses are shown in the following table.
Number of children
None
One
Two
Detached house
24
32
41
At least
three
23
Semi-detached house
40
37
88
35
200
Total
64
69
129
58
320
Total
120
(b)
Find:
(i)
P( D),
[1]
(ii)
P( D R ),
[1]
(iii)
P( D | R ),
[2]
(iv)
P( R | D),
[3]
(i)
[1]
(ii)
Determine whether the events D and R are independent. Justify your answer.
[2]