1) The document provides instructions for a mathematics assessment consisting of 8 questions covering topics including probability, matrices, differential equations, and combinatorics.
2) It instructs students to show working clearly and provide numerical answers to 3 significant figures unless otherwise specified.
3) The assessment is worth a total of 60 marks and requires answers for all questions.
1) The document provides instructions for a mathematics assessment consisting of 8 questions covering topics including probability, matrices, differential equations, and combinatorics.
2) It instructs students to show working clearly and provide numerical answers to 3 significant figures unless otherwise specified.
3) The assessment is worth a total of 60 marks and requires answers for all questions.
1) The document provides instructions for a mathematics assessment consisting of 8 questions covering topics including probability, matrices, differential equations, and combinatorics.
2) It instructs students to show working clearly and provide numerical answers to 3 significant figures unless otherwise specified.
3) The assessment is worth a total of 60 marks and requires answers for all questions.
1) The document provides instructions for a mathematics assessment consisting of 8 questions covering topics including probability, matrices, differential equations, and combinatorics.
2) It instructs students to show working clearly and provide numerical answers to 3 significant figures unless otherwise specified.
3) The assessment is worth a total of 60 marks and requires answers for all questions.
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)
In how many of these arrangements are all three Ds together?
[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)
Draw a fully labelled tree diagram to show this information.
[2]
(b)
Given that Maria did not have jam for breakfast, find the probability that she had toast.
[3]
The matrices P and Q are defined in terms of the constant k by
4 1 3 2 1 5
P= k 1 . 1 1 k and Q = 3 5 3 2 7 3 2
(a)
Express det P and det Q in terms of k.
[2]
(b)
Given that det (PQ) = 16, find the two possible values of k.
[4]
2 4.
It is given that y satisfies the differential equation
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)
Find the general solution of the differential equation.
[4]
Show that tan x is an integrating factor for the differential equation
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]
The matrix A is given by
1 A = 1 2
0 1 2 1 . 2 3
(a)
1 Given that A = 5 10
(b)
Given that A 3 6 A 2 + 11A 6I =
0, prove that
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)
Hence, or otherwise, find the solution of the system of equations
[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
A house on the estate is chosen at random.
D denotes the event the house is detached. R denotes the event no children live in the house. S denotes the event one child lives in the house. T denotes the event two children live in the house. ( D denotes the event not D.) (a)
(b)
Find: (i)
P( D),
[1]
(ii)
P( D R ),
[1]
(iii)
P( D | R ),
[2]
(iv)
P( R | D),
[3]
(i)
Name two of the events D, R, S and T that are mutually exclusive.
[1]
(ii)
Determine whether the events D and R are independent. Justify your answer.
