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Class Register Number Name

洋女子学校

NANYANG GIRLS' HIGH SCHOOL

Mid-Year Examination 2014Secondary 4

INTEGRATED MATHEMATICS 1 1.5 hours

Monday 12 May 2014 0845 – 1015

READ THESE INSTRUCTIONS FIRST

INSTRUCTIONS TO CANDIDATES

1. Answer all the questions.2. Write your answers and working on the separate writing paper provided.3. Write your name, register number and class on each separate sheet of paper that you use and

fasten the separate sheets together with the string provided. Do not staple your answer sheetstogether.

4. Omission of essential steps will result in loss of marks.5. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case

of angles in degrees, unless a different level of accuracy is specified in the question.

INFORMATION FOR CANDIDATES

1. The number of marks is given in brackets [ ] at the end of each question or part question.2. The total number of marks for this paper is 60.3. The use of an electronic calculator is expected, where appropriate.4. You are reminded of the need for clear presentation in your answers.

Setter: CMF This document consists of 8 printed pages.

N ANYANG GIRLS' HIGH SCHOOL [ Turn over

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BLANK PAGE

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1 Given that A

−=

k 1

03 and B

=

43

21, find the value of k if

A2 + 3B =

164

612. [3]

2 In a childcare centre, the number of children in the respective age groups are as

follows:

Age in years 3 4 5 6

No. of children 13 6 x 5

(a) If the modal age is 5, state the minimum value of x. [1]

(b) If the median age is 4, find the smallest possible value of x and the

corresponding position of the median. [3]

3

A box contains some cards of 3 different colours. Of these cards, p are purple, 15 are

blue and the rest are black. The probability of drawing a purple card and a blue card is

5

1 and

6

1 respectively. Find

(i) the value of p, [2]

(ii) the number of black cards to be removed so that the probability of drawing a

black card is27

16. [2]

4 It is given that A

−=

22

31, B

=

y

x and C

=

6

2 such that AB + AC =

− 4

14.

(i) Find A−1. [1]

(ii) Hence, by using A−1, find the value of x and of y. [4]

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5 The variables x and y are related by the equation 012

2 =−+n

ymx , where m and n are

constants. A straight line graph is obtained by plotting 2 y against 2 x .

(i) Given that the line passes through the point (5, 8) and has a gradient of 2,

find the value of m and of n. [4]

(ii) Another straight line can be obtained when

2

y

x is plotted against

2

1

y

. Using the

value of m and of n obtained in part (i), state the value of the gradient and the

vertical intercept. [3]

6 The following dot diagram represents the marks obtained by the students of class 4A

in a mathematics quiz.

•• • • •

• • • • •• • • • • • •• • • • • • • • •

1 2 3 4 5 6 7 8 9 10

(a) The data can also be represented in the box-and-whisker diagram below.

(i) State the value of a, of b and of c. [3]

(ii) Find the interquartile range. [1]

(b) State, with a reason, which statistical average you should use to comment on the

performance of the class in the quiz. [2]

1 a b c 10

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7 The histogram below shows the distribution of the heights of 200 adults.

The data is presented in the following frequency distribution table.

Height( h cm) 150< h ≤155 155< h ≤160 160< h ≤165 165< h ≤170 170< h ≤175

Frequency 12 p q 58 20

(a) State the value of p and of q. [2]

(b) Find the mean height. [2]

(c) Two adults are chosen at random from the 200 adults. Find the probability that

(i) both of them are taller than 170 cm, [2]

(ii) one of them is taller than 170 cm and the other is not. [2]

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Height (cm)150 155 160 165 170 175

F r e q u e n c y

20

40

60

10

50

30

70

0//

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8 The type and number of admission tickets sold at three tourist attractions on

Wednesday and Thursday of a particular week are shown below.

The information for the sale on Wednesday and Thursday is represented by the

matrices W

=

1124

2246

3143

and T

=

1829

2647

2940

respectively.

(i) If the matrix A is such that the elements of A represent the total number of each

type of tickets sold at each attraction over the two days, find A. [2]

(ii) In a particular promotion, for each of the three attractions, an adult ticket costs

$22 while a child ticket costs $15. Given C

=

15

22, evaluate AC. [1]

(iii) Explain what the elements in AC represent. [2]

(iv) Write down, in the correct order, a product of two matrices which will give the

total ticket earnings for all 3 attractions over the two days.(It is not necessary to evaluate the product.) [2]

Tourist

Attractions

Wednesday Thursday

Adult Child Adult Child

Singapore Zoo 43 31 40 29

River Safari 46 22 47 26

Jurong Bird Park 24 11 29 18

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9 There are two bags A and B. Bag A contains 2 red balls and 8 white balls and Bag B

contains 3 red balls and 6 white balls.

It is given that the probability of choosing Bag A is8

3.

(a) In an experiment, a person chooses a bag and draws a ball randomly from it and

replaces it before drawing another ball from the same bag. Find the probability

of choosing

(i) one red ball after one draw, [2]

(ii) at least one red ball after two draws. [3]

(b) In another experiment, a person chooses a bag and draws a ball from it and put

into the other bag. Then, the person draws a ball from the second bag.

What is the probability of choosing two white balls from the two draws? [3]

10 The equation of a circle, C 1, can be expressed in the form c y ym x x +

+=− 22

4

14 ,

where m and c are constants. When )4( 2 x x − is plotted against

+ y y

2

4

1, a straight

line is obtained which passes through the points (1, −3) and (−1, 5).

(a) Show that the circle C 1 has centre (2, −2) and radius 3 units. [5]

(b) Another circle, C 2, with its centre on the same vertical line of symmetry as C 1

but twice the radius is added such that the two circles meet at the point with the

lowest y-coordinate on C 1, as shown in the diagram below.

If ( x, y) represents any point on the circumference of the circle C 2, state the

range of values of x and of y. [3]

C 2

C 1

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Bonus Question

11 How many ways are there of choosing 3 different digits in increasing order from

{1, 2, 3, ... , 8, 9} so that no two of the digits are consecutive? [3]

End of paper

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