1. A four-sided die has faces with numbers 1, 1, 2, 3. Two such dice are thrown consecutively. (a) Copy and complete the possibility diagram. [2]

1 1 2 3 1 (1, 1) 1 2 (2, 1) 3 (3, 2)

(b) Find the probability that the sum of the two numbers is 4. [1]

2. A box of donuts contains 4 donuts with sugar coating, 3 donuts with chocolate coating and

2 donuts without any coating. Mandy randomly took a donut from the box and ate it. Nicholas then took a donut at random from the box and ate it. Find, as a fraction in its simplest form, the probability that (i) Mandy and Nicholas both chose a donut with chocolate coating, [1] (ii) Nicholas chose a donut without any coating, [2] (iii) one of them chose a donut with sugar coating and the other chose one without any

coating. [2] 3. A bag contains 15 identical beads, 5 red, 6 blue and 4 green. A bead is picked out at

random and not replaced. A second bead is then picked out at random. The tree diagram below shows the possible outcomes and their probabilities.

1st bead 2nd bead

Red

Blue

Green

31

52

154

w

Red

Bluev

73

(a) Find the values of v, w, x and y shown on the diagram. [2] (b) Expressing each of your answers as a fraction in its lowest terms, calculate the

probability that (i) both beads will be green, [2] (ii) one bead will be blue and the other will be red. [2]

(c) The second bead is replaced and a third bead is now picked at random. Calculate the probability that not all the three beads are blue. [2]

[ Turn over ]

Green72

514

72

Red

Blue

Green

y

x 145 Red

Blue

Green

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4. (a) Find the value of m, given that mp

pp

p=

× 52 [2]

(b) Solve the equation

32x+1 – 3x = 5(1 – 3x+1) [4] 5. Simplify the following, leaving your answer in positive index:

(i) ( )( )

4

341

32

923

24

82

⎟⎟⎠

⎞⎜⎜⎝

⎛×

×

−

−

bb

bb [2]

(ii) 2

21

23

21

23

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−

aa

aaaa [4]

6. A rectangular display panel ABCD is

supported to stand vertically by straight beams XY, XA, XB, XZ, YZ, YC and YD. Given that Z is the midpoint of BC, AB = 4 m, XA = 5 m and XY = 10 m, find (a) the length of XZ, [2] (b) the angle XYZ, [2] (c) the angle of depression of X from

B. [2]

D

C B

A

Y X

Z

4 m

5 m

10 m 7. In the diagram, A, B, C and D are four attractions

on level ground in a theme park, with B, C and D forming the three corners of a triangular pond.

North

The bearing of B from A is 128° and D is due north of B. Given that AB ⁄⁄ DC, AB = 95 m, BD = 86 m and CD = 74 m, calculate (a) the bearing of C from D, [1] (b) the distance BC, [2] (c) the bearing of B from C, [3] (d) the area of the pond BCD, giving your

answer to nearest m2. [2]

D

A

86 m 74 m

95 m C

B

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8. The cumulative frequency curve shown on the answer sheet provided shows the travelling time of 80 students from LMN school by bus from home to school every day.

20

80

40 5010 20 30 60

60

40

0 Time (mins)

Cumulative Frequency

(a) Use the graph to estimate (i) the median travelling time, [1] (ii) the interquartile range, [2] (iii) the number of students who take more than 45 minutes to travel to school, [1] (iv) the percentage of students who take less than 20 minutes to travel to school. [1]

(b) Two students are selected at random. Find the probability that both of them will take at least 40 minutes to travel to school. [2]

(c) The time taken by another group of 80 students from ABC school is represented by the box-and-whisker plot below.

60 50403020 10 0

Find (i) the median travelling time, [1] (ii) the interquartile range. [1] (iii) Which school may be perceived to be more accessible by bus? Give a

reason to support your answer. [1]

[ Turn over ]

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9. The distribution of the amount of time 50 students spent on the Internet per week is shown in the table below.

Time spent (hrs) 3 ≤ x < 4 4 ≤ x < 5 5 ≤ x < 6 6 ≤ x < 7 7 ≤ x < 8 8 ≤ x < 9 No. of students 5 6 11 9 h 10

(a) State the value of h. [1] (b) Draw a histogram to show the above information, using 2 cm to 1 hour for horizontal

axis and 1 cm to 2 students for vertical axis. [2] (c) Calculate an estimate of

(i) the mean amount of time spend by each student, [2] (ii) the standard deviation. [2] (iii) Find the fraction of students spent at least 6 hours on the Internet per week. [1]

Bonus Question

10. Simplify ( ) ( )babababa

babababa

++++

++−

−+2

2244

2

2244 3 . [2]

~ ~ The End ~ ~

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