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2019-2020 S6 Mathematics

Learn at Home（Round 2）

Please complete the following exercises: Paper 1 Section A(1)

Paper 2 Section A

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Paper 1

SECTION A(1) (35 marks)

1. Simplify 32

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)( nmm and express your answer with positive indices. (3 marks)

2. Factorize (a) 22 25309 yxyx , (b) xyxyx 325309 22 5y.

(3 marks)

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3. (a) Solve the inequality 3

13x ≤ 3x + 5.

(b) Write down the greatest integer satisfying the inequality 3

13x ≤ 3x + 5.

(3 marks)

4. For each positive integer n, the nth term of a sequence is sin1

180n

.

(a) Find the 3rd term of the sequence. (b) Write down two different terms of the sequence such that the product of

these two terms is equal to the square of the 3rd term of the sequence. (3 marks)

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5. In a summer camp, the ratio of the number of boys to the number of girls is 8 : 7. If 16 boys and 11 girls leave the summer camp, then the number of boys and the number of girls are the same. Find the original number of girls in the summer camp. (4 marks)

6. Three students, Amy, Bonnie and Cathy have $45.3, $33.5 and $38.6

respectively. (a) By rounding up the amount owned by each student to the nearest dollar,

estimate the total amount they have. (b) If the three students want to buy a cake of price $120, will they have

enough money to buy the cake? Use the result of (a) to explain your answer.

(4 marks)

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7. In a polar coordinate system, O is the pole. The polar coordinates of the points P and Q are (k, 34 ) and (10, 304 ) respectively, where k is a positive constant. It is given that PQ 26.

(a) Is OPQ a right-angled triangle? Explain your answer. (b) Find the perimeter of OPQ.

(5 marks)

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8. In Figure 1, AB // CD. E is a point lying on AD such that AC AE. Find x, y and z.

Figure 1

(5 marks)

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9. (a) For the set of data 18, 21, 12, 18, 18, 15, find (i) the median, (ii) the mean. (b) Two unknown data are combined with the six data in (a) to form a set of

eight data. It is given that the mean of the new set of data is 17.5. Tom claims that the new median must be equal to that obtained in (a)(i). Do you agree? Explain your answer.

(5 marks)

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Paper 2

Section A (30 marks)

1. 230 )4()4( xx A. 0。 B. 64x 。 C. 68x 。 D. 616x 。

2. If ba21 3, then b

A. 13a

a .

B. 13

2a

a .

C. a

a31

2 .

D. a

a31

2 .

3. If )2()1( 23 xxhhxx k, then k A. 5. B. 3. C. –1. D. 3. 4. 0.010 305 07 A. 0.01 (correct to 2 significant figures). B. 0.0103 (correct to 3 decimal places). C. 0.0103 (correct to 4 significant figures). D. 0.010 31 (correct to 5 decimal places).

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5. When xxxxf 2)(2()( 2 1) is divided by x k, the remainder is k + 2. Find the value(s) of k.

A. 1 B. 2 or 1 C. 0 or 2 D. 0 or 2 6. How many integral solutions do the compound inequalities 18 2x 10 and 3x + 1 4x

have? A. 1 B. 2 C. 3 D. 4 7. Consider the graph of y

22x x 1. Find the mean of the x-intercepts of the graph.

A. 41

B. 41

C. 43

D. 43

8. In the figure, the graph of y

2ax + bx + c touches the x-axis at one point only, where a, b and c are constants. Which of the following are true?

I. a 0 II. b 0 III. c 0 A. I and II only B. I and III only C. II and III only D. I, II and I I I

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9. The figure shows the graphs of ax + y b and cx + y d. Which of the following is true? A. a b B. a c C. ab 0 D. c + d 0 10. Consider a circular cone. If the base radius increases by 25% while the height decreases by

36%, find the percentage change in the volume. A. A decrease of 20% B. 0% C. An increase of 70% D. An increase of 112.5% 11. Helen cycles at a constant speed of 30 km/h for x hours, then cycles at a constant speed of

12 km/h for y hours. If her average speed is 20 km/h, then x : y

A. 4 : 5. B. 5 : 4. C. 16 : 25. D. 25 : 16. 12. The actual volume of a bus is 120 m3. If the volume of a model of the bus is 15 cm3, then the

scale of the model is A. 1 : 2. B. 1 : 8. C. 1 : 200. D. 1 : 500.

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13. It is given that z varies directly as the square root of x and inversely as y. Which of the following must be constant?

A. 2xy

B. yx

C. yzx

D. 2)(yzx

14. Consider the sequence 1, 21 ,

41 ,

81 … .

Which of the following may be the general term of the sequence?

A. n

n)1(

B. 2

1)1(n

n

C. n

n

2)1(

D. 1

21 n

15. The length and the width of a rectangular piece of paper are measured as

30 cm and 20 cm respectively correct to the nearest cm. If the paper is cut into n small squares such that the side of each square is measured as 4 cm correct to the nearest cm, find the least possible value of n.

A. 24

B. 28 C. 35 D. 37

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16. Which of the following figures has both rotational symmetry and reflectional symmetry? A.

B.

C.

D.

17. In the figure, the radius of the semicircle ABC is 6.5 cm. If BC 12 cm, find the area of the

shaded region correct to the nearest 0.01 cm2. A. 1.68 cm2 B. 3.36 cm2 C. 5.13 cm2 D. 6.17 cm2

18. Consider a wooden cube with side 6 cm. A sphere is made by cutting away part of the cube. Find the maximum possible volume of the sphere.

A. 24 cm3 B. 36 cm3 C. 72 cm3 D. 288 cm3 19. In the figure, BC 6 cm, DE 9 cm and AE 17 cm. Find the perimeter of ABCDE. A. 32 cm B. 34 cm C. 40 cm D. 60 cm 20. Find the maximum value of cos4cos2 . A. 3 B. 3 C. 5 D. 5

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21. Consider two lampposts A and B on horizontal ground. If the true bearing of B from A is 100 , find the compass bearing of A from B.

A. N80°E B. N80°W C. S80°E D. S80°W 22. In the figure, AC is a diameter of the circle. A chord BD intersects AC at E.

AB 13 cm, BD 10 cm and AC BD. Find the radius of the circle correct to 1 decimal place.

A. 5.5 cm B. 6.1 cm C. 7.0 cm D. 8.3 cm

23. In the figure, AD is a diameter of the circle. If

)AB)BC

)CD , then ABC

A. 90°. B. 100°. C. 105°. D. 120°. 24. The rectangular coordinates of the point P are (3, 33 ). If P is rotated clockwise about the

origin through 50 , then the polar coordinates of its image are A. (6, 250 ). B. (6, 350 ). C. (18, 250 ). D. (18, 350 ).

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25. The coordinates of the points A and B are ( 3, 2) and (5, 4) respectively. If P is a point lying on the straight line x + 2y 10 0 such that AP BP, then the coordinates of P are

A. ( 3, 16). B. (16, 3). C. (3, 4). D. (4, 3). 26. If P is a moving point in the rectangular coordinate plane such that the distance between P

and the line y 1 is equal to 3, then the locus of P is A. a parallelogram. B. a pair of parallel lines. C. a line perpendicular to y 3. D. a horizontal line with y-intercept 4. 27. In the figure, the equation of the circle is FEyDxyx 22

0. Which of the following must be true?

I. D E 0 II. D + F 0 III. E F 0 A. I and II only B. I and III only C. II and III only D. I, II and III 28. Two numbers are randomly drawn at the same time from five cards numbered 1, 4, 6, 8 and

9 respectively. Find the probability that the sum of the numbers drawn is even.

A. 52

B. 113

C. 259

D. 2513

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29. The stem-and-leaf diagram below shows the distribution of the weights (in kg) of a group of students.

Stem (10 kg) Leaf (1 kg) 4

5 6 7

0 2 3 8 2 3 3 5 7 8 2 4 4 4 8 0 0 1 1 5

Find the median weight. A. 57 kg B. 59 kg C. 60 kg D. 64 kg 30. The figure below shows the frequency curves of the examination score distributions A and B.

Let Am , Ar and As be the mode, the range and the standard deviation of A respectively while

Bm , Br and Bs be the mode, the range and the standard deviation of B respectively. Which of the following are true?

I. BA mm II. BA rr III. BA ss A. I and II only B. I and III only C. II and III only D. I, II and III

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