1

Instructions: 1 This question paper consists of 40 questions. 2 Each question is followed by the options A, B, C and D. Choose the correct option. 3 Answer all questions. 4 The diagrams provided in the questions are not drawn to scale unless stated. 5 You may use a scientific calculator.

Time: 1 hour 15 minutes PAPER 1

MATHEMATICSSPM Forecast Paper

1 Round off 0.0050287 correct to 3 significant figures.A 0.005029 C 0.005030B 0.00503 D 0.0050300

2 Express 5.7 × 10–6 as a single number.A 0.00057 C 0.0000057B 0.000057 D 0.00000057

3 Calculate 2.1 × 10–5 – 8.9 × 10–6.A 1.21 × 10–4 C 1.21 × 10–6

B 1.21 × 10–5 D 6.8 × 10–5

4 11012 + 11002 =

A 101012 C 110012

B 110102 D 110112

5 Convert 1100102 to a number in base 5.A 155

B 265 C 625

D 2005

6 In Diagram 1, ABCDEF is a regular hexagon. ABG and ACH are straight lines.

A B G

F

E D

CH

x°y°

70°

Diagram 1

The value of x + y isA 230 B 240 C 250D 260

7 In Diagram 2, PQR is an equilateral triangle and PRTU is a rhombus. PRS is a straight line.

60°

80°x°

U T

P

Q

R S

Diagram 2

Given that �RTS = 60° and �TSR = 80°, find the value of x.A 60 B 70 C 80D 85

8 In Diagram 3, the tangents ABC and CDE touch the circle with centre O at the points B and D respectively.

C

D

E

B

A

F

Ox°

50°

45°

Diagram 3

The value of x isA 5 B 10 C 15D 20

2

9 Diagram 4 shows five rectangles drawn on a Cartesian plane.

y12

10

8

6

4

2

–4 –2 2 4O

–2

–4

–6

C

D

B

A

x

Diagram 4

Which of the rectangles, A, B, C or D, is the image of the shaded rectangle under an enlargement with centre O(0, 0) and a scale

factor of 1—2

?

10 Diagram 5 shows two straight lines, PQ and P’Q’, drawn on an equilateral triangular grid. The straight line P’Q’ is the image of the straight line PQ under a certain rotation.

P

Q

A

B

CD

P′ Q′

Diagram 5

Which of the points, A, B, C or D, is the centre of rotation?

11 In Diagram 6, SPQ and PTQ are two right-angled triangles.

Q

T

P

S

y°

6 cm

2 cm x°

Diagram 6

Given that sin y° = 1—4

, then cos x° is

A – 3—4

C – 4—5

B – 3—5

B 4—5

12 Given that sin x = –cos 240° such that 0° � x � 360°, find the values of x.A 60° or 300°B 60° or 330°C 210° or 330°D 240° or 330°

13 Which of the following graphs is incorrect?A

x

y

0

1

90° 180°

y = sin x

B

y

1

–1

90° 180°x

y = cos x

0

C

y

90° 180°x

y = tan x

0

D

y

x

1

–1

45° 90° 135° 180°

y = sin 2x

0

14 Diagram 7 shows a cuboid. M and N are the midpoints of PQ and SR respectively.

U X

R

QMP

V

NS

W

Diagram 7

The angle between the plane UPM and the base PQRS isA �UPNB �UMSC �UMND �UPS

3

15 In Diagram 8, DA is a flagpole. The points A, B and C lie on a horizontal plane such that AB = BC.

D

A B C

12 m

Diagram 8

If the angle of elevation of point D from point B is 40°, calculate the angle of depression of point C from point D.A 20.0°B 22.8°C 67.2°D 70.0°

16 Diagram 9 shows three points, P, Q and R, which lie on a horizontal plane.

P

R

Q

100°

Diagram 9

Given that the bearing of point Q from point P is 220° and �QPR = 100°, find the bearing of point R from point P.A 050° C 120°B 060° D 130°

17 Diagram 10 shows three points, X, Y and Z, which lie on the surface of the earth. The points X and Y lie on the Equator. NXS is the Greenwhich Meridian where N is the North Pole and S is the South Pole. O is the centre of the earth.

N

Z

X Y

S

O 60°

70°

Diagram 10

State the position of point Z.A (60°N, 70°E)B (60°N, 70°W)C (70°N, 60°E)D (70°N, 60°W)

18 Diagram 11 shows two points, P and Q, which lie on the surface of the earth. PQ is the diameter of a parallel of latitude. O is the centre of the earth. N is the North Pole and S is the South Pole. The longitude of point P is 60°W.

N

Q

O

P

S

100°

Diagram 11

State the position of point Q.A (40°N, 120°E) C (40°N, 60° W)B (80°N, 120°E) D (40°S, 120°W)

19 Factorise (2p – 3)2 + 5p(2p – 3) completely.A (2p + 3)(7p – 3) C (2p + 3)(7p + 3)B (2p – 3)(7p + 3) D (2p – 3)(7p – 3)

20 6——–k – 2

– 3k + 6————k(k – 2)

=

A 3—k

C 3k – 6———k – 2

B 3——–k – 2

D 3k + 6———k – 2

21 Given that 3√ h – 1 = k, then h =

A k2 + 3——––3 C 9k2 + 1

B k2 + 9——––9 D k2

—9 – 1

22 Given that 9 – 2(3x – 1) = x, then x =

A 1—5 C 1 4—

7

B 7—–20 D 3 2—

3

23 Calculate 43 —2 × 8

– 4 —3.

A 1—8 C 64

B 1—2 D 128

24 Simplify (m2 n4)– 1 —

2 ÷ m5 n3.

A m4 n

B m4—–n

C n—–m4

D 1—––––

m6 n5

4

25 Which of the following number lines represents the solution to the inequalities –3t � 6 and 2t – 7 � –1?A

t–2 –1 0 1 2 3

B

3210–1–2

t

C

–2 –1 0 1 2 3

t

D

–2 –1 0 1 2 3

t

26 Diagram 12 is a pictogram showing the number of rubber, oil palm and cocoa trees in a plantation.

Rubber

Oil palm

Cocoa

represents 30 trees

Diagram 12

The ratio of the number of rubber trees to the number of cocoa trees is 5 : 3. Find the total number of rubber, oil palm and cocoa trees in the plantation.A 630 C 690B 660 D 720

27 Table 1 shows the height distribution of a group of students.

Height (cm) Frequency

156 – 160 6

161 – 165 10

166 – 170 4

Table 1

Calculate the mean height, in cm.A 160.5 C 163.5B 162.5 D 154.5

28 Which of the following graphs represents y = x2 – x – 6?A

x

y

O–2 3

B

y

x2–3 O

C

y

x3O–2

D

y

x2O–3

29 The Venn diagram in Diagram 13 shows the universal set ξ and the relation between the sets P, Q and R.

Pξ

Q

R

Diagram 13

The set which represents the shaded region isA (Q � R)’ � P C (Q � R)’ � PB (Q � R) � P D (P � Q) � R’

30 Given that the universal set ξ = {x : 2 � x � 12, x is an integer}, set R = {x : x is a multiple of 4} and set S = {x : x is a factor of 24}, calculate n(R' � S).

A 1 C 3B 2 D 4

31 A class has 35 students. It is compulsory for each student to join at least one uniformed body. Table 2 shows the number of students in each uniformed body.

Uniformed bodyNumber of students

Scouts 7

St John Ambulance 15

Cadets 18

Scouts and St John Ambulance 2

St John Ambulance and Cadets x

Table 2

5

The Venn diagram in Diagram 14 represents the data in Table 2.

Scouts Cadets

St John Ambulance

Diagram 14

Find the value of x.A 1 C 3B 2 D 4

32 In Diagram 15, LM is a straight line.

y

L

O M (15, 0) x

Diagram 15

If the gradient of the straight line LM is – 1—3

, find the y-intercept of the straight line.A 4 C 6B 5 D 7

33 Find the gradient of the straight line whose equation is –5x – 3y = 7.

A – 3—5

C – 7—3

B – 5—3

D 5—3

34 A bag contains orange, apple and strawberry flavoured sweets. The number of sweets of each flavour is as shown in Table 3.

Flavour Orange Apple Strawberry

Frequency 32 48 x

Table 3

If a sweet is drawn at random from the bag, the probability of drawing a strawberry flavoured sweet

is 1—5

. The value of x is

A 50 B 40 C 30D 20

35 Ten students of a class cycle to school. If a student is chosen at random from the class, the

probability that the student cycles to school is 2—5

. A

week later, five students who cycle to school join the class. If a student is now selected at random from the class, the probability that the student cycles to school is

A 1—2

C 3—5

B 2—5

D 4—5

36 Given that p varies inversely as q and that p = 4 when q = 8, express p in terms of q.

A p = 2—q C p = 1—–2q

B p = 32—–q D p = 1—2

q

37 Table 4 shows some corresponding values of the variables x, y and z.

y 72 p

x 4 6

z 3 2

Table 4

Given that y varies jointly as x and the cube of z, find the value of p.A 16 C 32B 28 D 36

38 Given that y � 1——x √ z

and that y = 1—5

when x = 4

and z = 100, find the value of y when x = 3 and z = 64.

A 1—6

C 1—3

B 1—4

D 1—2

39 If matrix A = �5 h3 –6� does not have an inverse, then

the value of h isA 30 C 30B –10 D –30

40 Given that (k –2) �1 0k 4� = (5 –8), calculate the

value of k.A 1 B –1 C 5D –5

6

1 On the graph in the answer space, shade the region which satisfies the three inequalities y � x + 5, y � 2x + 10 and y � 5.

[3 marks]

y

10

5

–5 Ox

y =

2x +

10 y = x + 5

2 Solve the quadratic equation m + 1—––—–m + 4

= 2m—–3

. [4 marks]

3 Find the value of m and of n which satisfy the following simultaneous linear equations:4m – 3n = –1

m + 1—3

n = 3 [4 marks]

4 Diagram 1 shows a right prism with a horizontal rectangular base CDEF. The prism has the isosceles triangular faces GCF and HDE as its uniform cross-section. P and Q are the midpoints of the sides FC and ED respectively.

G

P

C D

Q

E

H

F

12 cm

20 cm

10 cm

Diagram 1

Given that GC = 10 cm, GH = 20 cm and DE = 12 m, calculate the angle between the plane HFC and the plane GCF.[3 marks]

Time: 2 hours 30 minutes PAPER 2

Instructions: 1 This question paper consists of two sections: Section A and Section B. 2 Answer all questions in Section A and four questions from Section B. 3 Show your working. It may help you to get marks. 4 The diagrams provided in the questions are not drawn to scale unless stated. 5 The marks allocated for each question and sub-part of a question are shown in brackets. 6 You may use a scientific calculator.

Section A (52 marks)

Answer all questions in this section.

7

5 In Diagram 2, O is the origin, point Q lies on the x-axis and point S lies on the y-axis. The straight line ST is parallel to the x-axis. The straight line PQ is parallel to the straight line RS. The equation of the straight line RS is y + 3x + 8 = 0.

y

O Q

S T

P(–2, 7)

y + 3x + 8 = 0

R

x

Diagram 2

(a) Find the equation of the straight line ST.(b) Find the equation of the straight line PQ and hence, state the x-intercept of the straight line PQ.

[5 marks]

6 Diagram 3 shows a solid cylinder with a height of 21 cm and a diameter of 14 cm. The shaded portion, in the shape of a cone, is removed.

Diagram 3

If the volume of the remaining solid is 2874 2—3

cm3, find the height, in cm, of the cone that is removed.

�Use π = 22—–7 � [4 marks]

7 Diagram 4 shows two sectors of two concentric circles, OBC and OAD, with the same centre O. ADE is a semicircle with centre O and a radius of 7 cm. BAOE is a straight line. D is the midpoint of OC and ∠AOD = 120°.

AB

120°

7 cm

D

C

O E

Diagram 4

Using π = 22—–7

, calculate

(a) the perimeter, in cm, of the whole diagram,(b) the area, in cm2, of the shaded region.

[6 marks]

8 (a) State the converse of the following statement. State whether the converse is true or false. ‘If two straight lines never meet, then they are parallel.'(b) Complete each of the following statements using the quantifiers ‘All’ or ‘Some’. (i) multiples of 4 are multiples of 8. (ii) factors of 6 are factors of 36.(c) Complete the following argument: Premise 1: If ax2 + bx + c is a quadratic expression, then a ≠ 0.

Premise 2: Conclusion: ax2 + bx + c is not a quadratic expression.

[5 marks]

8

9 Table 1 shows the number of participants of the 1500 m open event during the sports day of a school according to category and house.

HouseNumber of participants

Category A Category B

Red 4 3

Green 3 4

Blue 5 3

Table 1

(a) If two participants are chosen at random from the Green house, calculate the probability that both of them are from category B.

(b) If two participants are chosen at random from category A, calculate the probability that both of them are from the same house.

[5 marks]

10 Diagram 5 shows the speed-time graph of a car for a period of 15 s. The distance travelled by the car at a uniform speed is 144 m.

30

18

t 12 15Time (s)

Speed (m s–1)

O

Diagram 5

Find(a) the value of t,(b) the rate of change of speed, in m s–2, of the car in the last 3 seconds,(c) the average speed, in m s–1, of the car for the whole journey.

[6 marks]

11 The inverse matrix of P = �6 –53 –2� is k �–2 5

–3 h�.(a) Find the value of h and of k.(b) Using matrices, find the value of x and of y that satisfy the following simultaneous linear equations: 6x – 5y = 4 3x – 2y = 7

[7 marks]

9

Section B (48 marks)

Answer four questions from this section.

12 (a) Complete Table 2 for the equation y = –x2 + 2x – 1. [2 marks]

x –3 –2 –1 0 1 2 3 4

y –16 –4 –1 0 –1 –9

Table 2

(b) For this part of the question, use a graph paper. You may use a flexible curve rule. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of

y = –x2 + 2x – 1 for –3 � x � 4. [4 marks](c) From your graph, find (i) the value of y when x = –0.6, (ii) the value of x when y = –11. [2 marks](d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation –x2 + 4x + 2 = 0

for –3 � x � 4. State these values of x. [4 marks]

13 Diagram 6 shows the trapeziums ABCD and JKLM.

y

C

B

MAD

J K

L

2

2 4 6

4

6

Ox

Diagram 6

(a) The transformations T, P and R are defined as follows:

T = Translation �–1–5 �

P = Reflection in the straight line x = 3 R = Clockwise rotation of 90° about the origin State the coordinates of the point C(1, 5) under each of the following combined transformations: (i) TP (ii) PR [4 marks](b) Trapezium JKLM is the image of trapezium ABCD under the combined transformation UV. Describe in full, (i) the transformation V, (ii) the transformation U. [4 marks](c) The transformations Q and N are defined as follows: Q = Reflection in the straight line y = x N = Reflection in the x-axis The point D(1, 3) is transformed under a combined transformation NQ. Describe in full, a single transformation which is equivalent to the combined transformation NQ. [2 marks](d) Calculate the area of trapezium JKLM if the area of trapezium ABCD is 15 cm2. [2 marks]

10

14 The data in Diagram 7 show the masses, in g, of 40 fish in a big aquarium.

80 53 29 50 45

78 65 54 34 70

63 57 65 67 56

76 56 36 47 84

35 77 38 28 60

63 55 48 34 64

75 44 83 62 53

61 62 73 50 77

Diagram 7

(a) Using the data in Diagram 7 and uniform class intervals of size 10 g, complete Table 3. [4 marks]

Mass (g) Frequency Midpoint

20 – 29

30 – 39

Table 3

(b) Based on your table in (a), (i) state the modal class, (ii) calculate the mean mass of the fish. [4 marks](c) Using a scale of 2 cm to 10 g on the x-axis and 2 cm to 2 fish on the y-axis, construct a histogram for the

given data.[4 marks]

15 (a) Diagram 8(i) shows a solid ABCDEFGH with a horizontal quadrilateral base ABCD. A cuboid DCPSRTHQ is joined to the solid ABCDEFGH on the plane TDCH. The edges FA, GB, ED, HC, QP and RS are vertical. It is given that FA = 5 cm, ED = 6 cm, QP = 3 cm, AD = 7 cm, BC = 4 cm, DC = 7 cm and DS = CP = 2 cm.

E

F

A

D

6 cm

S

R

T H

Q

7 cm

5 cm 7 cm

2 cm

3 cm

2 cm

4 cm

B

GC X

P

Diagram 8(i)

Draw to full scale, the elevation of the solid on a vertical plane parallel to ADS as viewed from X. [3 marks]

11

(b) The solid ABCDEFGH in Diagram 8(i) is partially removed so that the inclined planes FEB and EBH are formed, as shown in Diagram 8(ii).

E

F

A

D

6 cm

S

R

T H

Q

7 cm

5 cm 7 cm

3 cm

2 cm

4 cm

B

C

Y

P

Diagram 8(ii)

Draw to full scale, (i) the plan of the remaining solid, [4 marks] (ii) the elevation of the remaining solid on a vertical plane parallel to SP as viewed from Y. [5 marks]

16 P(0°, 50°W), Q(0°, h°E), R(20°N, k°W) and V(20°N, 50°W) are four points on the surface of the earth.(a) Given that the distance from P to Q, measured along the Equator is 4500 nautical miles, find the value of h.

[3 marks](b) An aeroplane departed from point P at time 0800 and flew towards point V with an average speed of 480

knots. Find the time, in the 24-hour system, the aeroplane reached point V. [3 marks](c) Another aeroplane flew due east from point V to point R. It flew with an average speed of 500 knots and it

took 3 hours to reach point R. Find the value of k. [6 marks]

12

ANSWERS

Paper 1 1 B 2 C 3 B 4 C 5 D 6 A 7 C 8 D 9 B 10 C 11 D 12 C 13 C 14 D 15 B 16 C 17 A 18 A 19 D 20 A21 B 22 C 23 B 24 D 25 C26 D 27 B 28 A 29 D 30 C31 C 32 B 33 B 34 D 35 A36 B 37 C 38 C 39 B 40 D

Paper 2 1

10

y

x

5

–5 O

y = x + 5

y =

2x +

10

2 m = 1 —2

or –3

3 m = 2, n = 3 4 ∠GPH = 68º 12’

5 (a) y = –8

(b) y = –3x + 1; 1 —3

6 7 cm

7 (a) 64 2 —3

cm

(b) 179 2 —3

cm2

8 (a) If two straight lines are parallel, then they never meet. True.

(b) (i) Some (ii) All(c) a = 0

9 (a) 2 —7

(b) 19 —–66

10 (a) 4 (b) –6 m s–2

(c) 17 4 —5

m s–1

11 (a) h = 6, k = 1 —

3

(b) x = 9, y = 1012 (a) x = –2, y = –9

x = 3, y = –4

(b)

y

x

−2

−4

−6

−8

−10

−12

−14

−16

−3 −2 −1 1 3 4

y = x − 3

O2−2.3 −0.6−1

−2.6

y = −x2 + 2x − 1

−11

(c) (i) y ≈ –2.6 (ii) x ≈ –2.3(d) Draw y = x – 3, x = –1 or 2

13 (a) (i) (4, 0) (ii) (1, –1)(b) (i) Clockwise rotation of 90° about the point

D(1, 3) (ii) Enlargement with centre (–1, 3) and a

scale factor of 2(c) Clockwise rotation of 90° about the origin(d) 60 cm2

14 (a)

Mass (g) Frequency Midpoint

20 – 29 2 24.5

30 – 39 5 34.5

40 – 49 4 44.5

50 – 59 9 54.5

60 – 69 10 64.5

70 – 79 7 74.5

80 – 89 3 84.5

(b) (i) 60 – 69 (ii) 57.75 g(c)

Frequency

0

Mass (g)

19.5 29.5 39.5 49.5 59.5 69.5 79.5 89.5

10

8

6

4

2

13

15 (a)

F

5 cm

3 cm 4 cm 2 cm

3 cm

3 cmG

E

H, TQ, R

P,SBA C,D

Elevation as viewed from X

(b) (i), (ii)

1 cm

2 cm

3 cm

E

F

T, R

B, C, DA, D, S

H, Q

7 cm

3 cm

Elevation as viewed from Y

7 cmR, S Q, P

2 cm

4 cm

2 cm

7 cm

H, C

B

E, T, D

F, APlan

16 (a) 25(b) 1030(c) 23° 24’