SMJK PHOR TAYMaths Revision Form 4 2013

Paper 2 Time: 2 hours 30 minutes

1 (a) Factorise 4(x2 9) 18x. (b) Solve the equation y = .

[2 marks]Answer:

2 In Diagram 1, DAE is the tangent to the cirlce ABC at point A.

Diagram 1

Find the values of (a) x, (b) y.

[2 marks]Answer:

3 Diagram 1 is a Venn diagram which shows the number of internet users in set A, set B and set C for a group of 53 users.

Diagram 1

Given that the universal set ξ = A ∪ B ∪ C, A = {New internet users who registered with Company A}, B = {New internet users who registered with Company B} and C = {New internet users who registered with Company C}.

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Given that the number of new internet users who registered with Company A exceeded the number of new internet users who registered with Company C by 8 persons. Calculate (a) the values of x and y, (b) the number of new internet users who registered with all the three companies, (c) the number of new internet users who registered with two companies only.

[2 marks]Answer:

4 A wardrobe contains a total of 40 shirts in three colours, blue, brown and purple. If a shirt is selected at random from the wardrobe, the probability of selecting a blue shirt is . Calculate (a) the number of blue shirts, (b) the probability of selecting a brown shirt if the wardrobe contains 16 purple shirts.

[2 marks]Answer:

5 In Diagram 2, the straight line AB is parallel to the straight line CD and the straight line DE is parallel to the x-axis.

Diagram 2

Given that the equation of the straight line CD is 14x − 9y = 76. Find (a) the equation of the straight line DE, (b) the equation of the straight line AB, (c) the x-intercept of the straight line AB.

[3 marks]Answer:

6 Diagram 3 shows a right prism with the isosceles triangle ABC as its uniform cross-section. G and H are the midpoints of BC and EF respectively.

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Diagram 3

Given that BC = 10 cm. (a) Calculate the length of BD. (b) Name the angle between the line DG and the plane DEF. (c) Calculate the angle between the plane DBC and the plane BCFE.

[1 mark]Answer:

8 In Diagram 4, the straight line AB is parallel to the straight line CD.

Diagram 4

Given that B(3, −7), C(12, 1) and the equation of the straight line AB is y = x − 13. Find (a) the gradient of the straight line BC, (b) the equation of the straight line CD, (c) the x-intercept of the straight line AB.

[3 marks]Answer:

9 The data in Diagram 6 shows the diameter, in cm, of 30 timber logs produced by a company.

35  73  71  80  55  64  31  7140  65  30  30  39  41  45  2433  31  65  47  51  70  50  36

79  53  45  37  68  49

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Diagram 6

(a) Based on the data in Diagram 6, complete the Table 1 in the answer space. [3 marks]

(b) Based on Table 1, calculate the estimated mean diameter of the timber logs. [3 marks]

(c) By using the scales of 2 cm to 9 cm on the horizontal axis and 2 cm to 1 timber log on the vertical axis, draw a frequency polygon for the data.

[5 marks]

(d) Based on the frequency polygon in (c), state the number of timber logs which has diameter more than 55 cm.

[1 mark]

Answer: Diameter (cm) Frequency Midpoint

20 − 28   24                              

Table 1

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10 (a) (i) Write a compound statement by combining the two statements given below using the word 'or'.

25 is a prime number.

26 is a perfect square. (ii) State whether the compound statement written in (a)(i) is true or false.

(b) Write down Premise 2 to complete the following argument: Premise 1: If nq + 3 is a quadratic expression, then q = 2. Premise 2:  Conclusion: nq + 3 is not a quadratic expression.

(c) Write down two implications from the statement below: Two straight lines are perpendicular if and only if the product of the gradients of the two

lines is −1. [3 marks]

Answer:

11 In Diagram 5, O is the centre of the circle PQR and QT is a diameter of the circle QSTU. PS and RUV are the common tangents to the two circles. STV is a straight line.

Diagram 5

Find the value of (a) x,

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(b) y, (c) z.

[3 marks]Answer:

12 Diagram 8 shows a right prism with right-angled triangle ABC as its uniform cross-section and rectangle ABED as its base.

Diagram 8

Given that AB = 7 cm. (a) Identify the angle between the line BF and the base ABED. (b) Calculate the angle between the line BF and the base ABED.

[3 marks]Answer:

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14 Diagram 6 shows a combined solid consists of a right prism and a pyramid which are joined at the plane PQUT. Trapezium PQRS is the uniform cross section of the prism. XUV is a straight line.

Diagram 6

Given PQ = 16 cm, SR = 14 cm, TP = 12 cm and XU = 14 cm. (a) Calculate the volume, in cm3, of the pyramid. (b) Calculate the length, in cm, of QR if the volume of the combined solid is 2696 cm3.

[3 marks]Answer:

15 In Diagram 10, CD is a common tangent to the cirlces with centres A and B.

Diagram 10

Given that the AD = 5 cm and BC = 2 cm. Calculate (a) the length of CD, (b) the value of x, (c) the length of minor arc CE. (Use π = 3.142)

[4 marks]Answer:

16 A brown marble, four yellow marbles and two black marbles are placed into a box. Ahmad draws a marble at random from the box and records its outcome. The marble is then placed

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back into the box before a second marble is drawn at random. The process is repeated 418 times and its result is shown in Table 2.

Outcome Brown Yellow BlackNumber of

times of occurence

162 150 106

Table 2

State the probability that (a) a brown marble is drawn, (b) a yellow marble is drawn, (c) a black marble is drawn.

[3 marks]Answer:

Answer:

1 (a) 4(x2 9) 18x= 4x2 18x 36= 2(2x2 9x 18)= 2(x 6)(2x + 3)

(b) y = 5(y) = 2(y2 6)5y = 2y2 122y2 5y 12 = 0(y 4)(2y + 3) = 0(y 4) = 0 or (2y + 3) = 0 y = 4 or y =

2 (a) ∠BCO =           = 13x = 65 − 13    = 52

(b) ∠BAC =           = 77y = 180 − 77 − 65    = 38

3 (a) (x + 14 + 1 + 13) − (1 + 13 + 1 + 7) = 8x + 28 − 22 = 8x + 6 = 8x = 2

2 + 14 + 1 + 13 + y + 1 + 7 = 53y + 38 = 53y = 15

(b) 1

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(c) 14 + 13 + 1= 28

4 (a) If number of blue shirts = x, = x = × 40x = 14

(b) Number of brown shirts = 40 − (14 + 16)= 10

Probability of selecting a brown shirt = =

5 (a) 14x − 9y = 7614(0) − 9y = 76y = −8The equation of the straight line DE is y = −8

(b) 14x − 9y = 76y = x − 8m = c = y − ()(x)    = 8 − ()(4)    = 1The equation of the straight line AB is y = x + 1

(c) y = x + 10 = x + 1x = −x = −×     = −1The x-intercept of the straight line AB is −1

6 (a) BD2 = 132 + 82

       = 233BD = 15.26 cmThe length of BD is 15.26 cm.

(b) The angle between the line DG and the plane DEF is ∠GDH. (c) The angle between the plane DBC and the plane BCFE is ∠DGH.

DG2 = BD2 − BG2

       = 233 − 25       = 208DG = 14.42 cmcos ∠DGH =        =        = 0.555∠DGH = 56° 17'The angle between the plane DBC and the plane BCFE is 56° 17'.

8 (a) m =

9

    = The gradient of the straight line BC is

(b) m = c = y − mx    = 1 − ()(12)    = −25The equation of the straight line CD is y = x − 25

(c) 0 = x − 13x = −(−13)x = ×     = 6The x-intercept of the straight line AB is 6

9 (a) Diameter (cm) Frequency Midpoint20 − 28 1 2429 − 37 8 3338 − 46 5 4247 − 55 6 5156 − 64 1 6065 − 73 7 6974 − 82 2 78

(b) Total (midpoint of class × frequency) = 24(1) + 33(8) + 42(5) + 51(6) + 60(1) + 69(7) + 78(2)= 1 503

Total frequency = 1 + 8 + 5 + 6 + 1 + 7 + 2= 30

Estimated mean diameter = = 50.1 cm

(c)

(d) Number of timber logs which has diameter more than 55 cm = 1 + 7 + 2= 10

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10 (a) (i) 25 is a prime number or 26 is a perfect square. (ii) True

(b) q ≠ 2 (c) Implication

1:If two straight lines are perpendicular, then the product of the gradients of the two lines is −1

Implication 2:

If the product of the two straight lines is −1, then the two lines are perpendicular

11(a) 2x + (360 − 220) + = 360

2x + 140 + 110 = 3602x = 110x = 55

(b) y = 90 − 55    = 35

(c) ∠PQS = 180 − 35 − 45         = 100∠RQS = 360 − ∠PQS − ∠PQR         = 360 − 100 − 110         = 150z = 360 − 90 − ∠RQS − y    = 360 − 90 − 150 − 35    = 85

12 (a) The angle between the line BF and the base ABED is ∠DBF. (b) BD2 = 72 + 242

       = 625BD = 25 cmtan ∠DBF =        =        = 0.64∠DBF = 32° 37'The angle between the line BF and the base ABED is 32° 37'.

14 (a) Volume of pyramid = × 16 × 12 × 14= 896 cm3

(b) Volume of pyramid = 2696 cm3

× (16 + 14) × QR × 12 + 896 = 2696180 × QR = 1800QR = 10 cm

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(a) AB = 5 + 2    = 7 cmAF = 5 − 2    = 3 cmFB2 = 72 − 32

      = 40FB = 6.32 cmCD = 6.32 cm

(b) cos x° =         =         = 0.43x = 64.53

(c) ∠EBC = 180 − 64.53        = 115.47Length of arc CE = × 2 × 3.142 × 2= 4.03 cm

16 (a) Probability that a brown marble is drawn = =

(b) Probability that a yellow marble is drawn = =

(c) Probability that a black marble is drawn = =

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