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SMJK PHOR TAY Maths Revision Form 4 2013 Paper 2 1

Time: 2 hours 30 minutes

(a) Factorise 4(x2 − 9) − 18x. (b) Solve the equation y = . [2 marks] Answer:

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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:

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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}. 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

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(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:

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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:

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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:

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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:

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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:

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The data in Diagram 6 shows the diameter, in cm, of 30 timber logs produced by a company. 35 73 71 80 55 64 40 65 30 30 39 41 33 31 65 47 51 70 79 53 45 37 68 Diagram 6

31 71 45 24 50 36 49

(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

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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: Frequen cy

Diameter (cm) 20 − 28

Midpoi nt 24

Table 1

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(a) (i)

Write a compound statement by combining the two statements given below using

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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:

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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, (b) y, (c) z. [3 marks] Answer:

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

Diagram 8

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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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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:

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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,

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(c) the length of minor arc CE. (Use π = 3.142) [4 marks] Answer:

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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 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. Bro wn

Outcome

Number of times of 162 occurence Table 2

Yell ow

Bla ck

150

106

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 − 12 2y2 − 5y − 12 = 0 (y − 4)(2y + 3) = 0 (y − 4) = 0 or (2y + 3) = 0 y = 4 or y = −

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(a) ∠BCO = = 13 x = 65 − 13 = 52 (b) ∠BAC = = 77 y = 180 − 77 − 65 = 38

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(a) (x + 14 + 1 + 13) − (1 + 13 + 1 + 7) = 8 x + 28 − 22 = 8 x+6=8 x=2 2 + 14 + 1 + 13 + y + 1 + 7 = 53 y + 38 = 53 y = 15 (b) 1 (c) 14 + 13 + 1 = 28

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(a) If number of blue shirts = x, = x = × 40 x = 14 (b) Number of brown shirts = 40 − (14 + 16) = 10 Probability of selecting a brown shirt = =

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(a) 14x − 9y = 76 14(0) − 9y = 76 y = −8 The equation of the straight line DE is y = −8 (b) 14x − 9y = 76 y=x−8 m= c = y − ()(x) = 8 − ()(4) =1 The equation of the straight line AB is y = x + 1 (c) y = x + 1 0=x+1 x=− x = −× = −1 The x-intercept of the straight line AB is −1

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(a) BD2 = 132 + 82 = 233 BD = 15.26 cm The 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 = 208 DG = 14.42 cm cos ∠DGH =

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= = 0.555 ∠DGH = 56° 17' The angle between the plane DBC and the plane BCFE is 56° 17'.

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(a) m = = The gradient of the straight line BC is (b) m = c = y − mx = 1 − ()(12) = −25 The equation of the straight line CD is y = x − 25 (c) 0 = x − 13 x = −(−13) x=× =6 The x-intercept of the straight line AB is 6

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(a)

Frequen Midpoi cy nt 20 − 28 1 24 29 − 37 8 33 38 − 46 5 42 47 − 55 6 51 56 − 64 1 60 65 − 73 7 69 74 − 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 Diameter (cm)

Total frequency =1+8+5+6+1+7+2 = 30 Estimated mean diameter = = 50.1 cm (c)

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

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(b) (c)

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(a) (i) 25 is a prime number or 26 is a perfect square. (ii) True (b) q ≠ 2 (c) Implication If two straight lines are perpendicular, then the product of the gradients of 1: the two lines is −1 Implication If the product of the two straight lines is −1, then the two lines are 2: perpendicular

2x + (360 − 220) + = 360 2x + 140 + 110 = 360 2x = 110 x = 55 y = 90 − 55 = 35 ∠PQS = 180 − 35 − 45 = 100 ∠RQS = 360 − ∠PQS − ∠PQR = 360 − 100 − 110 = 150 z = 360 − 90 − ∠RQS − y = 360 − 90 − 150 − 35 = 85 (a) The angle between the line BF and the base ABED is ∠DBF. (b) BD2 = 72 + 242 = 625 BD = 25 cm tan ∠DBF = = = 0.64 ∠DBF = 32° 37' The angle between the line BF and the base ABED is 32° 37'.

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(a) Volume of pyramid = × 16 × 12 × 14 = 896 cm3 (b) Volume of pyramid = 2696 cm3 × (16 + 14) × QR × 12 + 896 = 2696 180 × QR = 1800 QR = 10 cm

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(a) AB = 5 + 2 = 7 cm AF = 5 − 2 = 3 cm FB2 = 72 − 32 = 40 FB = 6.32 cm CD = 6.32 cm (b) cos x° = = = 0.43 x = 64.53 (c) ∠EBC = 180 − 64.53 = 115.47 Length of arc CE = × 2 × 3.142 × 2 = 4.03 cm (a) Probability that a brown marble is drawn = = (b) Probability that a yellow marble is drawn =

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Maths revision p2 f4 2013(edit)  
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