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Mathematical Formulae Compound Interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length = , where θ is in radians Sector area = , where θ is in radians Trigonometry Statistics Mean = Standard deviation =

EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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Page 1: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

Mathematical Formulae

Compound Interest

Total amount =

Mensuration

Curved surface area of a cone =

Surface area of a sphere =

Volume of a cone =

Volume of a sphere =

Area of triangle ABC =

Arc length = , where θ is in radians

Sector area = , where θ is in radians

Trigonometry

Statistics

Mean =

Standard deviation =

Page 2: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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2Answer all the questions.

1 Calculate

2−1 . 9083

√1050×5 .4 , giving your answer correct to 3 significant figures.

Answer ….……….…………...…..[1]

2 Giving your answers in standard form,(a) Evaluate 18 10 – 4 + 4.5 10 – 1 2.

(b) Express 8 × 1016 microseconds in minutes.

Answer (a)……….…………...…..[1]

(b)……….………minutes [1]

3 At a test flight, when an aircraft reached a height of 28 km, the temperature outside the aircraft was – 73.2oC. When it landed, the temperature outside increased to t oC.Find an expression, in terms of t, for(a) the difference in the two temperatures,

Answer (a)……….…………... oC [1]

(b) the mean temperatures of the two heights.

Answer (b)……….…………... oC [1]

If t = 11.5 oC, (c) calculate the temperature outside when the aircraft was at a height of 6 km, assuming

that the temperature changed uniformly with the height.

Page 3: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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3Answer (c)……….………….... oC [2]

4 Two maps of the same town are drawn. On the first map, a garden has an area of 2 cm2. On the second map, the area of the same garden is 8 cm2. Given that the scale of the first map is 1 : 25 000, find(a) the actual length, in metres, of a path represented by a line of 1.6 cm on the first map,(b) the scale of the second map in the form 1 : n.

Answer (a)……….……….…....m [1]

(b)……….…………...…..[1]

5 (a) Factorise the following completely.(i) 2x2 – 10x – 12

Answer (a)(i)……….………...…..[1]

(ii) p2y3 – y3 + 3p2x – 3x

Answer (a)(ii)……….……...….....[2]

(b) Simplify 16k2 + 1 – (4k – 1)2 .

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4

Answer (b)……….…………...…..[2]

6 Solve the simultaneous equations 2x + 3y = 7,y – 3x = 17.

Answer x = ..……...………..…...…..

y = ..……...………....…..[2]

7 (a) Expressing your answer as a power of 2, find

(i) 28 82 4– 2

Answer (a)(i).…….…………...…..[1]

(ii)

5125

√232

Answer (a)(ii).………………...…..[1]

(b) Simplify 16x2 4x– 3.

Answer (b)……….…………...…..[1]

(c) Given that 1−a−3×ak=0 , state the value of k.

Answer (c) k = .….…………...…..[1]

8 Ali has a length of string. The string is 4 m long, correct to the nearest 10 cm.(a) Find the least possible bound of the length of the string in centimetres.

Answer (a)……….……….…...cm [1]

(b) He cuts off ten pieces of string. Each piece is 5 cm long, correct to the nearest centimetre. Find the minimum possible length of string remaining in centimetres.

Page 5: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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5

Answer (b)….…….…………..cm [1]

9 (a) A bank exchanged Singapore dollars (S$) and South Korean Won (KRW) at a rate ofS$ 1 = KRW 917.9487. Calculate, in S$, the amount received for KRW 300 000.

Answer (a) S$…….…………..…..[1]

(b) Amy invests her $200 000 in a financial plan which paid an interest rate of r % per annum, compounded on a half-yearly basis. At the end of 2 years, she will receive an interest of $43 101.25. Calculate the value of r.

Answer (b) r = …….….……...…..[2]

10 (a) Find the range of values of x which satisfy the inequality 3 x+6≤5 x−1<4 x+7 .

Answer (a)……….……….….....…[2]

(b) Show the solution of part (a) on the number line below.

[1]

(c) Given that – 7 < x < 3 and 3 ≤ y ≤ 7, where x and y are integers, find

10 5 0 – 5 – 10

Page 6: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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6

(i) the smallest value of x4− y ,

(ii) the largest value of (2 x− y )2.

Answer (c)(i)…….…………...….. [1]

(ii)…............................. [1]

11 The time taken to fill a tank with water varies inversely as the square of the area of cross-section of the pipe. The time taken is 20 minutes when the area is 3 cm2.

(a) It is given that the area is A cm2. Find the expression, in terms of A, for the number ofminutes taken to fill the tank.

Answer (a)……….………minutes [1]

(b) Find the number of minutes taken to fill the tank when the area is 5 cm2.

Answer (b)………...……..minutes [1]

(c) Water flowed into the empty tank through a pipe of area 6 cm2. It flowed for3.5 minutes only. Find the fraction of the tank that contained the water.

Answer (c)……….………………..[1]

12 When written as the product of its prime factors, 500 is 22 53.(a) Express 1575 as a product of its prime factors.(b) Write down the least integer k for which 1575k is a perfect cube.

Answer (a)……….……….….....…[1]

(b)….…….…………...…..[1]

When written as the products of its prime factors, N = 2m×5r×7t

.

The highest common factor of N and 500 is 22×52

.The lowest common multiple of N and 500 is 23 53 7.

(c) Find m, r and t.

Page 7: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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7

Answer (c) m = …..………, r = …..………, t = …..….…….[2]

13

In the diagram, ABC is an equilateral triangle. The points P, Q and R lie on AB, BC and

CA respectively such that AP = BQ = CR. Prove that triangles APR and CRQ are congruent.

Answer In triangles APR and CRQ, …….……………………………………………..…

…….……………………………………………………………………………...

…….……………………………………………………………………………...

…….……………………………………………………………………………...

…….…………………………………………………………………………... [3]

14 The diagram shows part of a regular octagon FGHIJ… .

EF = EG = EH = EJ.EH meets GJ at B.

Calculate the angle(a) p,(b) q,(c) r,(d) s.

Answer (a)……….…………...…o [2]

BE

J

I

H

G

F

rst

q

p

R

Q

C

B

P

A

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8 (b)….………………...... o [1]

(c)…....….…………...... o [1]

(d)….………………...... o [1]

(e) Write down the special name given to the quadrilateral EGHJ.

Answer (e)…....….………….........[1]

15 (a) It is given that Q = {2, 4, 6, 8, 10), R = {5, 10, 15, 20},

15∈P, n(P) = 1 and P∩Q=φ . On the Venn Diagram shown in the answer space for (a), label each set with P, Q or R and indicate the elements of each set clearly.

Answer (a)

[2]

(b) Express, in set notation, the set represented by the shaded area in terms of A and B.

Answer (b)….………………......... [1]

16 The first four terms of a sequence are 55, 53, 49, 41.

The nth term of this sequence is 57 – 2n

.

(a) Calculate the 5th term.

(b) Write down the nth term of the sequence 57, 61, 67, 73…

BA

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Answer (a)….………………......... [1]

(b)….………………......... [1]

17 (a) (i) Sketch the graph of y=( x+1)2−4 .

Answer (a)(i)

[2]

(ii) Write down the coordinates of the minimum point.

Answer (a)(ii) (…….…, ……….) [1]

(b) A graph is drawn on the grid below. Points A and B are marked on the curves.

-2

-4

-6

6

4

2

0 7654321-1-2-3-4-5-6-7

B

A

0 x

y

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10

(i) Write down the coordinates of A and B.

Answer (b)(i) A (…….…, ……….) and B (…….…, ……….) [1]

(ii) The equation of the graph is xy = n. Write down the value of n.

Answer (b)(ii) n = ….………...…. [1]

Page 11: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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11

18 In the diagram, PQR is a straight line, QRT = 90o, QR = x, RT = y and QT = 3.

Write down an expression, in terms of x and / or y for

(a) sin PQT,

(b) cos PQT.

Answer (a)……….…………...…..[1]

(b)……….…………...…..[1]

19

The diagram above shows a sector of a circle of radius 8 cm. The angle θ is 0.25.

(a) Find the angle of the sector in terms of .(b) The perimeter of the sector can be written as (m + n) cm. Find the value of m and n.(c) Find the area of the sector in terms of .

Answer (a)……….…………...….. [1]

(b) m =.…….., n =.…........[2]

(c)…....….………….. cm2 [1]

3y

T

RQP x

θ = 0.25

8

Page 12: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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12

20 In the diagram, the point O and P are marked on the grid showing the vectors a and b.

(a) Write down an expression for OP in terms of a and b.

Answer (a)……….…………...…..[1]

(b) Mark and label clearly on the grid above, the point Q such that OQ = 3b – 3a. [1]

(c) Given that OP = ( 4−2)

, find |OP|.

Answer (c)……….………….units [1]

O

P

a

b

Page 13: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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13

21 Fifty loaves of bread of Brand A and Brand B are weighed. The cumulative frequency curves show the distribution of their weights.

(a) For Brand A, estimate the median.

Answer (a)……….………...grams [1]

(b) Estimate the value of x if 3

10 of the Brand B loaves weigh more than x g.

Answer (b) x = .….…………...…..[1]

(c) Both brands are of the same price and their packaging indicated 500 g. Which do you think is the better buy? Justify your answer.

Answer (c) Brand ……… is a better buy because …………………………………….……

…………………………………………………………………………………………….[1]

It was later discovered that the measuring equipment used to weigh the loaves from Brand A was faulty. This caused every measurement taken to be 2 g short. The diagram below shows the box-and-whisker plot of Brand A with the corrected values.

Write down the value of

(d) x2 ,

Weight (grams)

Cumulative frequency

0 490 491 492 493 494 495 496 497 498 499 5000

10

20

30

40

50

Weight (grams)x3x2x1

Brand A

Brand A Brand B

Page 14: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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14(e) x3−x1 .

Answer (d)……….…………...…..[1]

Answer (e)……….…………...…..[1]

22 Ben cycled from his home to a friend’s house. The diagrams below show the speed-time graph for the last part of his journey and the distance-time graph for the first 8 seconds of the same journey.

(a) Calculate the distance travelled from t = 8 to t = 18.

Answer (a) ................................. m [1]

He started from rest and cycled with a constant acceleration for the first 6 seconds.(b) Calculate, at t = 5,

(i) the speed in kilometres per hour,(ii) the acceleration in metres per square second.

Answer (b)(i)...........................km/h [2]

00 2 4 6 8 10 12 14 16 18

40

60

20

80

100

120

140

Time (t seconds)

Distance from home (metres)

3625

Time (t seconds)

Speed(metres/second)

0 2 4 6 8 10 12 14 16 18 20

2

4

0

6

8

10

12

Page 15: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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15 (ii)...........................m/s2 [1]

(c) On the grid of the speed-time graph, complete the speed-time graph of Ben’s journey for the first 8 seconds. [1]

(d) On the grid of the distance-time graph, complete the distance-time graph of Ben’s journey from t = 8 to t = 18. [2]

23 The triangle PQR is shown below.

Answer (a), (b) and (c).

(a) Construct the perpendicular bisector of PQ. [1]

(b) Construct the bisector of angle PQR. [1]

(c) Mark, by using a cross, and label a possible point S which is inside the triangle, equidistant from PQ and QR, and is nearer to P than Q. [1]

P

Q

R

Page 16: EM 4E 2012Prelim XinMin XMN2012 4E5N Prelim Math P1

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16Answer Key

1 0.001682a2b

9.018 × 10-1

1.33 × 109

3a3b3c

t + 73.2(t – 73.2) / 2–6.65

4a4b

4001 : 12500

5ai5aii5b

2(x + 1)(x – 6)(3x + y3)(p + 1)(p – 1)8k

6 x = –4, y = 57ai7aii7b7c

2-2

213/5

4x5

38a8b

395340

9a9b

326.8210

10a10c

3.5≤x≤8–7, 361

11a11b11c

180/A2

7.27/10

12a12b12c

32 × 52 × 7735m=3, r = 2, t=1

14 135,22.5,45,45,trapezium15b (A’∩B’)U(A∩B)16a16b

2557 – 2n + 2n2

17a17bi17bii

(-1,-4)A(-3, -4) B(2, 6)n = 12

18a18b

y/3- x/3

19a19b19c

1.75πm=16, n=1456π

20a20c

a-4b4.47

21 495.2, 497, B, 497.2, 222 72, 36, 2