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SEC 4 Os
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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 =
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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.
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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.
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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
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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.
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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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(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]
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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
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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
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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
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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
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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
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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