Answer All The Questions

Answer all the questions

1

(

)

m v2 − u . Express v in terms of g , m, p and u 2g

(a)

Given that p =

(b)

Simplify the expression

(c)

The distance from the Sun to Pluto is 7.38 terametres.

[2]

9 x 2 − 25( x + y ) 2 . 2x + 5 y

[3]

(i)

Express 7.38 terametres in metres, giving your answers in standard form.

[1]

(ii)

Light travels 1 metre in 3.3 nanoseconds. How many minutes does light take to travel from the Sun to Pluto? [2]

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2

(a)

A fruit trader realises he can earn a profit of $0.40 per fruit if he increases the selling price of each fruit from 20% to 45% above the cost price. What should his selling price be if he wants to make a profit of 22.5%? [3]

(b)

James wants to develop a roll of film. The total cost comprises 2 parts: a fixed cost of $3.60 per roll and a variable cost of 35 cents per photograph. A standard roll of film consists of 32 photographs.

(i)

Find the average cost of developing each photograph from a roll of film.

[2]

(ii)

A special promotion takes place in which there is a 40% discount off the variable cost. Calculate the percentage reduction in the total cost compared to the usual cost of developing two rolls of film. [3]

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[Turn over ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

3

The diagram shows a rectangular vertical board ABCD. AD is on the ground. It is supported by cables BE and CE where E is also on the ground. CD = 5 m, ∠CED = 36°, ∠ADE = 58° and ∆ADE is isosceles with AE = DE. C

B

5m

D 58°

36°

A E Find (a) (b) (c) (d)

DE, AD, ∠BEC, the area of ∆BEC .

[2] [2] [3] [2]

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4

In an annual field trip to Penang, a school chartered a bus for $1400 to take x number of students. Every student in the group is required to pay an equal amount for the hire of the bus.

(a)

Find an expression, in terms of x, for the amount of money each student has to pay. [1]

On the day of the departure, five students could not make it for the trip. The school decided to contribute $80 from the school fund to cover part of the cost of $1400.

(b)

Write down an expression, in terms of x, for the amount of money each student has to pay when five students cannot make it for the trip. [1]

(c)

When five students cannot make it for the trip, the amount each student who can make it has to pay $4 more than if all could go. Write down an equation to represent this information, and show that it simplifies to x 2 + 15 x − 1750 = 0 . [3]

(d)

Solve the equation x 2 + 15 x − 1750 = 0 .

[3]

(e)

Find the amount each student who makes it for the trip has to pay.

[2]

_______________________________________________________________________________ ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

5

Figure I shows a solid sphere with centre O and radius 15 cm placed completely inside an inverted open cone of radius, R cm and height, H cm. The sphere just touches the centre of the base of the cone. The reflex angle AOB is 210°. The upper portion of the inverted cone is then cut along AB and discarded. The resulting cone has height, h cm, diameter AB and radius, r cm as shown in Figure II.

R

15

210° O

15

15

B

A

H

210° O

15 r

A

B h

V Figure I

V Figure II

(a)

Show that r = 14.5 cm.

[2]

(b)

Find the height, h of the cone shown in Figure II.

[2]

(c)

Find the height, H of the original cone shown in Figure I.

[3]

(d)

Find

surface area of the sphere as shown in Figure I. Give your answer correct surface area of the cone

to 2 decimal places.

[3]

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[Turn over ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

6 Q

D

C

P

26

A

20

B

The diagram shows a square, ABCD. PQ is an arc of a circle with center A and its radius AP = 26 cm. AB = 20 cm.

(a)

Write down the value of sin ∠APC .

[1]

(b)

Prove that triangles ABP and ADQ are congruent.

[2]

(c)

Find ∠PAQ in radians.

[2]

(d)

Find the perimeter of the shaded region CPQ.

[2]

(e)

Find the area of the shaded region CPQ as a percentage of the square ABCD.

[3]

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ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

7

(a)

In ∆ABC, D, E and F are midpoints of AB, BC and AC respectively. The lines AE 2 and CD intersect at a point X such that CX = CD and AX = 2 XE . 3

C

E

F X

A

B

D

Given that AB = 10a and BC = 6b, express as simply as possible, in terms of a and / or b,

(i)

AE ,

(ii)

CD ,

(iii)

BX ,

XF . [4] Use your answers to parts (a)(iii) and (a)(iv) to explain why B, X, F lie in a straight line. [1] area of ∆ADX (vi) Find . [1] area of ∆ABC (iv) (v)

(b)

Consider the following number pattern. 77 × 80 + (76 − 1) 2 = 8705 2 79 × 82 = + (78 − 1) 2 = 9168 2 81 × 84 = + (80 − 1) 2 = 9643 2

u 77 = u 79 u81

(i)

Write down an expression for u 75 and show that u 75 = 8254 .

[1]

(ii)

Write down an expression for u1 and evaluate it.

[1]

th

(iii) Find an expression, in terms of n, for the n term, u n of the sequence. (iv)

Hence, find the value of n when u n = 445 .

[1] [2]

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[Turn over ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

8

A group of 120 students took an IQ test. The cumulative frequency curve below shows the distribution of their IQ.

(a)

Copy and complete the grouped frequency table of the IQ of the group of students.

IQ (x)

80 < x ≤ 90

90 < x ≤ 100

Frequency

100 < x ≤ 110

110 < x ≤ 120

120 < x ≤ 130

44 [2]

(b)

Using your grouped frequency table, calculate an estimate of

(i)

the mean IQ of the group of students,

[2]

(ii)

the standard deviation.

[2]

ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

(c)

The box and whiskers diagram of another group of 120 students who took the IQ test is shown below. 80

100

105

120

130

Describe how the cumulative frequency curve will differ from the given curve.

[1]

(d)

Use the graph to find the lowest possible IQ of the top 5% of the students.

[1]

(e)

Given that only two of the students were named Paul, what is the probability that both Pauls had an IQ greater than 120? [2]

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9

In the figure below, which is not drawn to scale, A, B and C are three points on a level ground. It is also given that the bearing of C from A is 023° and the bearing of C from B is 048°. AB = 35 km. N

C

48°

B 35 km 23°

(a)

(b)

A Calculate (i) the distance of BC, (ii) how far east is C of A, (iii) the bearing of A from C.

[3] [2] [1]

Peter leaves A at 0710 h and cycles directly to C at a speed of 18 km/h. (i) Find the time to the nearest minute, at which Peter is nearest to B. [3] (ii) If Tom cycles 50 km from A along AC to a point X, calculate the bearing, to the nearest degree of X from B. [3]

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[Turn over ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

10

Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation y = 2 x +

3 . Some corresponding x2

values are given in the following table:

x

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

y

13

5

4.33

4.75

k

6.33

7.24

8.19

(a)

Calculate the value of k.

[1]

(b)

Using a scale of 4 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit 3 [3] on the y-axis, draw the graph of y = 2 x + 2 for 0.5 ≤ x ≤ 4.0 . x

(c)

Find the gradient of the tangent to the curve at the point where x = 2.

[2]

(d)

Draw the graph of 4 x + y = 6 for 0 ≤ x ≤ 1.5 .

[1]

(e)

By drawing a suitable tangent to your curve, find the coordinates of the point at which the gradient of the tangent is equal to –4. [1]

(f)

By drawing a suitable straight line, use your graph to solve the equation [3] 2 x 3 − 7 x 2 + 3 = 0 in the range 0.5 ≤ x ≤ 4.0 .

(g)

State the range of values of x for 0.5 ≤ x ≤ 4.0 for which the gradient of the curve is negative. [1]

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ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

Answer Key 2 pg + mu m 12 1ci) 7.38 × 10 metres

1b) 7.38 × 1012 metres

2a) $1.96

2bi) $0.46

2bii) 30.3%

3a) 6.88m

3b) 7.29m

3c) 50.8°

3d) 28.0m2

1320 x−5

4c) Proving

4d) 35

5a) Proving

5b) 54.1cm

5c) 73.0cm

5d) 0.61

6a) sin ∠APC =

10 13

1a)

4a) $

v=±

1400 x

4b) $

1cii) 405.9 min

4e) $44

6c) 0.18448

6b) ∆ABP ≡ ∆ADQ ( RHS )

6d) 11.6cm

6e) 1.35%

7ai) AE = 10a + 3b

7aii) CD = −5a − 6b

7aiii) BX = 2(− a + b)

7av) BX = 2 XF

7avi)

5 3

7aiv) XF = − a + b

75 × 78 + (74 − 1) 2 = 8254 2 3n 2 − 5n + 8 7biii) u n = 2 7bi) u 75 =

5 3

7bii) u1 =

1 6

1× 4 + (0 − 1) 2 = 3 2

7biv) 18

8a) IQ (x)

80 < x ≤ 90

Freq

2

90 < x ≤ 100 100 < x ≤ 110 110 < x ≤ 120 120 < x ≤ 130

8bi) 111

8bii) 7.82

8d) 123

8e) 3/476

9ai) 32.4 km

9aii) 24.0 km

9bi) 0857 hrs

9bii) 061°

10a) 5.48

10c) 1.25

6

44

58

10

8c) The curve will be less steep and more to the left.

9aiii) 203°

10e) x=1

10g) 0.5 ≤ x ≤ 1.4

ACS(Independent)Math Dept/Y4E/EM2/2009/Prelim

10f) Draw y=7, x=0.75 or 3.35