Answer All The Questions. 1 A Map Is Drawn To

Answer all the questions. 1

A map is drawn to a scale of 1 : 25 000. (a) (b)

A highway is represented by a line of length 6.3 cm on the map. Calculate the actual length of the highway, giving your answer in kilometres. The actual area of a zoo is 5.2 km 2 . Calculate the area on the map which represents the zoo, giving your answer in square centimetres.

Answer (a) …………….…………..…..[1] (b)……………………………..[1] −3

2

2

(a)

5 5 Simplify   ×   .  x  x

(b)

Given that 3 42 × 3 4 ÷ 9 3 = 81t , find the value of t.

Answer (a) …….………………………..[1] (b)….………….………..… …..[1] ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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A school event is attended by 60 men, 96 women and 120 children. They are divided into small groups. What is the greatest number of groups that can be formed so that the men, women and children are distributed equally among the groups?

Answer …..…….………………………..[2] 4

(a) (b)

x is directly proportional to y 2 . It is known that x = 36 for a particular value of y. Find the value of x when this value of y is doubled. Given that p 2 is inversely proportional to q and p = 2 when q = 3, find the value of q when p = 6.

Answer (a) ……………………….…….[1] (b)……………………......……..[2]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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5

John invested $12 000 over a period of 3 years into each of two different plans, Plan A and Plan B. Plan A offered an interest of 5.25% per annum compounded annually. Plan B offered an interest of 5% per annum compounded monthly. Determine which plan offers a better return. Show your workings clearly.

Answer ………………….……..……..[3]

6

Solve the simultaneous equations

−

2

x + y = 10 ; 5 3y + x = 8

Answer x = …….……,y = ……..…….[3]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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7

Mr David marked the price of a dining table set at $805 so that he could make a profit of 15% on the cost price. During a sale, he gave a 5% discount to his customers on the marked price.

(a) (b)

Find the cost price of the dining set. Find his percentage profit on the dining set if he sold it during the sale.

Answer (a) ……………….….…..……..[1] (b)……………..…………….…..[2]

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20 women, 12 men, 8 girls and 4 boys attended a party. Two of them were chosen, one after the other, for a game. Find the probability that (a) the first person chosen is a female, (b) both were adults, (c) one adult and one child were chosen.

Answer (a) …….………………………..[1] (b)….………….….….…… .…..[1] (c)………….…..……...…….…..[1] ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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(a)

Given that x is a prime number such that 4 <

(b)

of x . Factorize ab + 2bx − 4ax − 8 x 2 completely.

3+ x 2

≤ 15 , find the smallest value

Answer (a) ……………………..…..…..[1] (b) ……………………….……...[2]

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(a)

Given that ξ={ x : x is an integer , 2 ≤ x < 17 }, A={ x : x is a prime number} and B={ x : 3 x ≤ 16 }. (i) List the elements of the set A. (ii) Find A ∩ B .

Answer (a) (i) …………..…………………………..[1] (ii)………………………………...……..[1]

(b)

On the Venn Diagram shown in the answer space, shade the set (A' ∩ B)'. Answer (b)

ε

A

B [1]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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11

A shop sells three types of rice, White rice, Brown rice and Mixed rice. Each variety of rice is sold in packets of two different sizes, small and large. The sales in two successive months are given in the table below.

July

August

Small

Large

Small

Large

White Rice

15

20

17

22

Brown Rice

13

21

11

9

Mixed Rice

18

12

10

6

 15 20    The information for July’s sales can be represented by the matrix A =  13 21  .  18 12   

 17 22    The information for August is represented by the matrix B =  11 9  .  10 6    1 2

(a)

Find C, where C = ( A + B) .

(b)

Given that P =   , find D = CP .  1 Hence, interpret the elements in matrix D.

(c)

 1

Answer (a) …….……………...........…..[1]

(b) …….……………...........…..[1]

(c)….………….……..…… ……………………………………….………………………….…….…….[1] …………………………………………………………………….……………………………………

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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The side AB = 6 cm of triangle ABC is drawn in the answer space. (a) (b) (c)

Complete the triangle with BC = 8 cm and CA = 5 cm. [1] Plot a point P, which is equidistant from the points A and B and equidistant from the lines AB and AC. [2] Measure the length of PC.

Answer (c) ……………………….…..[1]

13

Solve

2x + 3 2−x

=

4−x x+2

.

Answer …………………,………………[4]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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 − 3  , b = 2  

If a = 

(a) (b)

 7    , c = − 12  

 − 4   and d =  x 

 − 2   ,  5 

find | a + b |, find the value of x if d is parallel to c.

Answer (a) …..….……………………..[1]

(c)

(b)….………….….……… …..[2] 1 Using the value of x found in (b), draw the vector u = 2a + c, with O as its 2 initial point on the grid provided below.

[1]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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15

ABC is a triangle with vertices A(4,0), B(0,10) and C(x,y).

y

C

B

O

(a)

A

Find the coordinates of C, if the x coordinate of the midpoint of AC is 8 and the gradient of the line AC is

(b)

x

13 . 8

Find the equation of the line passing through C and parallel to AB.

Answer (a) …….………………….…....[2] (b)….……….….……….… …..[2] ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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OPQ and PRS are triangles. PQR and RST are straight lines. OQ = 8 cm, QR = 6 cm and RS = 5 cm. ∠PQO = ∠PRS = 90 o and sin θ =

1 . 2 T S

5 cm Q

P

6cm

θ

R

8 cm

O

(a) (b)

Find PR. Find tan ∠PST .

Answer (a) ……..…………………..…..[2] (b)….…………...………… …..[2] ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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A, B, C and D are points on a circle with centre O. BC = 15 units, ∠COD = 56 o , ∠BAP = 47 o and AC is the diameter. PT is a tangent to the circle at A. D

C o o 5656

15 3

T

O O B A

9 47 o P

(a) (b) (c)

Find the radius of the circle. Find ∠DCB . By showing your workings clearly, prove that the triangle DCB is not an isosceles triangle.

Answer (a) …….……………………..[1] (b)….………….………… …..[2]

(c)……………………………………………………………………………..………..………..[2] ………………………………………………………………………………………. ……………………………………………………………………………………….. .………………………………………………………………………………………. ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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A class of 12 boys and 8 girls took a mathematics test. The boys’ scores were 4, 8, 6, 7, 8, 11, 7, 8, 11, 7, x - 4, 11. (a) Find the value of x if the modal score for the boys is 8. (b) Given that the mean score of the boys is 7.5, (i) find the value of x, (ii) find the mean score of the whole class if x = 6 and the mean score of the girls is 8.

Answer (a) ……………………….……..[1] (b) (i)……….……………….....[2] (ii)…………….……………..[2]

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Peter has 2 similar conical flasks. The ratio of the heights of the flasks is 5:1. (a) If the volume of the bigger conical flask is 6250 cm 3 , find the volume of the smaller conical flask. (b) If the base area of the bigger conical flask is 187.5 cm 2 , find the height of the smaller conical flask.

Answer (a) …….….…...………..….…..[2] (b)….……………………… …..[3] ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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George drives home from Town A to Town B at a constantly increasing speed for 2 h and then travels at a constant speed of 50 km/h for an hour. He increases his speed uniformly for another hour until he reaches a speed of 70 km/h and then decelerates for an hour before he comes to rest. The speed-time graph of his journey is as shown.

(a) (b)

Calculate George’s speed at t = 3.5 hr. What was George’s acceleration during the last one hour?

Answer (a) …….………..………….…..[2] (b)….……………………… …..[1]

(c)

Draw the corresponding distance-time graph in the space provided. Answer(c)

[2]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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(a)

Express x 2 + 2 x − 15 in the form ( x − a ) 2 + b .

(b)

Hence, solve the equation x 2 + 2 x − 15 = 0 .

Answer (a) ………………………….…..[1] (b) …………………….………..[2]

(c)

Sketch the graph of y = x 2 + 2 x − 15 , showing the axes intercepts and the turning point clearly on the graph. Answer (c)

y

O

x

[2]

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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(a)

Find the value of x for the hexagon given below.

135 o 125 o 4x o + 20o

3 x o + 35 o

2 x o + 30 o

3 x o − 15o

Answer (a) …….…………………...…..[2]

(b)

The diagram shows an incomplete figure made up of a square, a regular hexagon and another regular polygon of n sides. Find the value of n.

Answer (b)….………….………..… …..[3]

END OF PAPER

ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

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Answers: 1. a) 1.58 km

12. PC = 3.7 or 3.8

13. x = 0.152 or -13.2 14. a) 10.8 b) x = 10

b) 83.2 cm 2 2. a)

x 5

b) t = 10

 − 8   9 

c) u = 

3. a) 12 groups 4. a) 144 b)

15. a) C ( 12 , 13) b) 5x + 2y = 86 16. a) 19.9 cm

1 3

5. Plan A : $ 13990.96, Plan B : $13937.67 . Ans : Plan A 6. x = -10 ; y = 6 7. a) $700 b) 9.25% 7 11 248 b) 473 192 c) 473

8. a)

b) -3.97 17. a) 11 cm b) 109 o c) 18. a) x = 12 b) (i) x = 6 b) (ii) 7.7 19. a) volume = 50 cm 3 b) h = 20 cm

9. a) 7 b) (a + 2x) (b - 4x) 10. a) (i) A = {2,3,5,7,11,13} (ii) A ∩ B = {2,3,5} b) ε

20. a) v = 60 km/hr b) acc = -70 km/hr 2 c) Distance 1 At t=2 --------- t = × 2 × 50 = 50 2 At t=3 ---------- t = 50 + 1× 50 = 100 At t=4 ---------- t =

A

B

16 21   1. a) C = 12 15  14 9     37    b) D =  27   23   

c) The three elements represent the average of the total of small and big packets of the three kind of rice sold per month. ACS(Independent)Math Dept/Y4E/EM1/2009/Prelim

1 100 + ( × (50 + 70) × 1 = 160 2 1 At t=5 -----------t = 160+ × 1× 70 = 195 2 21 a) (x – (-1)) 2 +(-16) b) x = -5 or 3 c) y intercept is -15 turning point : (-1 -16) x intercepts are -5 and 3. 22 a) x = 30 o b) 12

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