4e5n Math P1 Prelim 2009 With Ans

Name: _______________________________ (

)

Class: _______

MONTFORT SECONDARY SCHOOL ‘O’ LEVEL PRELIMINARY EXAMINATION 2009

Secondary 4 Express / 5 Normal MATHEMATICS PAPER 1 Monday

14 Sept 2009

4016/1 0830-1030

2 hours

READ THESE INSTRUCTIONS FIRST Write your name, register number and class at the top of this page. Write in dark blue or black pen in the spaces provided on the Question Paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question, it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π , use calculator value or 3.142, unless the question requires the answer in terms of π . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 80.

80 _____________________________________________________________________________ This document consists of 16 printed pages.

Setter: Ong CL

2 Mathematical Formulae Compound interest

r   Total amount = P1 +   100 

n

Mensuration Curved surface area of cone = π rl Surface area of a sphere = 4π r 2 1 Volume of a cone = π r 2 h 3 Volume of a sphere =

4 3 πr 3

1 ab sin C 2 Arc length = r θ , where θ is in radians Area of triangle ABC =

Sector area =

1 2 r θ , where θ is in radians 2

Trigonometry

a b c = = sin A sin B sin C a 2 = b 2 + c 2 − 2bc cos A Statistics Mean =

Standard deviation =

4E5N Math Paper1 Prelim 2009

∑ fx ∑f ∑ f x 2 −  ∑ f x  ∑ f  ∑ f 

2

3 Answer all the questions. 1

(a) Evaluate

38.89 − 6.03 , correct to 1 significant figure. 1.963 × 2703 1

(b) Simplify 27 x × 3−2 x ÷ 9 2

Answer (a)_______________________ [1] (b)_______________________ [1] _____________________________________________________________________________ 2 The distance between the Sun and the Earth is 1.5 × 108 km. (a) Light travels at 3 × 105 km/s. Calculate the time taken for light to travel from the Sun to the Earth. Give your answer in standard form. (b) Pluto is 6.0 × 109 km from the Sun. On a certain day, the Sun, Earth and Pluto are in a straight line with the Earth between the Sun and Pluto. The distance between the Earth and Pluto can be expressed as k billion km. Find the value of k.

Answer (a)______________________ s [1] (b)_______________________ [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

4 3

Solve the simultaneous equations.

5x + 2 y = 1 2 x = y + 13

Answer x = _________ , y = _________ [2] _____________________________________________________________________________ 4 Each interior angle of a regular 10-sided polygon is 84° more than each exterior angle of a regular n-sided polygon. Calculate the value of n.

Answer _________________________ [2] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

5 5

A shopkeeper bought a watch for $250. He still made a percentage profit of 80% despite offering a 25% discount to his customer. Calculate the selling price of his watch before discount.

Answer $_________________________ [2] _____________________________________________________________________________ 6 Given that − 2 ≤ x ≤ 1 and − 6 ≤ y ≤ −1 where x and y are integers, find (a) the greatest value of ( x − y ) 2 , (b) the smallest value of

x+ y . y

Answer (a)_______________________ [1] (b)_______________________ [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

6 7

(a) The temperature in Beijing at 0500 was −2°C. At 1400 the temperature was 16°C. Assuming that the temperature rose at a steady rate, find the time when the temperature was 10°C. (b) A feeder bus left the station 3p minutes after 1100 and arrived back at the station 20 minutes later at p minutes to 1300. At what time did the bus leave the station?

Answer (a)_______________________ [1] (b)_______________________ [2] _____________________________________________________________________________ 8 Given that ξ = {x :1 ≤ x ≤ 11, x is an integer} , A = { x : x is a prime number} , B = {x : x is a factor 24} , C = {x : x is a multiple of 3} . (a) List the elements in (i) A ∩ B ∩ C , (ii) ( B ∪ C ) ' (b) Find n( A ∩ B ') .

Answer (a)_______________________ [1] (b)_______________________ [1] (c)_______________________ [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

7 9

A man invests $1000 for 3 years at 10% per annum simple interest. How much greater would his return have been if the money had been invested at 10% per annum compound interest, compounded annually, for the same period?

Answer $ ________________________ [3] _____________________________________________________________________________ 10 A six-faced unbiased die was thrown 20 times. The table below shows the frequency that each number on the die appeared. Number Frequency

1 4

2 6

3 3

4 2

5 4

6 1

(a) After the 19th throw, the median number was 2. What was the least possible number that appeared on the 20th throw? (b) The die was thrown one more time and the mean number of all 21 throws was 3. Calculate the number that appeared on the last throw.

Answer (a)_______________________ [1] (b)_______________________ [2] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

8 11 During the last FA Cup Final, Coffee Shops A, B and C attracted more customers by providing live telecasts of matches on cable television. The average number of orders for soft drinks, beer and coffee for a match day are as follows: Coffee Shop

Cans of soft drinks

Bottles of beer

Cups of coffee

A

90

50

60

B

10

85

16

C

70

30

40

The price per can of soft drinks is $1.80, per bottle of beer is $6.00 and per cup of coffee is $1.50.

P is a 3 × 3 matrix to represent the number of soft drinks, beer and coffee sold by the 3 coffee shops and Q is a 3 × 1 matrix to represent the cost of soft drinks, beer and coffee sold by the 3 coffee shops. (a) If R = PQ, calculate R. 1 SR, calculate T. 3 (c) Explain what is the significance of your answer in (b). (b) Given S = (1 1 1) and T =

Answer (a)_______________________ [1] (b)_______________________ [1] (c)_______________________ [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

9 12 A football club decided to pay a certain amount of money as bonus to 3 strikers, David, Michael and Henry. They are to share the bonus in the ratio of the goals they score. David, Michael and Henry scored 21, 18 and 6 goals respectively. The mean amount of money each striker will receive is $4 500. (a) Find the amount of bonus Henry would receive. (b) The football club decided to include another striker Ronaldo, to receive the bonus. If Ronaldo received $3 375 and Michael received $1 350 lesser now, how many goals did Ronaldo score?

Answer (a) $______________________ [2] (b)_______________________ [2] _____________________________________________________________________________ 13 (a) Expand and simplify (2e − 3)2 − 5e(6 − e) . (b) Factorise completely m 2 n 2 + n 2 − m 2 − 1 .

Answer (a)_______________________ [2] (b)_______________________ [3] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

10 14 In the diagram, AOB is the diameter of a semicircle, centre O and radius 35 cm. OPQ is a quadrant of a circle and the straight line QR cuts AB at R. OR = 24 cm and QR = 25 cm.

22 , calculate 7 (a) the length of OP, (b) the perimeter of the shaded region. (c) the value of cos ∠QRB.

Taking π =

Answer (a)_____________________ cm [1] (b)_____________________ cm [2] (c)_______________________ [1] 15 (a) Given that y varies inversely as (2x + 3) and the difference in the values of y when x = 1 and x = 3 is 4, express y in terms of x. (b) If tap A can fill up a tank in 20 minutes and tap B can fill up the same tank in 15 minutes, how long will it take for the taps to fill up the tank together?

Answer (a) y = ___________________ [2] (b)____________________ min [2] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

11 16 A bowl of sweets contains 2 fruit gums, 3 mints and 5 toffees. 3 sweets are to be chosen at random and without replacement, from the bowl. Calculate the probability that (a) all 3 sweets will not be mint, (b) of the 3 sweets chosen, the first 2 will be the same and the third a toffee,

Answer (a)_______________________ [2] (b)_______________________ [2] _____________________________________________________________________________ 1 + mn 17 (a) Given that p = , express m in terms of p, q and n. mq (b) Solve the equation x( x − 2) = 18 + 5 x .

Answer (a)_______________________ [3] (b)_______________________ [2] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

12 18 (a) The equation of each of the curves below is of the form y = ax n , where n is an integer such that −1 ≤ n ≤ 2 . Each of the curves passes through the point (2, 1). Write down the equation for each of the curves.

(b) (i) Sketch the graph of y = ( x + 3)(1 − x) .

y

0

x

[2] (ii) Write down the coordinates of the maximum point of the curve.

Answer (a)Fig. 1:__________________ [1] Fig. 2: __________________ [1] (b)(ii) (_________, _________) [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

13 19 The diagram shows the speed-time graph of car A and car B. The distance travelled by car B in T seconds is 720 metres. speed (m/s)

12 Car A Car B 0

20

50

90

T

time (s)

(a) Calculate (i) the value of T, (ii) the speed of the cars at the instant when they were both travelling at the same speed. (b) Sketch the distance-time graph for car B for T seconds. distance (m)

0

20

50

90

T

time (s)

[2]

Answer (a)(i)______________________ [2] (ii)__________________ m/s [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

14 20 3 corners P, Q and R of a plot of land PQRS are drawn to scale in the diagram below. Q is 1 km south of P. The corner S is 900 m from R on a bearing of 055°. (a) Find the scale used in the form 1 : n. (b) Hence, complete the scale drawing of the plot of land PQRS. (c) On the plot of land, construct (i) the bisector of angle PQR, (ii) the perpendicular bisector of the line QR. (d) A hut H lies on the intersection of these 2 lines. A footpath is constructed from the hut to the corner Q. Find the actual distance, in metres, of the footpath.

[3]

Answer (a)__________:____________ [1] (d)_____________________ m [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

15 21 The diagram shows the points P(−5, −3), Q(−1, −3) and R(5, 5). y 6

R

4

2

-6

-4

-2

0

2

4

6

x

-2

P

Q -4

Find (a) the gradient of the line PR, (b) the equation of the line which is parallel to PR and passes through the point Q, (c) the area of ∆PQR, (d) the shortest distance from the point P to the line RQ produced.

Answer (a)_______________________ [1] (b)_______________________ [2] (c)___________________ units [1] (d)_______________________ [2] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

16

uuur uuur uuur uuur 22 In the figure, OA = 4a , OB = b , BC = 3OA and D is the intersection of OC and AB. E B

b

O

C D

4a

A

(a) Express in terms of a and b, uuur uuur (i) AC , (ii) AD . uuur (b) Given that E is a point on AB produced and AE = ka − 5b, find the value of k. (c) Find the value of each of the following: area of ∆OAD area of ∆OBD , (ii) . (i) area of ∆CBD area of ∆ACD

Answer (a)(i)_____________________ [1] (ii)_____________________ [1] (b)_______________________ [2] (c)(i)_____________________ [1] (ii)_____________________ [1] _____________________________________________________________________________ 4E5N Math Paper1 Prelim 2009

17 Answer Keys: 1.(a) 0.08 (b) 3x−1 2 2. (a) 5×10 (b) 5.85 3. x = 3, y = −7 4. 6 5. $600 6. (a) 49 (b) 0 7. (a) 1100 (b) 1215 8. (a) {3} (b) {5, 7, 10, 11} (c) 3 9. $31 10. (a) 3 (b) 4  552    (b) (490) 11. (a)  552   366    (c) It represents the average earnings of the 3 coffee shops. 12. (a) $1800 (b) 15 2 13. (a) 9e −42e+9 (b) (n + 1)(n − 1)(m2 + 1) 24 14. (a) 7 cm (b) 185 cm (c) − 25 45 15. (a) y = (b) 8 4/7 min 2x + 3 7 5 (b) 16. (a) 24 36 1 17. (a) m = 2 (b) x = 9 or −2 p q−n 1 2 18. (a) Fig. 1: y = x 2 ; Fig. 2: y = (b)(ii) (−1, 4) 4 x 2 19. (a)(i) 105 (ii) 6 m/s 3 20. (a) 1 : 20 000 (d) 1040 m 4 4 11 21. (a) (b) y = x − (c) 16 units 5 5 5 22. (a)(i) 8a + b (ii) ¼ (−4a + b) (b) 20 1 (c)(i) (ii) 1 9

4E5N Math Paper1 Prelim 2009

(d) 3.2 units