chs 4016 2010 prelim iii p2

CHS /4016 2010/PRELIM III P2

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R

Q

P

N

38 km

28 km

1. (a) (i) Factorise 12 2 + nn . [1]

(ii) Hence, find the value of 199992 2 + . [2]

(b) Express as a single fraction in its simplest form 22 3 3

2 13 15 5x

x x x

+ +

. [2]

(c) It is given that uvuw

+

=1

.

(i) Find w when u = 49 and v = 17. [2] (ii) Express u in terms of w and v. [2]

2. A ship leaves P on a bearing of 315 for a point Q that is 28 km away. It then sail

from Q on a bearing of 168 to R which is 38 km from Q.

Calculate

(a) the distance PR. [3]

(b) the bearing of Q from R. [1]

(c) the area of triangle PQR. [2]

(d) S is a point on the path QR which is nearest to P.

(i) Calculate the distance PS. [2]

(ii) A helicopter H is hovering at a distance of 22 km vertically above P. Calculate the largest angle of elevation of the top of the

helicopter as seen from the path QR. [2]


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3. (a) After an 11% pay-cut, Mr Tans pay was reduced to $ 2,800. Find his original pay. [2]

(b) Mr Heng bought a car whose cash price is $144 000.

Since he could not afford to pay the total sum by cash, he paid a down payment of $30 000 and took a loan to pay for the balance at a compound interest rate of 2.5% per year for a 5-year period. Calculate how much more Mr Heng paid for the car in compound interest loan terms compared to cash terms. [3]

(c) Amanda brought a total of 3000 Euro dollars for his trip to Italy. She

spent 45

of it on sightseeing tours, and13

of the remainder on food. Upon

returning to Singapore, she changed what she had left back to Singapore dollars.

(i) Calculate in Euro dollars (), the amount Amanda spent on food. [2]

(ii) Calculate, to the nearest ten cent, the amount of Singapore dollars Amanda managed to get back, given that the exchange rate was S$1.78 to 1. [3]

4. Consider the following sequence

Line 1 2+6 = 8 =

Line 2 2+6+10 = 18 =

Line 3 2+6+10+14 = 32 =

(a) Write down Line 10 of the sequence. [1]

(b) State the sum 2 + 6 + 10 + 14 + + 50. [1]

(c) Given the sequence for Line x is 2 + 6 + 10 + 14 + + p = 648 = 22 q .

State the values of x, p and q. [3]

(d) Write down a formula, in terms of n, for the sum, Sn, of the nth line of the sequence. [1]


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5. The distance between two countries, A and B, is 286 km. David travelled by car from A to B at an average speed of x km/h.

(a) Write down an expression, in terms of x, for the number of hours he took to travel from A to B. [1]

(b) He returned from B to A at an average speed of )4( +x km/h. Write down an expression in terms of x, for the number of hours he took to travel from B to A. [1]

(c) The total time he took to go from A to B and to return from B to A was 8 hours.

(i) Write down an equation in x and show that it simplifies to

02861352 2 = xx . [3]

(ii) Solve the equation 02861352 2 = xx , giving each answer correct

to 2 decimal places. [3]

(iii) Calculate, correct to the nearest minute, the time he took to travel from A to B. [2]

6. In the diagram, 90EDC CEB EAB = = = , 5 , 8 ,BE cm EC cm= =46AEB = . Find

(a) AB, [2] (b) ,BCE [2] (c) BC, [2] (d) DE. [2]


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7.

Seven small circles, each with the same radius r cm, are enclosed within a larger circle.

The small circle, centre O, touches the other six small circles.

These six small circles each touch three small circles and the large circle, the centre of

which is also O. A and B are the centres of three of the small circles.

(a) Find, in terms of r, the height of the triangle OAB, given that its base is

2r cm. [2] (b) Given that the radius of each of the small circles is 5 cm, calculate the area

of triangle OAB. [1] (c) (i) Find the value of angle COD, in radian. Hence, find the area of sector OCD. [3]

(ii) Find the shaded area. [3]

O r r

r r

r r

A

B

C

D


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8. (a) Solid I is consists of a hemisphere solid of radius 30 cm attached to a

conical solid of radius 30 cm and height 90 cm.

(i) Calculate the total surface area of this solid, in m2. [4]

(ii) Find the volume of Solid I, in litres. [3]

(b) Solid II, whose cross section is a trapezium has parallel sides lengths 30 cm and 50 cm respectively. The height of the trapezium is 40cm in length and is 80 cm long, as shown in the diagram below. Calculate the volume, in litres, of Solid II. [2]

40

30

50

80

Solid II

Solid I

90

30


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(c) Solid III is a cylindrical solid of radius 20 cm.

Solid I was melted and shaped into the exact size of Solid II.

Being smaller in size, the leftover of Solid I was shaped into

Solid III. Find the height of this Solid III.

[3]

9. The diagram shows a circle XBWZ with centre O where line segment ABC is tangential to the circle at the point B. The chord BZ intersects XC at Y.

XOZ = 120o, ABX = 58oand ACX = 28o. (a) Show that XCB is similar to BCW. [2]

(b) Find (i) ZBC [1]

(ii) BYX [1]

(iii) OZB and hence, conclude if points BOZC lies on a circle and why? Explain your answer clearly. [2]

30 120

Solid III

20


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10. A class of 34 students was surveyed on how much money they brought to school on a particular day. The cumulative frequency curve below shows the result of the survey.

(a) Use the curve to estimate

(i) the median amount of money, [1]

(ii) the interquartile range, [1]

(iii) the 80th percentile. [1]

(b) In the class, Kok Meng and Jessie brought the most amount of money to school. Find the difference between the amounts of money they brought. [2]

(c) One student is chosen at random. Find the probability that the student brought more tan $20 to school. [1]

(d) Two students are chosen at random. Find the probability that one brought more than $50 to school while the other brought at most $10 to school. [2]

0

5

10

15

20

25

30

35

0 20 40 60 80 100 120

Cum

ulat

ive

freq

uenc

y

Amount of money students brought to school


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Five students from another class were surveyed on how much money they brought to school. The results are listed below.

Name of student Amount of money

Abdullah $ 54.20

Bin Bin $ 13.00

Cecilia $ 2.50

Drew $ 56.10

En Ci $ 43.40

Calculate the mean and the standard deviation. [3]

Another five students were found to have a standard deviation of $15.51. Comparing with the results you obtained in (i), explain what had caused this difference.

11. A particle moves along a straight line PQ so that t seconds, the velocity v m/s in the direction PQ is given by

= 22 2

and corresponding values of t and v are given in the table below.

t 0 0.5 1 2 3 4 5 6 v 0 0.9 1.5 2 1.5 0 2.5 6

(i) Taking 2 cm to represent 1 unit in each axis, draw the graph of = 22 2 for values of t in the range 0 6. [3]

(ii) By drawing a tangent, find the acceleration of the particle when t = 4s. [2]

(iii) Use your graph to find the value of t when 0 6 for which 2 = 22 2. [2]

(iv) By drawing a suitable line on the graph, solve 22 8 = 6. [2]

chs 4016 2010 prelim iii p2

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