954/2 2009

PEPERIKSMN PERCUBAAN S11>M

MATHEMAnCS T

PAPER 2

Three hours

. SM ST. MIOIAEt IPOH

Instructions to candidates:

Answer all questions.

All necessary working should be shown clearty.

Non-exact numerical i!nswers may be glYen correct to three significant figures, or one decimal place In the case of angles In degrees, unless a different level of accuracy Is specified In the question .

..................................................................................................................... ...............................................................

1. Find all the values of x. where - 1! < x < 1C , which satisfy the equat ion

cos 3X- C05 x = sin x (4)

2. Four coplanar forces (I + 3j)N, (21 - 3j)N, (21 + 3j)N and ( -1 - j)N actat a point. . .

3.

Determine the magnitude of the resultant force. Show that the resultant force is perpendicular to the vector ( -I + 2j)N. (6)

AP~:::::::"Q~_--,.T

In the given diagram, POT is a straight"line parallel to SR. RT is a tangent to the circle with centre O. If L POS = 2LPTR. prove that

(I)

(II)

Rp .. RT, .

QR=QT. (3) (2)

4. The diagram shows points A, Band C on a horizontal plane, with AB = Sa. AC = 3a and angle CAB = 120°. The bisector of angle CAB meets the line Be at the point P. 'I} Prove that 8C = 7a.

21 Ii) Prove that CP .. -ga.

c~ A sa 8

iii) A vertkal post CV is placed at C. The angle of elevation of V from A is 3ft. Find the angle of elevation of V from P. (8)

5. Two particles A and B move freely on a plane with velocities (-31 + 29j ) m/s and

( 31 + 21) )m/s respectively. ~

(i) Find the velodty B relative to A and also the vector AB at time t, given that when ~

t=O, AB=(-56i+8j)m. (4) (Ii) Show that the distance between A and B is the shortest when t = 4s, and find this

distance. (4)

6 . . (a) Show that the differential equation c(y + y l = - 2y may be reduced by means' of dx . x

v dv v2

substitution y = -, to - =--x dx x 2 '

Hence, or otherwise, find yin tenns of x, given that y = 1 when x = 1. (8)

dv ( )' (b) Find the general solution of the differential equation . dx = 1-Y expressIng y In

terms of x and sketch the solution curve which y = 0 when x = O.

7. A game is played by throwing a pair of unbiased dice, one red and one green. The score

Is obtained as follows It the red die does not show a six, the score Is the number showing on the green die.

I.f the red die shows a six, the score is twice the number showing on the green die.

A player throws the pair of dice once.

(6)

(a) Find the probability of the score being 4 or less. (4) (b) Find the conditional prohabllity that the red die does not show a six. given that the

score Is more than 4. (4)

8; The random variable X has the following probability distribution

x 1 2 P(X = x) • 2.

. I where 0 < a < -. Show that E(X) = 3~a and find Var{X) in terms of a.

3

3

The random variable S Is the sum of n Independent values of X. Write down E(S)

(4)

and V.rjS) In teons 01 •• nd n. (2)

The random variable T Is defined by T = P + qS. The values of p and q are such that

I E(T) = a for all a in the lnteryal 0 < a < -. Find the values p and q. (4)

3

9. The cumulative' distribution function of the continuous random variable X Is given by

{

0 x<2 F{x}= ax'+bx 2"x ,, 4

I x >4

(I) Find the values of a and b. (2) (Ii) Find the values of c such that P( X > c) = 0.88 (3) (III) Find the probability density function ofX. (2) (Iv) Determine the mean and variance of X. (S)

10. The probability that a lamp Is defective is 0.01 and 100 lamps can be packed In a box.

(a) Find the probability that in a boxL

(I) there Is no delectlvelamp. (2)

(II) there are 2 delectlvelamps. (1)

(iii) there are more than 3 defective lamps. (2)

(b) A buyer will buy 50 boxes of lamps If, when he choose 2 boxes o,f lamps, there are not more

than 2 defective lamps. Find the probability that he will buy the SO boxes of lamps. (3)

11. A and 8 are two Independent events with

(5)

12. The following stem plot shows the marks obtained by a group of Form 6 students.

2 0 J 3 1 4 6 4 1 2 3 5 7 5 I ~ 4 4 9

6 0 4 S 7 7 2 3 8

8 :1 6

9 4 Key: 2/4 = 24 martts

tl) Find the percentage of students having less than 40 marks and percentage of students

with at least 80 marks. (2) (!I) Find the mean and standard deviation for the above data. (4)

till) Find the median and seml-Interquartile range. (3)

(Iv) Draw a box plot to represent this data. (3)

Prepared by: ... .k..: ............... . &i4~ Checked by: .•.•..... ...... ... ....... , ••• " .. ................... .

. Formulae:

. . A . B 2' (A+B)' .r A-B) sm +sm = sm -2- 00'\--"2

. A . B 2 {A+B). (A-B) sm -sm = co -2- ~m ~

{A+B)..J A-B) eosA + eos B = 2eo ~ ~"\.-2-

. 2 . (A+B) . (A-B) cosA:-cosB=- sm ~ SID -2-