stpm trials 2009 math t paper 2 (malacca)

8/11/2019 STPM Trials 2009 Math T Paper 2 (Malacca)

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PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA

SEKOLAH MENENGAH MALAYSIA

CAWANGAN MELAKA.

PENlLAlAN PENGESANAN PEPERIKSAAN

SIJIL TINGGI PELAJARAN MALAYSIA 2009

Instruction to candidates:

Answer all questions, Answers may be written in either English or Malay,

ll necessary working should be shown clearly.

Non*exact numerical answers may e given correct to three significantfigures, or one decimal

place in the case

o

angles in degrees, unless a different/evel ofaccuracy is specified in the

question.

This question paper consists

of

4 printed pages,


2/4

CONFIDENTIAL

I. Use

the substitution y = u - 2x,

find

the general solution

of

the differential equation

dy 8x

+

4y

+

1

= =

dx x

+

2y

+

1

2. Express3sinO 4cosO in

the

form rsin(O+a) wherer>Oand Osin a-13)

C

A

In

the above diagram,

BC is

a diameter

of

a circle with centre 0 and

D is

a point

on

the circumference. If

CD is

produced

to

A so that CD

=

DA prove that

[5]

a) the triangle CDB and ADB are congruent.

[3]

b) Ois parallel toAB [4]

Given that BC =

12 cm

and L DCB 40, calculate the area of the triangle ABC

[4]

954/2

CONFIDENTIAL


3/4

5. At time t

=

0 a ship A

is

at the point 0 and a ship B

is

at the point with position vector 10j

referred to O. The velocities of the two ships are constant. Ship A sails at 34 m h-

I

,

in

the

direction

of

the vector 8j + 15j and ship B sails at 30

m

h

1

in

the direction

of

the vector

3 + 4j

(a) Determine the velocity

of

B relative to A [3]

(b) Find the position vector

of

B relative to A at time t hours [3]

(c) Given that visibility

is

10 km

show that the ships are within sight

of

each

other for 3 hours [4]

6. One model for the spread of a rumour

is

that the rate of spread

is

proportional to the product

of

the fraction y of the population who have heard the rumour and the fraction who have not

heard the rumour.

(a) Write a differential equation that

is

satisfied by y [2]

(b) Given that a small town has 1000 inhabitants. At 8 am, 80 people have heard the

rumour. By noon, half the town has heard it. At what time will 90 of the population

have heard the rumour. [10]

7.

It is known from experience that the probability that an individual will suffer a side effect

from a given drug is 0.003 . By using suitable approximation, find the probability that,

out of2000 individual taking the drug,

(a) exactly 2 will suffer a side effect,

(b) more than 3 will suffer a side effect.

[2]

[3]

Find the probability that

in

5 groups of2000 individuals, 3 groups will have exactly 2

individual suffer a side effect. [3]

8. The table below shows the time taken by a group of students to solve a mathematics

question.

Time taken (seconds)

No of students

; 10

8

x ;

2

27

::; 3

69

;

4

138

;

5

172

;

6

195

::;

7

200

(a) Calculate the mean and the standard deviation

of

the time taken by the group

of

students. .

[5]

(b) Plot a histogram for the above data. State the shape

of

the distribution

of

the

b o ~ e data. [4]

(c) Estimate the mode of the time taken from the histogram. [1]

(d)

If O of

the students take more

thany

seconds to solve the mathematics question,

estimate the value

of

y from the histogram. . [3]

954 2

CONFIDENTIAL


4/4

9..

Boxes of sweets contain toffees and chocolates. Box A contains 6 toffees and 4 chocolates,

box B contains 5 toffees and 3 chocolates, and box C contains 3 toffees and 7 chocolates.

One of the boxes is chosen at random and two sweets are taken out, one after the other,

and eaten.

i) Find the probability that they are both toffees. [3]

ii)

Given that they are both toffees, find the probability that they

both came from box

A

[3]

10. The discrete random variable X has probability density function as shown:

2

3 4 5

0.15

0.25

0.3 a

0.1

a) Find the value of a

b) Find P 1:O;; x:O;;

3

c) Sketch a graph to represent the probability distribution ofX.

d) Sketch a graph of the cumulative distribution funct on of

X.

[ ]

[2]

[2]

[2]

The lifespan, T hours),

of

an electric item has the probability density function given

by

f l) =

k

J[

t

SIO 0:0;; t ;

18

3600

o

otherwise

a)

Determine the value ofk. [2]

b) Determine the probability that

an

electrical component which already lasted

for

at least 1200 hours will have a lifespan ofmore than 1500 hours. [4]

12.

A student has a choice of two routes for travelling to school each day. Travel times for each

route may

be

assumed to

be

normally distributed, with parameters

in

minutes) as shown

in

the table, and the time taken for any journey on either route may be assumed to be independent

of

the time for any other journey

Route A

Route B

Mean

25

30

standard deviation

5

2

a) Find the probability that ajourney on route A will take longer than 32 minutes [3]

b) Two students set out from the same place at the same time to travel to school, one using

route A and the other using route

B.

Find the probability that the student using route B

wiil arrive first. Find also the probability that one of the students will arrive more than

5 minutes after the other

[7]

954 2

CONFIDENTIAL

stpm trials 2009 math t paper 2 (malacca)

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