trial stpm mathematics t2 johor

8/8/2019 Trial STPM Mathematics T2 JOHOR

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*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

2

T

R

X

YPO

1 Givenf(x) = 3 sin 2x + 2 cos 2x. Expressf(x) in the form rsin(2x + ) where

r> 0 and 00

0) in terms ofx for whichy = 4 when x = 2. [7 marks]

4 Three forces, F1 = (4i+ 5j )N, F2 has magnitude 13Nin the direction parallel to

-5i+ 12j and F3 has magnitude10Nin the direction parallel to 3i4j,act on a particle,

with i andj as unit vectors in the directions to the east and to the north respectively.

Calculate the magnitude and direction of the resultant forces. [8 marks]

5 The diagram below shows three straight lines from an external point P to a circle

with centre O.

The lines from P touch the circle at the points TandR whereas another line from

P intersects the circle at pointsXand Y. Show that

(a) TP = PR. [4 marks]

(b)PTXand PYTare similar. [4 marks]

(c) PT2

= PXPY. [2 marks]


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6 (a) Prove that cosx + cosy = 2 cos2

yx cos

2

yx . [2 marks]

Hence, show that cos+ cos2+ cos3 + cos4 = 4cos2

cos cos

2

5.

Find all the values of between 00

and 1800

such that

cos + cos 2+ cos 3= cos 4 [6 marks]

(b) Prove that sinx sin

3

2x sin

3

4x =

4

1sin 3x.

Hence, deduce the maximum and minimum values of the expression

sinx sin

3

2x sin

3

4x for all values ofx. [6 marks]

7 The lifespan, in hundred hours, of an electrical devise is a random variable Twith

the probability density function as follows

f(t) =

otherwise

0fore50

1

0

50 t

t

Find the probability that the device will function after 600 hours. [3 marks]

8 Each different arrangement of all the letters of the word SELECTION be called a

word.

(a) How many of these words the vowels are together? [2 marks]

(b) What is the probability that in such a arrangement bothEs are not side by side?

[3 marks]

9 From past experience, it is found that the number of typing errors,X, made by Alion a randomly chosen page follows a Poisson distribution with E(X2) = 0.24.

(a) Calculate the mean ofX. [3 marks]

(b) Calculate the probability that no errors made on a randomly chosen page.

[2 marks]

(c) 10 pages are selected randomly calculate the probability that at least 8 pages

contain no error. [3 marks]


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10 The mass of a red apple is normally distributed with mean 116 g and standard

deviation 16 g whereas the mass of a green apple is normally distributed with mean 106 g

and standard deviation 12 g.

Determine the probability that the mass of a red apple is more than that of a green

apple. [4 marks]

One day, Yap bought two apples, but one of the two apples he bought is rotten and

the apple left has an equal chance of being red or green, determine the probability that its

mass will be more than 118 g. [5 marks]

11 When a protozoa is subjected to radiation the protozoa may die, survive as a

single protozoa or divide into two protozoa with probability2

1,3

1and

6

1respectively.

Two protozoa are independently subjected to radiation. The random variableXrepresentsthe total numbers of protozoa still survive after this experiment.

(a) Show that P(X= 2) =18

5. [2 marks]

(b) Find the probability distribution forX. [4 marks]

(c) Calculate E(X) and show that Var(X) =9

10. [5 marks]

12 (a) From the basic definition of standard deviation, prove that the standard

deviation of a set of data consist n observed values with mean x

is given by

22

xn

x

. [3 marks]

(b) The data below shows the age of 24 residents staying in a village.

34 59 62 37 40 48 52 49

48 32 99 77 79 33 54 57

35 64 52 52 61 79 61 90

(i) Display above data in a stemplot. [2 marks]

(ii) Calculate the mean and standard deviation for the age of the residents.

[5 marks]

(iii) Determine the median and semi-interquartile range. [4 marks]

trial stpm mathematics t2 johor

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