Stpm Trial 2009 Matht Q&a (melaka)

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1. Show Ihal IA· r> (1 v(8'r> C) v (A r> B r> C) " C.

2. find Ihe volume oflhe solid fonned

wh~n Ihe ellipse wilh equation ~: +~

[ISl '0 201-']andN __

[- I

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m

I i5

/5 marks1

mll!ed eomplc~dy about !he y-axis.

J.Given!halM ..

(3 marks1

2

-4

-, -'} " "

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Find ~he m31ri~ N - 6.11 and show thai M(N - 6M) " kl. where k is an in!cgennd I is the idemity ma1ri~ . Sla!e [he value of~. Hence. find !he inverse of M. 15marh1 4. Show thai the equation of the locus of PI et H· e.

.r _~'()a$l varies is given by

x ' y' - - - ., I and sketch its graph.

[6 marks)

• •

5. If A is Ihe pOint on Ihe parabola x '

~

t

y which is nearest 10 the line y " 3x - I,

find Ihc coordinates of p
[6 marks [

,

,

6. Solve lhe following inequalilies.

I') - - - - >0 x - 5 ..,+2 (b)

x' - x - 6 <0 x ' +2.>0+2

[7 marks1

7. Prove tbal (log .h) /log . c) (log
[6 marks]

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8. (a) ' The polynomial P(x) gives a remainder of 5 when divided by (x - 2) and a remainder of9 when divided by (x - 4). Find the remainder when P(x) is divided by (x - 2) (x - 4). [5 marks] (b)

Show that for all real values ofy, the expression

3y2 -2y-1 always lies y2 + y+2

4

[5 marks]

between - - and 4. 7

9. Given that e'y

= sin x .

d'v dy (a) Show that - ' , + 2 - + 2y = O. . dx· dx (b) When y

v

[4 marks]

= ~ , show that the equation has a root that Iies between 0 and I . 10

By using Newton-Raphson method, find the root of the equation correct to four decimal places. f7 marks]

10. (a) Find the domain for the function f : x ~ ~ .

[2 marks]

,

(b) Function X is defined as follows. 1 1 g(x)= mx· +3, -"3:::;x:::;"3

1N=I.

otherwise

[f g is continuous for all values of x, find the value of the constant m.

(c) Function h is defined as hex) =

[4 marks 1

~ . 2 9x -I

State the domain of h. State the asymptotes of h. Hence sketch the graph of h.

[I mark] [2 marks] [3 marks]

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II. The equation of a curve is given by y = xe- h • (a) Find the stationary point of the curve and state its nature.

[4 marks]

(b) Find the point of inflexion of the curve.

[3 marks]

(c) Sketch the graph of the curve.

[2 marks]

(d) Show that the area of the region bounded by the curve, the x-axis and the . I. e-2 [5 marks] I me x =-IS - - . 2 4e

12. Given that f(x) =

5x (I - 2x)(2 + x)

(a) Express f(x) in partial fractions. Hence, show that 5 15 2 65 3 !(x)=-x+-x +-x + ... . 2 4 8 and state the set values of x for which the expansion f(x) is valid. [12 marks] I (b) When x = - , calculate the error correct to significant figure in the use of the 10 expansion off(x) up to the term in x 3• [3 marks]

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CONFIDENTIAL * I. Use the substitution y = u - 2x, find the general solution of the differential equation dy 8x + 4y + 1 -=:----=-dx 4x + 2y + 1

[6]

2. Express3sinO+4cosO in the form rsin(O+a) wherer>Oand O
[4]

a) State the maximum and minimum values of 3 sin 0 + 4 cosO

[2]

b) Hence solve the equation 3 sin 0 + 4 cosO = 2 , giving values of 0 in the range 0::; 0::; 27r in term of 7r .

[2]

3. PQR is a triangle that lies on a horizontal plane, with Q due west of R. The bearing of P from Q is a , and that of P from R is f3, where a and f3 are both acute. The top of the tower PM is at a height m above the ground, and the angle of elevation of M from R is O. The top of the tower QN is at a height n above the ground, and the angle of elevation of N from R is ¢. Show that ntan8cosa tancj rel="nofollow">sin(a-13)

[5]

m==-----

C

4.

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 (a)

the triangle CDB and ADB are congruent.

[3]

(b) DOis parallel toAB.

[4]

Given that BC = 12 cm and L DCB == 40°, calculate the area of the triangle ABC.

[4]

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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 I

referred to O . The velocities of the two ships are constant. Ship A sails at 34 km h- , in the 1 direction of the vector 8j + 15j and ship B sails at 30 km h· in the direction of the vector

31 + 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. [2] (a) Write a differential equation that is satisfied by y (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,

[2]

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

[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) x ::; 10 x::; x ::; x ::; x ::; x ::; x ::;

20 30 40 50 60 70

No of students 8 27 69 138 172 195 200

(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 abo~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]

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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. [3] (i) Find the probability that they are both toffees. (ii) Given that they are both toffees, find the probability that they [3] both came from box A.

10.

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

0.15

2 0.25

3 0.3

4

a

5 0.1

(a)

Find the value of a

[I]

(b)

Find P(1:O;; x:O;; 3)

[2]

(c)

Sketch a graph to represent the probability distribution ofX.

[2]

(d)

Sketch a graph of the cumulative distribution functIon of X.

[2]

II . The lifespan, T (hours), of an electric item has the probability density function given by

f(l)

=

. J[ t k SIO - - 0:0;; t ::; 1800 3600

o

otherwise

a) Determine the value ofk. b) Determine the probability that an electrical component which already lasted for at least 1200 hours will have a lifespan of more than 1500 hours.

[2]

[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 standard deviation Mean 5 25 Route A 2 30 Route B (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]

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