stpm trial 2009 matht q&a (melaka)

8/14/2019 STPM Trial 2009 MathT Q&A (Melaka)

1/13

1. Show Ihal IA r> (1 v(8 'r> C) v (A r> B r> C) " C. (3 marks1

2. find Ihe volume oflhe solid fonned w h ~ n Ihe ellipse wilh equation m I i5mll!ed e o m p l c ~ d y about !he y-axis. /5 marks1

[' 0 -'] [IS.Given!halM .. 021 andN __- I I 2 -4 -,-'}" .. "Find ~ h e m 3 1 r i ~ N - 6.11 and show thai M(N - 6M) " kl. where k is an in!cgennd I isthe idemity m a 1 r i ~Sla!e [he value o f ~ . 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 byx ' y '- - - ., I and sketch its graph. [6 marks)

5. If A is Ihe pOint on Ihe parabola x ' t which is ne arest 10 the line y " 3x - I,find Ihc coordinates of p


2/13

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

3y2 -2y -1(b) Show that for all real values ofy, the expression always liesy2 + y+24between - - and 4.7

9. Given that e'y =sin x .d'v dy(a) Show that -' + 2 - + 2y = O .dx dx

[5 marks]

[4 marks]

(b)When y = , show that the equation has a root that I es between 0 and I .v 10By using Newton-Raphson method, find the root of the equation correct to fourdecimal places.

f7 marks]

10. (a) Find the domain for the function f :x . [2 marks]

(b) Function X is defined as follows.

1, 1 1g(x)= mx +3, -"3:::;x:::;"3

N=I. otherwise[f g is continuous for all values of x, find the value of the constant m. [4 marks1

(c) Function h is defined as hex) = .9x 2 - IState 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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3/13

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. I . e-2me x =-IS -- .2 4e [5 marks]

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 8and 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 the10expansion off(x) up to the term in x3 [3 marks]

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4/13

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5/13

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6/13

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7/13

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8/13

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9/13

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10/13

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11/13

CONFIDENTIAL*I. Use the substitution y = u - 2x, find the general solution of the differential equation

dy 8x + 4y + 1-=:----=--dx 4x + 2y + 1

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


12/13

5. At time t = 0 a ship A is at the point 0 and a ship B is at the point with position vector 10jreferred to O. The velocities of the two ships are constant. Ship A sails at 34 km h- I , in the

direction of the vector 8j + 15j and ship B sails at 30 km h 1 in the direction of the vector31 + 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 eachother for 3 hours [4]

6. One model for the spread of a rumour is that the rate of spread is proportional to the productof the fraction y of the population who have heard the rumour and the fraction who have notheard 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 populationhave heard the rumour. [10]

7. It is known from experience that the probability that an individual will suffer a side effectfrom 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 2individual suffer a side effect. [3]8. The table below shows the time taken by a group of students to solve a mathematicsquestion.

Time taken (seconds) No of studentsx ::; 10 8x::; 20 27x ::; 30 69x ::; 40 138x ::; 50 172x ::; 60 195x ::; 70 200

(a) Calculate the mean and the standard deviation of the time taken by the groupof students. . [5](b) Plot a histogram for the above data. State the shape of the distribution of thea 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 ofy from the histogram. . [3]

954/2 CONFIDENTIAL*

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13/13

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 boxA. [3]

10. The discrete random variable X has probability density function as shown:2 3 4 50.15 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 functIon ofX.

[I][2][2][2]

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

f( l) =k . J[ tSIO -- 0:0;; t ::; 18003600

o otherwisea) Determine the value ofk. [2]b) Determine the probability that an electrical component which already lasted forat 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 eachroute may be assumed to be normally distributed, with parameters( in minutes) as shown inthe table, and the time taken for any journey on either route may be assumed to be independentof the time for any other journey

Route ARoute BMean25

30standard deviation5

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 usingroute A and the other using route B. Find the probability that the student using route Bwiil arrive first. Find also the probability that one of the students will arrive more than

5 minutes after the other [7]

954/2 CONFIDENTIAL

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stpm trial 2009 matht q&a (melaka)

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