Maths T Trial STPM Kuching Sarawak 2008 [Edu.joshuatly.com] [C9D36FD8]

7/29/2019 Maths T Trial STPM Kuching Sarawak 2008 [Edu.joshuatly.com] [C9D36FD8]

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Sarawa Kuc ing STPM 2008http://edu.joshuatly.com/http://www.joshuatly.com/


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By using the laws of algebra of sets, prove that (A - 8)' ., (nw C) = 1.

2. Obtain the first four terms of the expansion of (Z + 3x)* in ascending powers of x. Statethe range of values of x for which the expansion is valid t41

3.ltz=x+ iv andz2 = a+ibwherex, y, aand b are real numbers, provethat2x'=^{q\b'*t4. Express (k +1)(k+2) in partial fractions.

IIHence, evaluate the value of Ir,(k +1)(k +2)A variable point P lies on the curve yt = xt and is joined to a fixed point, A with coordinate(2, 0). Frove that the equation of locus of the mid-point of AP is y 2 = 2(x -l)3 t6I

6. lntegrate le' sinxdx

Find the solution set of the inequality lr - Zr1 . I I8IxSketch the graph of y = r *:By sketching a suitabl" gr.if, on the same coordinate axes, show that the equation

t5le'-4 =0 has exactly 1 real root, o.Using the Newton-Raphson method with ao=1.5 as the first approximation, find thevalue of o, correct to 2 decimal places. I5l

(r o o)(a) Given that matrix M is given by I 1 -2 0I. Find the real numbers sand f such[-r -3 t)

that M2 = sM + fl, where I is the 3 x 3 identity matrix. Deduce that Ma = -5M + 61. t41(b) There are three commodities X, Y and Z which are bought and sold by three dealersP, Q and R. Dealer P purchases 2 units of X and 5 units of Z and sells 3 units of Y.Dealer Q purchases 5 units of X, 2 units of Y and sells 7 units of Z and anotherdealer R purchases 3 units of Y, 1 unit of Z and sells 4 units of X. ln the process Pearns RM 11 and R earns RM 5 but Q loses RM 12. lf the earning while purchasingequals the losing while selling for each respective units find the earning of each of

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the commodities X, Y and Z. t6l

Sarawak Kuching STPM 2008http://edu.joshuatly.com/http://www.joshuatly.com/


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-10. Functions g, h and q are defined byg:x-+lnx, xeR,x>0h:x-+1+X, xeRgrX--+*-4x, xeR*The function f is defined bYf:x--+gh(x), xeR,x>-1

(a) Sketch the graph of y = f(x).(b) Write down expression for g -11x) and n - t(x)(c) Write down an expression for g - t h - t(").

/,(d) State the domain of q such that q is a one-to-one function.

11. Sketch the curves of y = x'15x - 6) and y = x(4x - 5) on the same diagram.Show that the coordinates of their points of intersection are (0, 0) and (1, - 1).Find(a) the area of the region enclosed between the two curyes.(b) the volume generated, when the region is rotated 360o about the x-axis.

12. Given that the polynomial p(x) = 6x' +17x3 + axz + bx -6 where a and b are constantshas a factor (x-1) and when divided by (x+1) the remainder is 4, Find the value of aand b. Hence factorise P(x) completely.

1Using substitution y = ;, solve the equation6ya+19y3 -2y'-17y-6=0

END OF QUESTION PAPER

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Sarawa Kuc ing STPM 2008http://edu.joshuatly.com/http://www.joshuatly.com/


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1. Express 3 cos 0 + r..3 sin g0o 0 and3cosg+xEsing=3 for 0os0 j2n.

2. A, B and C are three points on the horizontal ground. B is to the north of Awhile the bearing of C from B is 060". The angles of elevation of the top of atower right above B from A and from C both eqr.lal to the angle B.P is a point on AC such that AP : PC = 1 :2.Show that the angle of elevation 0 of the top of the tower from P satisfiestan o = 16 tan B.

3. By using the substitution y = vX , show that the ditferential equation

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dy - x'+2y2dx 3xy can be reduced to * il =ax 1-v2 I3l3vHence, find the general solution of the differential equation.

4.showthat a.2=lt4'and y!= 2.2.Hence, prove that the angle at the circumference of a semicircle is a right angle.5. ln the figure below, LT is the tangent at L to the circle LMN, O is the centre of the

circle LMTX and MX is perpendicular to MN.Prove that triangles LMN and TMX are similar.Y is a point on circle LMN, such that YM produced meet the tangent LT at T.lf LM = MT and 4MLT = 370, calculate the angle YLM and findlhe ratio of areaof triangles LMN and TMX.

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6. Water leaks from a full cylindrical tank of radius 2.5 m and height 3m, through around hole of radius 25 mm in the bottom of the tank. Given that water flows fromthe hole with velocity approximately v = 2.5Ji ms-1, h being the height of the waterin the tank.(a) Write down the volume of water running out of the tank through the small holeper second. t1l(b) Write down the decrease of volume per second in the cylindrical tank. t1l(c) Show that the differential equation of the rate of decrease of height h, of the

cylindrical tank is O! = - 2.5 x 104 ^n. Hence solve this differential equation. t6ldt(d) What is the height of the water in the tank after t hour? t2J(e) Find the time (in hour) required for the tank to be empty. I3I

7. Every day, a fisherman can choose to fish at sea, by the river or at a lake.The probabilities that he fishes at sea, by the river and at the lake are + +

land . respectively.4lf he goes out to sea to fish, his chances of catching some fish is 80% whilehis chances of catching some fish at the river and the lake are 40o/o and 600/orespectively.(a) Find the probability ihat the fisherman catches some fish on a randomlychosen day.(b) lf one day, the fisherman does not catch any fish, determine the place hemost probably has chosen to fish.

8. The discrete random variable X has a probability distribution as shown in thetable below, where p is a constant.X 0 1 2 3

P(X = x) t'\YI-p2'. I-p4' I-p0'

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(a) Showthat p= l.9(b) Find E(X) and Var(X), giving your answers as exact fractions.



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9. The diameters of circular rods are normally distributed with mean 1 cm andstandard deviation 0.003 cm. The inside diameters of washers are normallydistributed with mean 1.005 cm and standard deviation 0.004 cm.The rods and washers are randomly paired.Find the percentage that they do not fit.10. Given that 3% of a large consignment of mangoes produce by a plantation areover-ripe.(a) A total of ten mangoes are selected at random. Calculate the probability thatnot more than 2 are over-ripe.(b)An independent sample of 200 mangoes are selected at random and packedinto boxes. By using normal approximation, determine the probabilitythat at least five mangoes are over-ripe11. A continuous random variable X represents the waiting time, in minutes,for a commuter train. The cumulative diskibution function of X is given by

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Maths T Trial STPM Kuching Sarawak 2008 [Edu.joshuatly.com] [C9D36FD8]