_stpm trials 2009 math t paper 1 (chung ling butter worth)

8/9/2019 _STPM Trials 2009 Math T Paper 1 (Chung Ling Butter Worth)

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SEKOLAH MENENGAH JK CHUNG LING BUITERWORTHPEPERlKSAAN PERCUBAAN STPM 2009

MATHEMATICS TIS

PAPER 1

3 hours

Instructions to candidates :Answer all questions.All necessary working sh ould be shown clearly.Non-exact numerical answers ma y be given correct to three significantfigures, or one decimal place in the case ofangles in degrees, unless adifferent level ofaccuracy is specified in the question.

Prepared byChecked by

This question paper consists of 4 printed pages.

Ms Lee Phaik LeanMr Lee Hock Leong


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1.

2.

3.

(a) Solve the equation (3 H1 )(4 x+,) = 5'< - 1, giving your answer correctto four significant figures . [ 4 I

(b) 1 xProve that - - ::; , ::; 1, for all x E R.11 x" - 5x+9a bThe complex numbers z and ware such that z = - and w = - - ,l+ i 1+2i

where a and b are real. Given that z+ w = 1, find a and b.Hence, express z and w in the form x+iy, where x and yare realnumbers.Find also the distance between the points which represent z and w in

[ 3 I

the Argand diagram. I () J(a) . . x' -dx ' -6x+ (d +4)Find the value of d If has 2 as its remainder.x - 2

r 3 J(b) Given [(x) = 6x' + ax 3 + bx ' + x - 3, where a, b are constants.

I f (3x - 1) is a factor of [(x) and (x+ 1) is a factor of ['(x) ,determine the values of a and b. [6 J

4. Given that a and fJ are roots of the quadratic equation x ' +5x +3 = O.

5.

Without explicit caiculation of a and fJ, find(a)(b)(c)

(a)

fJ ' a '--+--2a - 3 2fJ-3 [ 4 J(Fa - fiN [ 2 Ja quadratic equation ax' +bx+c = 0 whose roots are and !!....fJ aGiven a sequence 1,2,2,3 ,3,3,4,4,4,4,5,5,5,5,5, ... i.e, the integerk occurs k times as consecutive terms.

[4 J

(i) Find the 1989 th term. [ 3 J(ii) Find the sum of the first 1989 terms. [ 3 J

(b) Find the first three terms of the expansion of (8+3x)3 in increasingpowers of x.


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Alex plans to send X'mas cards to 38 friends. He chooses 3 types of cardswith different prices from a book shop. The price of a piece of type II cardis the mean of the price of a piece of type I card and a piece of type III card.If Alex bought 20 pieces of type I cards, 12 pieces of type II cards and (,pieces of type III cards, the total cost of the cards is RM 18.10. I f Alexchooses 19 pieces of type I cards and 19 pieces of type II cards only, thetotal cost of the cards is RM 17.10.Let x, y and z be the prices of one piece of type I, type II and type IIIcards respectively, write a system of linear equations base on the aboveinformation given.Rewrite the linear equations in matrix form and solve to find the price ofeach type of cards. I 9 ]A circle touches the straighlline 4y = 3x-8 at the point P(4 , 1) andpasses through the point Q(5, 3). Find the equation of the circle and showthat it touches the y-axis. [ 10 ]Given the function [ wheref(x) = f :lxl, x < 01 2x , x ~ O

lim limFind _ [(x) and .. f(x).x ~ O limHence, what is [(x)?x ~ O

Sketch the graph of y = [(x) .

(a) I f y = nJ J y , find the value of dy when y =1.dx[ 5 J

l 3 1(b) Show that the equation In x +x - 3 == 0 has a root between 1 and e.

Show that the Newton-Raphson iterative formula for this root is. x (4 - l n x )gIven by x"+J = " " .l +x"

With Xu = I, use the formula to obtain the root correct to twodecimal places. I H I


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h h . . 1 2 4 ( 0)10. Acurve as t eparametncequatlOnsx=I-- , y= 1+ - lot .I I -

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Find dy in terms of t. and obtain the equation of the tangent to the curvedxwhen t = 2.(a) Find

(i)(ii)

Jx 2 lnxdxJ +l--dx~

(b) By using the substitution u = 4-3cosO , evaluate]J sinO dOo (4 - 3cosO) 2 .

! 71

[ 3][ 3)

[ 4)

12. Water in an upright cone is leaking through its vertex. After one second,the water leve l has lowered by 0.2 em. If the side of the cone makes anangle of 30 with the vertical, find the rate of change of its volume when

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2.

3.

4.

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the water level is 4.2 cm high. [ 6 )

Mathematical Formulae" 12> = - n(n +l)

r I 2n , 12>-= - n(n+l)(2n+l)

r = I 6n 12> ' = - n 2 (n+lf

r = I 4( ) n - 1 n(n - l) 2 n(n-l) ... (n-- r+l) T Ixl < 1.l +x - +n x+ - - - x + ...+ x + ..,2! r!

f(x.)Newton-Raphson iteration for f(x) = 0: x".! = Xn - f '(xn)

_stpm trials 2009 math t paper 1 (chung ling butter worth)

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