STPM954/1I2009

SULIT 95411 Mathematics SIT Paper 1 ~September 2009 Three hours

SEKOLAH MENENGAH (J.K.) lIT SIN BUKIT MERTAJAM

PULAU PINANG

TRIAL EXAMINATION 2009 Upper Six

MATHEMATICS SIT PAPER 1 3 hours

0745 - 1045 am 10 - 09 - 2009 (Thursday)

Instruction to candidates. Answer all questions. The marks for questions or parts of questions are given in brackets [ ]. All necessary working must be shown clearly. For questions which do not require answers in any specifiedform, the answer may contain Jr, e, surds, or fractions.

This question paper consists of 4 printed pages.

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STPM954/112009

Answer all questions.

1. If A, B and C are arbitrary sets,:' show that [CA - B) n (A - C)] u B u C = Au B u C. [4]

2. Using the trapezium rule, with five ordinates, evaluate C .J1 + e dx, giving your answer correct to three decimal places.

3. The functions g and h are defined as follows: g:x-+x3'-1 , x>O; h:x-+5e-X, x>O

(a) Show that the composite function gh exists and define it in a similar form.

(b) Detennine the range of gh.

2

4. Determine the positive value of p if z = p - 3i and z is a real number. I-J)i

[4]

[5]

Hence, find the real number. [6]

5. Using the factor theorem, show that (3x + 1) is a factor of f(x), where f(x) =: 6x3 + 23x2 - 5x - 4. Hence, factorise f(x) completely. [4] Determine the range of x such that f(x) > O. [2]

6. Find

(a) J 2cosx dx, sin x + cosx

[Hints : You may use 2 cos x = (sin x + cos x) + (cos x - sin x)]

(b) Jln(2x) dx

7. Find the constants A, B, C, D' and E such that

4X4 +3x 2 -x-3 E A B Cx+D =: + -+-+-.,,---

x2(2x2 +1) x x2 2X2 +1 .

-2-

[3]

[3]

[8]

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STPM954/112009

I

8. (a) Expand (1 + 2X)3 up to and including the term in x3. Use your series to obtain a I

value for (1.04)3 giving five decimal places in your answer. [6]

(b) The first term of a geometric progression is 5, its last term is 640 and its sum is 1275. Find the common ratio. [4]

9. PQRS is a rhombus and tangent of the angle between PR and PQ is 3. Given that P(-I,O) and R(3, 8), find the coordinates ofthe points Q and S. [11]

10. (a) Find the equation of the asymptotes of the curve 2x-3

[lJ v=-----J (x-1)(x+3)

(b) Find the points where the curve intersects the x-axis and y-axis, and find the stationary points of the curve. [7]

(c) Sketch the curve.

(d) Determine the values of k for which the equation 2x - 3 = k(x - l)(x + 3)

has only one real root.

11. Show that the equation xex+1 = 4 has only one root and that this root lies in the

[3]

[1]

interval 0 < x < l. [6] Taking x = 0.6 as the initial value, apply the Newton-Raphson process to find this

root correct to three decimal places. [6] .

12. (a) Matrix M is given as

M~ l~~ ~1 ~ J Find the values of p, q and r such that M3 +pM2 + qM + rI = O.

I . Deduce ~ . [10]

-3-

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STPM954/1/2009

(b) Matrices A and B are given as

A ~r ~l ~ -43

J' l-i 1 - 1

B ~ ( -~ 6

= ~ ~:J Find AB. Hence, solve the equations

6x+y - 14z=42 x-2y-l1z=-149 7x - y - 12z = 23

~~ The End ~

Prepared by,

" '~"' .. (LEE GIM LEONG) 25 August 2009

-4-

AJiisted by,

lu~ !No'jOC)Slm) I

[6]

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S.M.J.K. JIT SINBukit Mertajam.

Soalafl Markah

JumlahNombor Pusat / Nama

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l l .z+l

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)" Vqp Le*'' -Y Wat srLll ane rlol

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LPtt fc*) = Le t* ' -P

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fco). f [ , ) = -V(3.3&q): -_ lg, ss6

CMcc f(ol . f ( t )

) . Tt\e na+ liu M the v*-etval 0

l t . b) A-- 5 z + p= l -z - l lI , " /

r' - ) r

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s z q. l l I -L- l

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f tL = -13 B- '

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a J -1 Iy )" r'['/

/X r /-l 7

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fz>l?L\ t?/

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u

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Dq = (() ,e) , / :> By 9 D,t - /So, qh cx) exilt [sltor"lvr ] ./ I

qh(x) : ( 5.-")- - |l lse-rr I(

gh'x -+ - 5)(

t+5 e >(> O

b) gh ( t ) = t lS , - t ' - ' l , x >0

^1"t*) gh(tl = L-r, .

e7* |

h[x)= o ) t=0, 4[ ' (x)= fJS-- |l+f _r

7\C I " tw11 V

r . to9 x = { .60JY'>x,91^'( : r )+ - l

ra\te *f ,qh{x) . y €t?-, ( - l , l>+) /

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facto r' Il,,-zo((M, i( (Zxtr) ,3 a fa'tor, tla-e't ft-ll = o

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= ( :x+t)()y '+ +x-*)

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f[:.) 'tr(3r+ lX).v -r) fx* V) > o

{_" ' " eW, -V"t i orx>\

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Angka Giliran

Soalah Markah

Jumlah

Nombor Pusat / Nama@

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t ,0 l3t6

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5)= tr+5'

dr^ ' t ' - 6\ tA

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t- -- ttB 635 r = l lTo

, ' " - - tzt , -o) 65r*rwon rr^li o -z

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t0, a) Y'4.4 /

1x -l Yv+r) \

Veftft^l 6J yn^ffttet Y=|. g-- -3

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r r . . r3

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S.M.J.K. JIT SINBukit Mertajam.

Soalah Markah

Jumlah

Perkara Angka Giliran

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)x-3 = V (rr- t ) (xt i ) , l

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f ( 0.6)

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f'( e,+j.q+)

a, l t la)

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S.M.J.K. JIT SINBukit Mertajam.

Soalait Markah

Jumlah

Nombor Pusat / Nama

Perkara Angka Giliran

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nL re1I ' \5o 6|" l - '

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r= 3- B(r)rz( '6)

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A, tSM-/ = M'r>M,6I

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L l - t t - r +t ) I I\ | + r /

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Qz*+ gt -L3 -L- L , . a++v>(2->"+ t) * 7t?+ 1

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ts

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f f r

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STPM Trials 2009 Math T Paper 1 (Jit Sin)STPM Trials 2009 Math T Paper 1 (Jit Sin)

STPM Trials 2009 Math T Paper 1 Student's Answer (Jit Sin)