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MAJLIS PEPERIKSAAN MALAYSIA(ueravsnN EXAMTNATToNS couNCrL)

SIJIL TINGGI PERSEKOLAHAN MALAYSIA(varavsn HrcrrER scHooI. cenrm'rcerr)

Instructions to candidates:

DO NOT OPEN THIS QUESTTON pApER UNTILYOUARB TOLD TO DO SO.

Answer all questions. Answers may be written in either English or Bahasa Malaysia.

All necessary working should be shown clearly.

Non-exact numerical answers may be given conect to three significant fgures, or onedecimal place in the case of angles in degrees, unless a dffirent level of accuracy is specified inthe question.

A list of mathematicalformulae is provided on pages 6, 7 and 8 of this question paper

Arahan kepada calon:

JANGAN BUKA KERTAS SOALAN INI SEHINGGA ANDA DIBENARKAI\BERBUAT DEMIKIAN.

Jawab semua soalan. Jawapan boleh ditulis dalam bahasa Inggeris atau Bahasa Malaysia.

Semua kerja yang perlu hendaklah ditunjukkan dengan jelas.

Jawapan berangka tak tepat boleh diberikan betul hingga tiga angka bererti, atau satutempat perpuluhan dalam kes sudut dalam darjah, kecuali aras kejituan yang lain ditentukondalam soalan.

Senarai rumus matematik dibekalkan pada halaman 6, 7, dan 8 kertas soalan ini.

II

EI

I

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This question paper consists of 8 printed(Kertas soalan ini terdiri daripada I halaman

pages.

bercetak.)

@ Majlis Peperiksaan Malaysia 2011

srPM 950lr, g54tl*This question paper is CONFIDENTIAL until the examination is over.*Kertas soalan ini SULIT sehingga peperiksaan kertas ini tarnat.

[Turn oYer (Lihat sebelah)CONFIDENTIAL*

SULIT*

COI{FIDEIiITIAL*

-z d2y - ,, dYrY- dxz d"r

b are real numbers.

14 marksl

14 markl

16 marlcsl

16 marlcsl

implicit differentiation,

16 marlcsl

13 marlcsl

13 marksl

12 marlcsl

19 marlcsl

+)r: o

EA,(

(o)

(b)a and b,

9501r,9541r*This question paper is CONFIDENTIAL until the examination is over.

{ Solve the equation ln x +ln (x + 2): 1.

ir'tr- t _ ,'Show that # r' * r n+ 1'

3//+Using the substirution u-lnx,evaluat. I W*/

* ("-

lines /, : ! = mx + a and lr: y = - *. *b, where m * 0 and b > a >0, intersect atR.

Find the coordinates of R in terms of a, b and m. l2 martrsf

The line /, cuts they-axis at P and the line /, cuts the x-axis at Q. If m: 1, find, in terms ofthe perpendicular distance from R to line fg, anddetermine the areaof triangle @-*.-

15 marksl

\E - 3i, where z* denotes the conjug ate of z.The complex number z LS such that z - 2z* :(") Express z in the form a * bi, where a and

(b) Find the modulus and argument of z.

(c) Represent z and its conjugate in an Argand diagram.

Differentiate e" with respect to x.

Hence, determine integers a, b and c for which

l,'*'"*dx - f""

4Findthesetofva1uesofxsatisffingtheinequa1ity2x_1<

5 Given that .y is differentiable and yG: sin x, where x + 0. Usingshow that

COI{FIDEI\TIAL*

COI\FIDENTIAL*

9

,/

Functions f and g are defined by

r:x* ffiforx +*tg: x e axz + bx a ct where a, b and c ate constants.

(o) Find f o f, and hence, determine the inverse function of f .

(b) Findthe values of a,b andcifg o f : #

(,)Giventhatp(x):x,_2,eXpreSsh(x):#intermsoffandp.

l0 )/and

B are two matrices such that

,/ l-4 -3 o\ l-z 6 o\

":l-, -z of-.o,":1, o 4l\, 2 -3l \o 4 ,l

(a) Find the determinant and adjoint of A. Hence, determine A-r. 16 marksl

(b) Using A-1 obtained in(a),find B. 14 marlu)

y',fn"polynomialp(x): ax3+bxz-4x* 3,where aandb areconstants,hasafactor(x+ 1)./When p(x) is divided by (x - 2), it leaves a remainder of -9.

(a) Find the values of a and b, and hence, factorise p(.r) completely. 16 marksl

14 marksl(b) Find the set of values of x which satisfies * u O

(c) By completing the square, find the minimum value of *, x * 3,and the value of x at

which it occurs.

12 The function f is defined by

(") - tn ?*, where .r > o.

xt/

(") State all asymptotes of f.

(b) Find the stationary point of f, and determine its nature.

(r) Obtain the intervals, where

(i) f is concave upwards, and

(ii) f is concave downwards.

Hence, determine the coordinates of the point of inflexion.

(d) Sketch the graph y - (x).

14 marlcsl

95011,9541r*This question paper is CONFIDENTIAL until the examination is over.

14 marlcsl

14 marlal

12 marlrsl

12 marksl

16 marksl

16 marlcsl

12 marksl

CONFIDENTIAL*

COl\FIDEI{TIAL* 6

MATHEMATICAL FORMULAE

Sets of nambers

Nl The set of nafural numbers, {I,2,3, ...}.Z The set ofintegers, {..., -3, -2,-1, 0, I, 2,3, ...}.

0 The set of rational numbers, It, a, b e Z, b + 0l,.

R The set of real numbers

C The set of complex numbers

Logarithms

logfr- log'"Iogu a

Zr:tn(n+t)(2n+r)

2':!n'1n+ r7'

(a + b)" = d + (i)n u +(i)pu,+ ... + (X)o,-,u,+ ... + bn, where n G N.

Coordinate geometry

The coordinates of the point which divides the line joining @r, !r) ^d @r, yr) in the ratiom:nis

Series

n

(W,"r;9,)The distance from (*r, yr) to ax + by + c - 0 is

lo*r+ by, + cl:{a'+ b2

9501t,95411*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

CONFIDENTIAL*/SULIT*

Integration

j"*dx-uv-|,ffa.

lpdr-hl(x)l*,J (x)

Maclaurin expunsions

(l +x)' = t *nx.To+... + %lr + ..., Fl < r.

Numerical methods

Newton-Raphson iteration for (x) : 0: .r-,, : t- - $'l'"n+r '"n f,(X,)

Trapezium rule:

f'f{r)Ar = L hfyo+ 2(y,, + y, * ... r y,-r)*./,], where y, : f(o + rh), h : U, o .

Joz

Pengamiran

h*":tw- I'#"J#":mt(x)t+c

Kembangan Maclaurin

(1 +x)' : I t nx* ry-2 + ... +Wf + ..., Irl < r.

ffi

ffiffi

Kaedah berangka

Lelaran Newton-Raphson bagi ("r) : 0: xn*t: xn #

Petua trapezium:

!:*0. = L' hlyo+ 2(J, + y, r . . . * y,-r) + y,f, dengan y, : f(o + rh), h : b ;

o .

9501r,95411*This question paper is CONFIDENTIAI until the examination is over. CONFIDENTIAL**Kertas soalan ini SULIT sehingga peperiksaan kertas ini tamat. SULIT*