com selangor stpm trial 20111 maths t

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PEPERIKSAANPERCUBAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA

NEGERI SEMBILAN DARUL KHUSUS 2011

Instruction to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.

Answer all questions. Answer may be written in either English or Bahasa Malaysia.All necessary working should be shown clearly.

Non-exact numerical answer may be given correct to three significant figures, or one decimal

place in the case of angles in degrees, unless a different level of accuracy is specified in the

question.

Mathematical tables, list of mathematical formulae and graph paper are provided.

Arahan kepada calon:

JANGAN BUKA KERTAS SO ALAN INI SEHINGGA ANDA DIBENARKAN

BERBUAT DEMIKIAN.

Jawab semua soalan. Jawapan boleh ditulis dalam Bahasa Inggeris atau Bahasa Malaysia.Semua kerja yang perlu hendaklah ditunjukkan denganjelas.

Jawapan berangka tak tepat boleh diberikan betul sehingga tiga angka bererti, atau satu

tempat perpuluhan dalam kes sudut dalam darjah, kecuali aras kejituan yang lain ditentukan

dalam soalan.

Sifir matematik, senarai rum us matematik, dan kertas graf dibekalkan.

This question paper consists of 3 printed pages.

(Kertas soalan ini terdiri daripada 3 halaman)

http://edu.joshuatly.com



CONFIDENTIAL */SULJT* 2

I Ex2 - 3x . . I fi .

. xpress m partm ractiOns. [5 marks]x2

-2x+1

5 2

2. By using the substitution x = I + u, find the exact value ofJ( x ~ 1)2 dx.

3. If lzl; .J5, find the modulus of z + ~ , where z' is the complex conjugate ofz.z

[5 marks]

[6 marks]

4. Given the polynomial f{x) ; x4

- 3x3 + kx

2 + 15x + 50, where k is a constant and that

(x - 5) is a factor of f(x). Find the value of k and hence factorise f(x) completely into

exact linear factors. [6 marks]

5. Use the binomial theorem to expand J4 + x as a series of ascending powers of x up to1 - X

and including the term in x2

Find the set of values of x such that the expansion is valid.

[7 marks]

6. Given that function J(x) = 2 •{ll+xl, if x :S 0

x , i f x>O

(a) Sketch the graph ofy = f(x). State the range of f(x).

(b) Hence, find the set ofvalues ofx for which f (x)- I ::: 0 . [7 marks]

7. Find the set of values of x which satisfies ~ + I ::: 4 - _ I _ .x 2 -x

[7 marks]

8. Given that matrices A= and B = r ~ ~ 4 2 1 0

0 5 J1 -7 .

2 -1

(a) Show that A is a non-singular matrix.

(b) Find matrix AB and deduce A- 1•

(c) Given matrix C = ( ~ J , find matrix X in terms ofn ifAX= C. [9 marks]

STPM 950/1, 954/1 Turn over [Lihat Sebelah)

This question paper is CONFIDENTIAL until the examination is over CONFIDENTIAL*

Kertas soalan ini SULIT sehingga peperiksaan kertsa ini tarnal SULIT*




CONFIDENTIAL */SUL!T* 3

9. A circle touches the straight line 4y =3x- 8 at the point P(4, 1) and passes through

another point Q(5, 3). Find the equation of the circle and show that it touches the y-axis.

(10 marks]

I 0. Find the coordinates of the stationary points on the curve y = e-'x'

nature. Sketch the curve.

and determine their

[11 marks]

2

II. Sketch on the same coordinate axes, the curve y = 2x 2 and the ellipse x2 +L = 1.9

(a) Calculate the area of the region bounded by the curve y =2x2

and the line y = 1..:'..2

(b) Find the volume of the solid formed when the region bounded by the curve and the

ellipse is rotated through 180° about they-axis. [12 marks]

12. (a) Express ( X ) n partial fractions. Hence, obtain an expression for3k- 2 3k+ 1

n 1 .sn = "" X ) and find 11m sn.

L... 3k-2 3k+1 n->®k=1(8 marks]

(b) Find the least value ofn for which the sum of the first n terms ofthe geometric series

I+ 0.75 + (0.75/+ (0.75)3+ .. .

is greater than ~ o f its sum to infinity. [7 marks]4

STPM 950/1, 954/1 Turn over [Lihat Sebelah)

This question paper is CONFIDENTIAL until the examination is over CONFIDENTIAL*

Kertas soalan ini SULIT sehingga peperiksaan ke1tsa ini tamat SULIT*




Skema Pemarkahan Mathematics T 1/S l Pepcriksaan Percubaan STP:Vl 2011

x 2- 3x . . .

l. Express m parllal fracttons. [5 marks]

x2 -2x+1

x2 -3x x+1---- . :1

x2

-2x+1 . x2 -2x+1Ml

X +1 A 8

Let (x -1Xx -1)"' x- 1 + (x -1)281

x+ l =A (x - l ) +B Ml

X = 1, B = 2; X= 0, A = I Ml

x2

-3x "' 1__

_x2 -2x+1 x-1

2 AI(x -1)2

5 2

2. By using the substitution x = 1 + u, find the exact value of f. ~ , - d x . ' (x- '

x = 1+ u, u = x - 1, du = dx,

X = 2, ll = ] ; X = 5, ll 4

[,,

= 4 - + 21n u u,

Bl

Ml

= 4[(- 21n4 +4)- (- 1 + 21111+ :)l= 15+161n7.

AI

I\11

AI

[5]

[5 marks]

[5]




2

3. If jz[= 15. find the modulus u t ' z + ~ , where z' is the complex conjugate ofz.z

[6marks]

Let z =a+ bi

/zl = .Ja':b2 151 . 1

z + = a + b1 +---z' a -bi

. a +bi=a+bi+--

a2 + b 2

= a+ bi + -1

(a+ bi)5

= El_(a + bi)

5

Bl

Ml

Ml

AI

Ml

Al [6]

4. Given the polynomial f(x) = x4

- Jx3

+ kx2

+ 15x + 50, where k is a constant and that

(x -· 5) is a factor of f(x). Find the value of k and hence facto rise f(x) completely into

exact linear factors. [6 marks]

f(5)=0 :=:, (5)4. J , s / + k(5)

2 + 15(5) +50= o:::;> k=-15

f(-2) = (-2)4

-3(-2)3

- 15(-2/ + 15(-2) +50= 0

:. x + 2 io a factor of f(x)

f(x) = (x + 2) (x - 5)(x2

- 5)

•= (x + 2) (x - 5)(x - --/S)(x + -iS)

MlAI

Ml

AI

MlAI [6]

5 Use the binomid theorem to expand ~ L \ + x as a series of ascending powers of x up· 1-x

to and including the term in x2

. Find the set of values of x such that the expansion isvalid. [7 marks]

1

(4+x)l =2(1+)(l)z\ 4

1 r ~ y _ q ]= i , + i ( ~ } i ' l ; : _ 2 2 ( ~ r + ..

I .L

Ml

1

+ - X ~ ~ - X2 -i . A!

64




3

M1

A1

M1

5 55 2" '2+-X+-X AI

4 64

. . Set of values of x for the expansion to be valid is { x: -I < x < I} A I [7]

( ){

11 + xj, if x :<:: 06.

Given that functionf x

= , .x- , zf x > 0

(a) Sketch the graph ofy = J(x). State the range of f(x).

(h) Hence, find the set of values ofx forwhichf(x)-1?: 0. [7 marks]

(., \")

y

y = [1 + x[D1 y =[ I+ x[

y =_L_ __ -----].

D 1 all correct

Range off= {y: y 2': 0) Bl

f(x)-1::>0 =:>f(x)::>1

Solving y = I andy= -(x + I) =:>X= -2 M1

. 'So!vmg y = I and y = x- => X= 1 Ml

Tl'" set of values of x. is {x: x:::; -2 or x ::> 1 or x = 0) AI [Tj




4

7. Find the set of values of x which satisfles2 +I:: 4- _I

2(2- x) + x - 3x(2- x) >0

x(2- x)

3x2

- 7x + 4 ;::0

x(2- x)

(3x- 4)(x- 1) > 0x(2- x) -

0 + I - 4 +

3

x 2-x

Ml

Ml

Ml

Ml

X

2 - Ml

The solution set is { x: 0 < x S I or i > x < 2} AI

[7 marks]

AI [7]




5

8. Given that matrices A= [ ! and B = [

(a) Show that A is a non-singular matrix.

(b) Find matrix AB and deduce A_,.

(c) Given matrix C = [ l ind matrix X in terms of n if AX = C.

(a) de tA=J ( l - 6 ) - 2 (2 -12 )+ 1 (4 -4 )=5 Ml

Since IAI i ' 0, :. A is a non-singular matrix. AI

~ ] [ ~ ~ 2 I 0 2

-1 0 11 7

= 25 5

02 1-5 5

( c )X=k1

C

-1 0 11 72 -- l ~ l

02--n51-n5

52--5

51

5

5 j rs a oJ-7 = lo s o- ] 0 0 5

Ml

Ml

AI

Ml

Ml

AI

AI

[9 marks]

[9]




7

I0. Find the coordinates of the stationary points on the curve y = e-' x' and determine

their nature. Sketch the curve. [I I marks]

dy 2 -X -X 2- = xe - e xdx

=xe-x(z-x)

Let dy = 0, xe -x (2- x) = 0dx

x = 0 or x = 2

MI

MI

4y= 0 or y= - MI

e '

:. the stationary points are (0, 0) and (2, 4) A Ie

d'y = e-x(2- 2x)- e-x(zx- x2 ) MIdx2

= e·x(2- 4x + x2)

X= 0 _c:i_'y - 2 > 0' dx 2 -

d2 y 2x=2, --=--" <0 MI

dx 2e"

:. (0, 0) is a minimum point and (2, ~ - ) is a maximum point.Y e

X -7

+oo, y -7 0

tX -7 - W, y -7 + W

AI AI

DI (shape)

01 (critical points)

(2, 4/e2) D 1 (all correct)

---4-L.________· ~ = - - - = - - o _ ; , , X [II]·)I

l I




8

"ll . Sketch on the same coordinate axes, the curve y = 2x 2 and the ellipse x2 + y_·_ = 1.

9

(a) Calculate the area of the region bounded by the curve y = 2x 2 and the line y = 11_2

(b) find the volume of the solid formed when the region bounded by the curve and

the ellipse is rotated through 180° about the y-axis. [12 marks]

y2 4x 4 .[3 3

X +-=1=>X=±--=>y=-9 2 2

Ml AI

D I (parabola)

Dl(ellipse)

-1

1-

(a) y'X=-123 1 1 1

Area =f 12 y2 jdy Ml

3

2 [2 T Ml- -Y '12 30

4(3[3_ \=12 22 --

0J Ml

=13 AI

3

(b) Volume= fn(±y fY + H ~ - ~ - } y Ml

2

3 J3' ' r Y3'Ml[-ny] +l{y---1

4 , L 21; "2

[( 9 , r 3 ( 3 r Jl--0 +l3-1- -+ i ·:.:· +2716 j 2 \ .L ;

MJ

19AI ['-1 t [ Lj

16




9

112. (a) Express (:lk-=2X:Jk +

1) in partial fractions. Hence, obtain an expressron for

" 1 .s, =L( X ) and find lrm S,. [8 marks]k=l 3k - 2 3k + 1 HW '

(b) Find the least value of n for which the sum of the first n terms of the geometric

series

1+ 0.75 + (0.75)2

+ (0.75i + ...

is greater than 2of its sum to infinity.4

1 A B

(a) Let (3k-2X3k+1) 3k-2 + 3k+1

=> I = A(Jk + J) + B(Jk- 2)

1 1 1

(3k- 2XJk + 1) = 3(3k- 21 3(3k + 11

" 1

s. =6 3k- 2X3k + 1)

"1[ 1 1 J=6 3k - 2 - 3k + 1

M1

Ml

Al

Ml

= _311(1--41)+ (-41- _71) +-71 - 11o) + ...(-1-- ~ _ 1 _ __ ) + ( ~ L ____1 _ ~ 3n .. 5 3n - 2) 3n -· 2 3n + 1 /

= H- 3 n ~ 1 J = 3nn+1lim S =im[·!J1--

1 ) ] = ~ n--+w n n--+aJ 3l 3n + 1 3

(b) S = 1(1-0.75") Ml

" 1- 0. 75

1- 0.75"

0.25

a 1s =-=-w 1 - r 1-0.75

= ~ 1 ~ = 4 0.25

s. > ~ ( 4 ) => 1-0.75"

0.25

AI

Ml

Al

>3

=> 1- 0.75" > 0.75

=> 0.75" < 0.25

n > 4.819

The least value ofn is 5.

AI

Ml A I

M1

Ml

AI

[7 marks]

Ml

[8]

[7]


com selangor stpm trial 20111 maths t

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