stpm trial 2009 matht q&a kelantan

8/14/2019 STPM Trial 2009 MathT Q&A Kelantan

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I. Using alcebraic laws ofsct. prove that I(A u B)" C]u [(A u B) " C']- B=A-B. [6 marb]2. .IiFind 1 3x dxZTx + 1 [4 marb]3. On the same diagram, sketch the graphs of y =x : I and Y - 2x - J. [3 morks)

A region R is bounded by the two graphs and the y-axis. Find the area of region R.(7 marh]A solid is fonned by rotating the region R complete ly about y-axis. Find the volume ofthcsol id fonned . [5 marb]

4. The functionfis defined by fIx) = x 3 .O x =3

(i) Find ! ~ T - fIx) . ! ~ T - f [7 marb]Ixl

7. [Y - 2

Findthevaluesofx.yandzif 4..1'; 4 x ' 4< -1x is symmetric.2: - 1

[5 marb)

8. Given thatf(x) ... f + a:/ + 8x + b, where a and b are constant. f '(x) - 0 when x z : -2. When1(..1') is divided by x + 2, the remainder is 2.

9.

Dctennine the \'alues of a and b.(8) Show that the equationf(x) - 0 has on ly one real root and find the set of values of xsuch thatf(x} > 0(b) Express x + 4 in partial fructions.fIx)

(I I

Given the matrices A - P 47 - 3

Find the values of p. q and r.

-I) (Iand 2- I I

[5 marb]

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2/29

954 /1

10.

Hence. so lve the simultaneous equationsx+ y - z : :-22y + Z a.5.r - 97x - 3y - : - 14

Given that y z> , showthat (I + X2)d1y 3 x dx' dx

2009

[4 marks)

) I. Express _ -1_ - as partial fraction.(r + I)( r +2) (4 marks), . J - IHence or otherwise. find the value of ) ---.:-=, (r + l )( r + 2)

, .. 2.. IFind also the value of lim L ---....... . .. .4 1 (r + l )( r +2)

(6 marks)

[4 marks)

12 . By using sketching graph. show that the upproximate value of the root can be derived by thefollowing Newton - Raphson iterative fonnulaX . . 1 =- X . - f(x ,, ) . [4 marks]f(x . )

Show that the equation Ian x "" 2.r has a root in the interval ( ; . ~ ) . [3 marks]Hence. find the root ofme equation for ( %). correct to three decimal places.[7 mor.tsJ

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3/29

1. Ex press 3sinO-cosO inthefonn Rsin{O-a) ,with R >O andOo


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6. lb e figure shows a quadrilateral which inscribed two isosceles triangles ABC and ADE with baseAB and AE respectively. Each trianglcs has the base angles of75 o . Be and DE are parallcl andequal in lengths.

~ , ~ - - - - - - - - - - - - - - - - - - - - - - ~ ;

Show that(a) LeBE =LBED =90'(b) ACD is an equilateral triangle [ 4 )r31

7. The random variable X is normally distributed with mean jJ and variance 400 . It is known thatI'(X > 1159) ,; 0.123 and P(X > 769) 0.881 . Determine the range of the value of I ' . 15 )

8. In a certain city . records for burglaries show that the probability of an arrest in one week is 0.42 .l bc pro bability of an arrest and co nviction in that week is 0.35.(a) Find the probability that a person arrested for a particular week for burglary will be co nvicted .(b) What is the probability that there wi ll be 3 weeks of successful arrested and conviction in oncparticular month . .[5 }

9. The continuous random variable X has probability dcnsity function given by

{k(x+I), O';x S 2

lex) = 2k 2 S x S b .() otherwise

Where k and b arc constants.(a) Given that P(X > 2.5) =.!.. find the val ue of the constants k and b.6(b ) Sketch the graph ofthc probability density function [(x).(e) Find 1'(1 .5 S X S 2.5)

16)13112 1

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5/29

10. A discrete random variab le X has a probabil ity functionP(X) ={ k(9-x) ' , x = 5,7,8 .o otherw ise

With k as 0 constant.(0) D e t U ' the value of k.(b) Find th e cumulative dis tribu t ion f ooction F( X).ond sketch the graph of F(X)(e) Find the mean and variance fo r X(d) Find E(4X-2} and Var{4X - 2)

[2 J[ 4 )[3 J[7 J

II. The successful sale of T-shirts in a local shop is bi nomially distributed with the probability ofselling one T-s hirt (i f thc customer enter the shop) is 0.35.(a) If 12 customers visit the shop in onc particular day , find thc probabi lity thatat least two T-shirts were sold. (2 ](b) CaJculate the least number of c u . ~ o m nceded in order the probability of getting at

least one sale is greater than 99 %. r3 J12. The fo llowing data are the handphones prices . to the nearest RM 100, of40 hand phones taken atrandom from a handphone shop, arrange in ascending order.

100100014002 1003300

4001100150023003400

5001100160024003900

6001200170025004 100

9001300170026004500

9001400180030004500

10001400190030004900

10001400210033005300

(a) Display the data in a .templot [ 2 :(b) Find the median and interquartile nUlge. [ 4 I(c) Calculate the mean and the standard deviation. r 5 '(d) Draw a box piotto represent the data ( 3 I(e) State the sJ lape of the fTequency dis tribution and givc a reason fo r your answer. , j

Elfd0/ questions

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6/29

Stpm2009 mls: IUs ing algebraic laws of set, prove that [(Av B)nCJv[(AVB)nC 'J- B =A -B . [6 mar.lsl

So lution: [(Av B)n CJV[(AvB)nC 'J - B=[(A v B)nCJV [(A V8) n C'ln B' BI= (A vB) n (CvC) n B' BI

n ~ = (A v B) n B' BI= (AnB') v (BnB') BI=(AnB') v ;= (A n B') BI=A-B BI

2.:;Find r 3x dxi 2:.rr:i 14 mar'; .]

So lution : r 3x 'f 3Ul 2J7:i


7/29

Slpm2009 mls:2~ = 2 x -H IS= 2x2+X - 12xl +x - 6 = O(2x - 3)(x+2) = 0

MIAI,Area - j . . . ! . . . d x X 3 X ~ MIAIox+ 1 2 2 2, 9=5 In [x+ IU -3 +4 MIMI

= I AI2 4

'n5 )' I (3'o lum e = "i ly- ' dy +"3 " 2" (3)X25 10) 9,.. If 1 '" - -+ 1 dy+-1t1 y Y 4( 25 ) ' 9= f - - - 101ny+ y +-IfY , 4

M IA I

AI

= ,,[( - - 10In5 + 5) - ( - 10In2 +2H1= I O l n AI4 5

4. The function/ is defined by I(x) = { ~ . x ';C 3.O. x = 3

MI

(i) Find lim I(x) . lim I(x) and hence detennine iflhe function/i s continuous al x =3.....,. . ...r [3 marls)(ii) Sketch the g .ph off [2 marls]So lution:

lim lex) = lim x - 3=I. -.,. .-d x - 3lim / ( x) = lim - H 3 =_ 1 MIAI....,. . ..,- x - 3Since lim I{x) "'$. lim I(x). _., ' . - 0 ) ':. lis not continuous at x = 3. A I

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8/29

Stpm2009 mls:3

0---

DIDI

5. Prove that, for all va lues of k, the line /l!x +y = 8k is a tangent to the hyperbola xy::: 16.' -Icnee, find the coordinates of the point of contact. [5 marks1Solution:

y=8k-.I"..I"(8k - 6 ) =16k'x' - 8h+ 16 - 0(h - 4Xh - 4) = 0(h - 4)'- 0

4:. x=!

.:::) There is only one point of contact

MIAI

MI

:. The line Itx +y "'" 8A: is a tangent to the hyperbola .ry "" 16 . A IWhen x = ! . x=4kk:.The coordinates of the point of contact =- ~ . 4 . t ) BI

6. Sketch the graphs of y = and y "" 12x - l ion the same diagram. Hence. so lve theIxlinequality 12x - I I> [7 mOTu]Ixl

So lution:

...!.. =f 2.1" - 11Ixl:::) .!. =2x - 1.I"

=> 2..' - .1" - 1= 0

y Ilx - II

010101

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9/29

Slpm2009 mls:4=>- I)(2x+ ) ~ O

Ix : J - 2' MIAIFor 1 2 x - I > ~ , the so lution D {x:x< I} MIAI

(y - 2 x'

7. Find the values of x, y and z if 4x - 4 2xY y+z

4X- I)X is symmetric.

2% - 1So lution : Xl = 4% - 4 ,, - 4x - l, x"'y+z. MIAI

=> 0(b) Express %+ 4 in partial fractions.I(x)

So lution: I(x) - x' + ax' + 8x + b['(x) - lx' + 2ax + 81 (-2) - 12 - 40 + 8 = 040 20a - 5Whena - 5. I(x) - x' + 5x' + 8x + b

1(-2) - -8 + 20 - 16 + b - 2-4+b - 2b - 6:.I(x) - x' + 5 0But ,r2 + 2.t + 2::: (x + 1)2+ 1> O. since (x + 1)2 > 0 for all real va luesof x

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10/29

Stpm2009 mls:5Hence (x + 3)(x2 + 2x + 2) > 0Whenx+3 > O::::)x > -3The set of values ofx such that}{xO is/x: x >-3).b) x+4 _ x+4

f(x) (x +3)(x' +2.


11/29

10.

mls:6q - 2pq+227q-14

3p+ 16+2r = nl7- r J9 - r

q - 2pq+227q - 14

3p+16+2r = 0 n 0-r J (n 0 OJ9 - r 0 0 n

p+ 10 = 0, q - 2 = oand 7 - r = O. N o t e t h a t np=IO,q=2and r=7( _ : O 4 ~ I J

7 - 3 - I 7 - 3 - 1Also, AB :: BA "'" 2Jx+y-z "' -22y+z = 5x - 9. -IOx + 4y+2z = - 187 ,- 3y-z c 14The 3 simultaneous equations can be written as a matrix equation as fo llows

HO ~ : J ( ; ) = ( ~ ~ 8 JPremultipllying with matrix B

BAUJ=B~ ~ 8 l V ( ~ J =B( ~ ~ 8 ) 2 [ ~ } = ; J U ~ 8 J S l o c e /

2.r = 4, 2y c - 2 and 2z =6x == 2.y = - 1andz ; )

Given that y = . show that O+ x2}d2y 3 x dx' dx (4 marks]

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12/29

Stpm2009 mls:7xAnswer: y = - - - ,

( l + x1)1,y( I + X2)2 - xy(1 +x, )- i (.!.X2x)+ (1 +x , )i !!l. = 12 dxy ' + (1 u , )i !!l. = 1dx2y !!l. + (I +x, )i d'y + !!l.(.!.Xlu , )- i (2x)= Odx dx' dx 2(I+xl)i d 2y +2[2Y + (I + x2)-i x1 = Odx' dx(I +X, )i d1y + 3(__X ~

(/xl ! fix(I +X1 )1(I+z ' / 'y + 3z!!l. = Odx' dxII . Express --1-- as partial fraction .(r + IXr +2) ._2. 1

Hence or otherwise, find the value of L ---, , (r+IXr +2 ). , 1.. I

Find also the value of lim L ---....... , .... , (r + IXr +2)Answer: Let ____ = _ A_+ _ B_(r+ IXr + 2) (r+l ) (r + 2)

I - A(r+2) + B(r + I)Substitute r ... -2 I = - B, B :: - Ir '"" -I if .. I

(r + IXr + 2) (r + l) (r + 2)'f ____ ;; 'f,.... , (r + IXr + 2) , .... , r + 1 r +2(_ I__ ~ ) + ___ ) + (_ I__n + 2 n + 3 n +3 n + 4 n + 2 n+3

(_ I__ .......+ (-,-___ ) +n + 4 n + 5 2n 2n + 1(_1____ ) .

2n + 1 2n + 21n +2 2n + 2

[4 mar,u]

[6mar,u]

[4 mar,u ]

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13/29

S!pm2009 mls:S2("+1)("+2)

~ _ l . . 1IimL ----.40, , (r + I)( r + 2) lim n - 2 ( "+ 1)("+2)l im (_ I - - _ I-)..... n+2 2n+2o

12. By using sketching graph, show that the approximate value of the root can be derived by thefollowing Newton - Raphson iterative fonnulax , =x. - f(x . ) . [4 marks]f(x.)

Show that the equation tan x = 2x has a root in the interval ( ~ . %). [3 marks1Hence. find the root of the equation for ( ~ , ~ ) , correct to three decimal places. [7 marks]

Answer.frx)

f{x.)]

Q(x" . ,) x,.01Refer to the graph above , Q is the point of intersection of tangent at P and the x-axis.Gradien! ofll>e langen!.! P = f(x.)

:. Equation of langen! . ! P is:y - j(x.) = f(x.) [x - x.lA!Q.y= O.O - j(x.) =f(x.)[x", - x. l MIAI:. x, = x. - f(x.) , f'(x.) 0 A If(x.)

l i ~ t a n x = 1 X ! "

MI

:. The equation tan x =2r has a root in the interval %. A Ij(x) - lanx - 2xf(x) = ec ' x - 2Letxo "" I , XI = X(J - !(xo)

f '(x. )

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14/29

SIpm2009 mls:9x = 1_ tanl - 2 = 1.3 105 MIAI

I secl l - 2= 1.3105 _ tan(I.3105) - 2(1.31 05) - 1.2239

X, sec'(1.3 105) - 2= 1.2239 _ tan(1.2239) - 2(1.2239) _ 1.1760

X, sec'(1.2239) - 2x, = 1.1760 tan(1.I760) - 2(1.I76O) = 1.1659 MIsec'(1.I76O) - 2x, = 1.1659 tan(1.I659) - 2(1.1659) = 1.1656 MIsec' (1.1659) - 2

:. x = l.l66 AI

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15/29

G rz. ,c:; .. ({! - c (;'{3 R COl 0( 1 ot!'I (.l f in 0(

fah 0( v:0( l {j. 43 0

>f;,,(j- CorB : . J I o { " , , , ( { 7 - I ~ i f 3 ) t / IB _ cor &" I! , '" ( /? - l ' l ' ' 13 ' I = Jf in ( fJ- ,g ' lf']') "

../TO

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16/29

u I _VA" : _,,-

. 1.! ,/.

V x : V -I ">( dv"oix

o l e) = ':t -I- :>o/ 'x. 3 "('

V ' ." dvoIx

d,(( rlV) I - V :::

I " ( 1 - " )

1 1:Jv - V-'1 + ':>.\1 - 3 V

3I - V /' I( A.?s..) 3"1 (

1. If) " ., c.1 1' 1 ) " 1 I" ( I - v) =- - C3 l"x .1 1, , ( 1_ \;)1 _7 , ( .\" [ X - v) 3J B

X( I - V)J = e. li4 \ ,

)( ( 'X.-;'_L) J A(" " (1 ) , - fj,

xJf>(,( 'J) ' - A ( 'II (." ' I ) I

/v '

(/}

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17/29

:: k(11 -)(0c1{cI k ( M - .,.. )oi l( .fx =( k ..J t .. /.'JM ->< - \f I ( ( v 1- / ,., (IIA) =C

M - >(')( =

- K f /.\M -e - k -(M - M e - I,, !

,'i- / ( I - -i> "'..; r,I ~ . , f II I IW I /1 : AI

i I' f /11", 1\ '4/ lufV.' I. -' . /H. i ,: I i/ "I'V I -: 1/1_ "t(' ./(

,1< c{ (A, I _>< ).Ti/( ( hi - ( .-1" 'iF"/-

\

X M ( I - e - /


18/29

J /


19/29

rt?-i\:011 - - 'o A : q pn

--'OJ> = ?OP 0 " ( p lYh1,,,,, VfCW- )

n -->PBo (r - 4 ) = -= (b - rl

I lr _ n tl "" N)b - rn rn r t (Yl r- =- fY' h I n l'1.-.( IVl-ln ) f " I t \ ~ fY'b

f'" "'4 + r'YI b

"\ - i>( , / --, /_ - 7 r

r'l'1 -J n- ' (2..-r,-- -

I , . " ' ' ') j ' lf}.1 1 On rl l" /. .')11'/


20/29

w{Mt

e, . ' c0,1/ ~ 9 0 - 11-" (5 otr 4. "Ii 0 );, ,/'(

H = '\.(1'>(\.19' hller ' 1 0 ~ .

-}( H =- .{.(!>[O = 1 0 '1 f?> I 0 - 4 /tt0 1 1.(IU J

> 15; + 1


21/29

..if I 1I ,q ) :;:: 0 / :> 3

ilS-'1 - A.i >;;>..01/ ,.. q - .,( ;:> : l 3 . .:lII ,- 1 - J 3 . :) ;>.,{( /'

.4{ / () .8'8 /I' ( 'Z' :> 76:

0- , . ( / ) >/0 gg I

r ( z > () ) " 8'il I1- i?(z < ( f ) > /0 8 1r ( '2 I - 0 8 i S l /

? ( ;Z


22/29

n )

I, )

,?Go 1\

/ '

' l 1 :"'--f c....\P,C Il / IC ) ( ) . J

0 ' 14 7 ....---;/

x .- ( 4 / (; "f r) c.--;/? ' (:, ( ( ! 14- 1 ) ! ((" 3 _c ) I /,

( " (> J ? / 'v,

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23/29

(j) tI ) ( X > ; J C'5

G !:>k d )( - '. I / '- :6 v I[ :>kx ] .b , :

b -V: :>)/:2 ) I - (P(O) i- p(I))I - (" (.0 Q.") r O {, S- I >

1 - ( O ' 6 ~ 4 - ) 0'1-- I ) > 0 -11 .) v:

.I - p (X ;0 ) > () ')


28/29

tl)

th

' ; f ~ ( Y f o [,,6Itv,,-,

J (jiJO I 1dlJ"O20,..,1 007) /

V I

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1- 0 . ,;. i J V - II OV c /I: /111 I '100. 1'1 vi

stpm trial 2009 matht q&a kelantan

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