stpm trial 2009 matht q&a (pcghs, penang)

8/14/2019 Stpm Trial 2009 Matht Q&A (Pcghs, Penang)

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(a) .If X c Y and Z 'c Y' . show that X c Z(b) Find the solution set of the inequality - x 2 + 6 > I Ir+ a x::; 0Given I(x) = bx 2 +c o


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8. Find the equation of the tangents to the circle x 2 + y2 == 2(a) -at (-1, -1)(b) that passes through the point (3, 1)

9. (alif x = -! and y =2t + !, where t is a non-zero parameter,t t

10.

dy 3prove that - =2 - -2- - .dx t + 1Deduce that -1 < dy < 2

dx d 2 dM.Given y == e- 2x cos3x , prove that -f+ 4. .z + 13y = 0/ - dx dx

(a) Show that f 3 ;'+4 dx = In 4J72X2 - 3x -2 3

[3][8]

[4]

[3]

[5]

[6]

(b) By using the substitution t = tan 0 or otherwise, fmd the value of [8]tr

_ 6f 1 dO2cos 2 0-1 -o11. Show that the equation 2 In x = has only one real root that liesxbetween x = 2 and x =3. [3]

5Use the Newton Raphson method to find the root of 2Inx = - [5]xcorrect to 3 decimal places by using 2.5 as the first approximation.

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12. Find the coordinates of he turning points of each of the curves [5]3 1y=-x +3x+2 and y= - ----x 3 - 3 x - 2Sketch the curves in separate diagrams and give the equations ofthe [6]

asymptotes of the second curve.Find the set of values of k where the equation x 3 - 3x - 2= - has [2]

(i) one real root(ii) three real and different roots

Prepared by :

Teh Siew Hong

Checked by:

h Guat BengHead athematics DepartmentPenang Chinese Girls' High Sch n 1\

4

Approved by : .

r.,VolreCilewGuru K .- SIins ellaMatetnalikSMJK ~ a m t A P.PINANGshteh pcghs270809

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4/14

No. :Da te :

I l4J ~ 1 c . r . l ~ ,re It'll @ ~ : c " " ' - i- + .. : 6 - Ln .xc ' ( .1 ' r c ~ Let xty ', )C.. g2 =x-.yi.( x + y ~ ) ( x - y L ) - 2 ( X . t ~ i . ) t 2 { x / ~ L ) ~ 5 - l f ~ /tl.(' ) _)(2.+6 >/xl )( 1+ Y ._ :u: -l 'y i. + .X - ~ ~ ( . :. 5 -4-(. MIte,. ~ Q . 1 i V { J.J. P. : l ( . 1 . - + ~ = _ ) ( p ~ : -xl.+.'=x )( 11 2. _ ijy t -::. ' ; - l{-L....... . ' 1 ~ ' X 'l._x_ 6= 0 X ot x-6 ='0 Co "-lp ore of i r\ GU)t" P ...r.(: oJII .. . L . I'... .. 'I .. . txt 1.)('( - 3 ) ~ 0 ( X - l ) ( ~ t 3 ) : > O - l t ~ ;-; 4-n. .~ : - - &ttAto x= -2 X.: 2. ::-\ M,TO .J Co "'-Pcu-e the r etU pGVr1)fk . mtS) \. ( ) l l - \ - ~ 1. =-5

" ./ r-...... /" x2.:. 6 -\L\P' )L. =If-/ \ : X : tl N,M, """.r 2::2.1"( . .I - 2 f i._xl. t6 >1x,1 = t . ~ C ,J-'l. I -.).-l Ai.fx -2


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No.Dote '

M-( 'L) j( - l J a ~ 'HI){ 2.) Tt:3 t -fxCi- of --::1'""" " ' / ; ; J c / ~ )( :: ..l.tt) - :l I. J ,-2t14

....... ~ X C ~ 1 l . ) / - X ~ I = 3.. -1.. r 2 0 0 ) +3 1 IU-.1 :t :lR '+ l (IOi"IXIO-t2JJ. -

:: 5L - :L r -J J ' -:0. - 1 f t 4' " ~ / 2 IVi / ' f ~ f 6 A-, 1M - M ~ : : I M.,~ ( J ) . J i ~ ~ : : . ( I - X y ~ ( t f J t ) - ~ ; ' :: 1 It, :: ( 1 ( - ~ ) t t -1: t X ) ~ 1 i / (- . t -:;; , - -; 1t ' .

~ ( t - t ) - i ) ( ) . ) U - t ) ( t - t ) ( z . J ... '---/.It\t1-M).:: I 'Co / - f;)( t ..J-)(2 T t J r ~ - t J ( ~ - t " X t .. ..lMrn-' - Mlrn- ' = M'""' fl,A, , / - ) ( -f z.M-1= .).}-1\1 o pr;. b:.1.( 10 ) _ ( a 2 , )0 1 -1 . -1 rr-:;; 1::. (-I 1 ""Ttf =- 1/ 9 -


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I\A :: ' /J 'j =-t\'\ X + (,1-3)( ... (1-0 (IJ]

::'X-2. ttl - -I\A : : . - - ~ J ~ : : ~ t ) ( ... r r - J ( ~ ) ] -:.-.1.:X+

-

_ "\ 3 /'Ie- ,J .- ~ z . , IT ,

+----+----') V

-:2


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cJ.y .. :. -3 X t .3 hA 1when :It: I"_11 .'7J - . . -3) ( ~ - 0 )( I I I f) ax t ) 0 /1\.t 1+,

wh'eh x=-I"o I l j - .." I' 'A 'c{X 2 . \. ) \ "'.J( 'J 0 ) /V..in- ,poiht If ,,

.."'.J! .:: (p t - .jGf ")( I . . .. r\.\.,l i t -0 -L )( I, if) I"V\I'n POo\t A-,

o p j _-- r >n'cpV'K :. -I 7x"'zr ft ,~ : : . o J

,tI, '1-)"t / :\ f',\\ II 1 \ ,\ T \\ / I \ " 'LY . ;\, I

y ~ - ) ( 3 + 3 X t 2 ': _()( +,) - ( )( -2)

...J, -" 7T PiIJ I ~ / I ..../ ....-If I1I,

III I II , II

-

,...

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No.Do te :

")( -3)( -2 =- t, -=-1:'" a- ... _"'\'1 - ~ , . P Lro t1A R"'-P J }c < u. 4-; ,To J . . . o . . . v ~ d di(f(ero...f rook

k'>fz ll...v I , ,

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f \

'fr-ofV..

- -

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SMJK Perempuan China Pulau PinangPeperiksaan Percubaan STPM 2009Matematik T Kertas 2Masa: 3jam

Instructions to candidates:Answer all questionsAll necessary working should be shown clearly.Non exact numerical answers may be given correct to three significant figures,or one decimal place in the case of angles, unless a different level of accuracy isspecified in the question.A list ofmathematical formulae is provided

/ Prove that 8 58. cos0 +cos20 +cos 30 +cos40 =4 cos-cosO c o s - 22Hence find all the values of e between O{l and 180{l such thatcos8 + cos2B+cos3B=-cos4B [6 marks]2. Find all the values of x, where 0 < x < 360, that ;atisfy the equationcosec x + cot x = 3. Give your answers correct to the nearest 0.1 D.

[6 marks]

3. In the given diagram, AP and AQ are tangents. QBR is a straight line. Provethat RP is parallel to AQ.

p [6 marks]

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9/14

4 ,/p, Q and R are three points such that PQ = 2 i + 3 j and QR = 4 i - j .F i m r ~ " ~ l e P Q R . [3 marks](b) The force R = 6 i + 2 j is a resultant of force P which is parallel to i and

force Q which is para]]el to i + j. Find the magnitudes ofP and Q.(3 marks]

5. A hlp. which was at a position vector of 12 i + 21 j nautical miles reJative tothe position of a destroyer, was traveling with a velocity of 15 i + 6 j knots. Thecaptain of the ship realized that the destroyer was pursuing it with a velocity of24 i + u j knots. If the destroyer managed to intercept the ship, find the value ofu and also find the actual velocity of the destroyer. Find the time taken by thedestroyer in its successful pursuit of the ship. [7 marks]6. The rate of change in the temperature ofwater is given by the differentialequationdxdt = - k(x - fJ) where x is the temperature ofwater at the time t

( measured in hours), f} the room temperature and k a positive constant.Boi!ing water at the temperature of 100C is left to coo1 down in the roomtemperature of 30 C. It takes 1 hour for the boiling water to cool down to 80 C.7Show that k= In - [4 marks]5When the temperature of the water is 60C, it is placed in-the freezer oftemperature - 8 dc. Find the time taken for the water to form ice at 0 DC.

[6 marks]

7. (4 ompany has two factories X and Y. Factory X employs 30 workersa r i { ' f u ~ ; o r y y employs 40 workers. The mean-pay of the workers in X is RM1250, with standard deviation ofRM 280. The mean pay of the workers in Y isRM 1190, with standard deviation ofRM 290.(i) Calculate the mean pay of all workers in the company, [2 marks](ii) Calculate the standard deviation of the pay ofall workers in the company[2 marks]

2

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10/14

7 (b) The table below shows the number of hours per week spent readingmagazines by 18 students in a class.I 25 I 31 8 4850 24 47 3234 I 42 48 4034 40 40 44I 16 45L--

(i ). / (ii)

(iii)(iv)

Construct a stemplot for the data aboveFind the med ian and interquartile rangeCalculate the mean and standard deviationDraw a boxplot to represent the dataComment on the skewness of the distribution

[2 marks][2 marks][2 marks][2 ;narks][ l mark]&(f(factory produces ball bearings using machines A, Band C. The percentages

A t - bearings produced using machines A, Band Care 20 %, 35 % and 45 %r


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(a) Determine the value of a(b) Write down the cumulative distribution function ofX(c) Find P ( jX - E(X)j < 1)(2 marks][3 marks][4 marks]

11. ( /P ive patients need to undergo an operation that has a success rate of 0.8./ . B y assuming that each operation is independent ofone another, find(i) the probability that none of the operations fail { 2 marks](j ) the probability that at least 4 patients will be saved [ 2 marks]

(b) In a factory which produces an electronic component on a large scale,the production of each unit ofthe component is independentof oneanother and the probability of producing a unit that is defective is0.0006. By using the Poisson approximation, find(i) the probability that from 500 units of the component produce.,d, noneis defective, [3 marks](ii) the probability that from 500 units of the components produced,

there are, at mest, only 3 defective units. [3 m a ; ~12 . The probability of getting a rotten apple in a supermarket isJl&5'. Byusi g the Normal Distribution as an approximation, find the probability that outof 500 apples in the supermarket,(i ) more than 80 apples are rotten,(ii) at most 50 apples are rotten [3 marks][3 marKs]

',,-(b) The mass of an apple follows a normal distribution with m e ~ n 50 g and .,,-standard deviation 5 g. Find the gn:lbahility that 4 apples havera-total mass ofmore than 180 g. . [6 marks]

Prepared by: Lau YF Verified by:

Approved by: f\t1Tay Yoke Che\"Guru KIIIIIl Sains danM temalikSMJK PEREMPUANCHINA ,'.PINANG

4

C BengHe?d of hematics DepartmentPenang Chinese Girls' High School

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12/14

CD u:! e tc 1 "t 0 i" Cc-{ e te-Oj ife2 (0 f ). (} c.o-:r t} t 2.Col S6(os t/Yi ,,,I.....- 2. C0"16 l (err s e ftrrs &)2.(61 l 2 ( ~ C01 ! J Prt":! 4- (01 -t Ca l (j (OJOJ ce-;r C(ir $'}- 0

6- r80 ) J S"to 1 ,6- &l6 j 108 J::: 36 J 0'" IoJ:) ~ 2 - J r 0

@1 .L + C6""!1C.,..,:-.. -'"11 --:--:;; .::a.. a;fll\ J( --co"!){.:; , M/L e t 91)1 '}! ~ c C J r ) ( -= R-Sf/t1 ( i . 0 ( )IZ tt l

)(. :; $Lf ' /80" ft 2-\ I --.@ / - ~ ". .I / ~ \ \.1 I \ i\ I

'\Zl / L' ~ / /J "",- LiN Ai6-, :: &l (1- I) , CJJFe.rM t r


13/14

No. :Da te :

'-\ 2i)t'Z. + ~ I ~ / 0''123 lJO-O-o M, Ii> + c ~ ) clA:::-1 I\J ,( ) ~ )('2 + , 2. I'" r- "'1 i'2. i Ih

J -< /J - "-::. 2k>:r . J / A-, ( ftJ ) t=()c)-=- . Xf(;K;)c/XB,

b) 0 g , 1I 6 I E ( X2.) :: )(,2. f ( )(,)qX B...} 4- 5 IJ I 2.4-'+ \1, Va.r( -t) =f(X2-)-Cf;(x) r flA-.it 000 .2. ,+ 5 :=t g g _ J... Pr-il..5 0

116 (\/1 ..Ov(f S r , k-():; tU A-I ," .0,

- '".........l"ni

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14/14

No.Date:

(Ii) (tt)(i) X - ' V ~ C S I OP( I ! ~ S ) If-C S D.g!; 0,2 0'::- 3 ~ 8 Bi

U O P ( X 3 1 l p(X::4-)rp(X:.:s) u,-=. fh

to X", I?, U 0 0 I o0000A IIp:: O 3 13 1u) r )( 0) =0 : r ~ J

( f t ) P l ~ 3 ) ... PX 0) f P X -=- [) t P X:; 2-) "(\X I-tPex::.J )

( ~ 9 ~ 1 I},-:...j

,0 [cd "( "tE ( I,. bVan't:U1 ce - f\-p 7r 6 1- \k'" N ( ~ S / b 3 1 POr) 8-0)- PC ( )80 -S ) rVL

..:: P(:t )O68Q) ft,..,.. 0 - .ll/-b IV,Pex 6S0 )

-= P x< O .S ) /\..A ,PC r-z - J .068)":- 0 ' 0 /08 Pr,

iJ,) l ~ " V N i . . r O 2 } ) I( X, t Xd. -t )( ! J- Xli-'){""N( ~ o ) f-h-f(t- ) - l.-)- o-'i'i-t A-2

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stpm trial 2009 matht q&a (pcghs, penang)

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