2013 Bukit Panjang SS Copy

8/21/2019 2013 Bukit Panjang SS Copy

1/21

/ 80

Name of Candidate: ( ) Ctxs: Calculator Model:

BUKIT PANJANG GOVERNMENT HIGH SCHOOL

Preliminary Examination !"#$

%SECON&AR' ( ) Ex*re+ , Normal A-a.emi-/

Paper

(lntr0-tion/

&0ration1 ! 5o0r

Time1 "6"" 7 #" ""

$rite %o& name' class and reister numer on all *e &or*%ou hand in+ $rite in dar* lue or lac* pen on oth sides of thepaper+$rite the model numer of %our calculator in the top riht hand comer of the,rst pae of %our ans&er paper+

-ou ma% use a soft pencil for an% diarams or raphs+.o not use paper clips' hihlihters' lue or correction uid+

Ans&er A 1uestions+$rite %our ans&ers on the ans&er paper pro2ided+3i2e non4exact numerical ans&ers correct to " sini,cant ,ures' or 5 decimalplace in the case of anles in derees' unless a di6erent le2el of accurac%is specif ed in the 1uestion+

7he use of a scienti,c calculator is expected' &here appropriate+ -ou are reminded of the need for clear presentation in %our ans&ers+

At the end of !e examination' fasten all %our &or* securel% toether+ 7he numer of mar*s is i2en in rac*ets 8 ) at !e end of each 1uestion or part

1uestion+

7he total numer of mar*s for this paper is 80+

Setter1 (M On3LK8 7urn o2er9

This paper consists o/8 printed pagas


2/21

Quadratic: Equation

For the equation ‹ir' + b× + c= 0,

Binomial Expansion

n

2

where n is a positive integer and

Identitie s

2T CONOMXIRY

sin :d ;cos A

-- I sec :l =

5;tan :f cosec

:d = 5;cot :d

sin (A?@)= sin Acos £?cos Asin Bcos(t::tl)=

coslcos@7sintsin B

sin l = sin A nos A

cosA = cos lBsin ’A-- cos A- I -- 5Bsin ’A

sin A +sin 4 = 2sin

sin A —sin B = 2cos —

2'

cos A+ cosB = 2cos—

A -B

2'

A - B2'

A - B

2'

cos A -nosB = —2sin — sin

— A -B

Formulas for 6

ABC

: 2'

a ! "# c

s$in A sin B sin%

a ’--b '+ c '— 2bc cos A

&age 2


3/21

"x4x ; !

(a) *he curve = ‹' + n' + b + c has s-tionar points at = and . /

"t aso passes -rough -e point (0,0) Find the vaues o1 ‹i, b and c 3

() ence s*etch the cur2e sho&in !e intercepts and the turninpoints clearl%+

+ (a) Dind all the anles et&een 0E and "F0E inclusi2e&hich satisG the

e1uations

(i) tan: % ; sec % = 5 '

(ii) H cosIsin x 45 +

(b) Find a the vaues o1 between 0 and 2 radians 1or which

2cosec(/ — ) = /

!9

!9

4

/ (a) 5iven a triange ABC has area (2

)in and side AB e1uals2 6

(2 + v1i )7 8press the perpendicuar distance 1ro7 % to AB inthe

1or7 (‹i + b ) where 9i and b are rea nu7bers

() Jol2e the follo&in e1uations+

"

(ii) lo lo (!x: 4F)) = loK !

!9

#"9

[3]

!+ (a)

Lxpressin partialfactions+ !9

"9

+ (a) 3i2en *at the coecient of x and the coe3ientof x’ in the

expansion of ( ; O) is in the ratio — Find the2alue of ii+

/

#!9

(b) :ence, with the vaue o1 n 1ound in the 1irst part, 1ind the coe11icient o1 ;'

in the epansion o1 (2 + )


4/21

x ;"

(a) >-etch the graph o1 = "n( +)

(b) ?n the sa7e graph, draw a suitable snaight line that 7ust be added

to ooin *e solufons to !e e1uafon aQ= a(I; ) +

Jho& %our &or*ins cear

H+ (e) Jho& !at the expression 4I ;x B is

al&a%s neati2e for all real 2alues ofx+

() ence or other&ise' ,nd !e rane of 2alues of @ for&hich *e

"9

)

ine1uali%

Ax:4F

4x ; x

B R

for all reS 2alues of x+

8+ .iTerentiate

with respect to 8press ew answer in 4e 1o7

@rite doin the vaues o1 a and b.

U+ (a) 3i2en the cur2e has the e1uation % = x:4 "I ;"I;H + Jho& !at it is

an increasing 1unction 1or a vaues o1 ecept at ——'

() Jho& that x =B5 is the onl% re@ solution &hich satisG!e e1uation

(!9

9

[3]

Pae !

0+ 7he circle &ith e1uation x ; %B IB F% ; U = 0 touchesthe %4axis at point A.

7he line touches !e circle at point @+

(a) Find the centre and the radius o1 the circe

(b)


5/21

Find the vaue o1 7

(c) Find the coordinates o1 point A and point B.

(d) Jho& that afVWC = tanQB+

*he tabe shows eperi7enta vaues o1 two variabes and

2

/A

0 3 20 23 /0 /3 40

30 /33 2.50 2 0CC 0.63

Xsin the 2ertical axis for f% and the horiontal axis for x:'plot f% aainst x and otain a straiht line raph+

Dsing the graph drawn,

(a) epress in ter7s o1 ,

(b) esti7ate the vaue o1 when . /

LN. WDPAPLY

&age 3

[4]

2


6/21

0 ECl

2 9 : ;


7/21


8/21


9/21

SECONDAR !E"SN# ADD $A%&

'eliina* E+ainati,n 20-3

*EB8 ?F >&8%"F"%E*"?G

%,i/s

J*etchin ofloarithmic function

Qn

(a)

Mar*s 7%pe of

allocafon 1ueaion Zno&lede

uadnti/ roots o1equation B(a) 2 Hnowedge>urds .&robe7 soving /(a) 4 Hnowedge%oordinaI geo7eJ . %irce. K(a) 2 Hnowedge1inding radius and centre>oving trigo equa1ons

(a)(ii)(

Hnowedge

Eppication o1 di11erentiation . "Jationar% Points Jol2in trioecuationsJol2in e1uationin2ol2in indices

C Eppication

4 Eppication

/ Eppication

Jol2in lo e1uation "[)(ii) "Application Partial fractions! ! Application

6inomial theorem M 8 Application.iTerentiation 8(a) M

Application Coordinate eometr%B circle U()'(c)\(d) U ] Applicationinear la& 0 9 8 ]Application

] ] Application5 5

S,l1ing uadati/ ineualit* (b) / L%o7prehensionand decreasing C(b) 3 %o7prehension

" i;;;;;,gMerivingsoaightineequation =J) 4 / %o7prehension 4 4 %o7prehension /3N'e

4 4 *ota;C0 -004567 Ga7e o1 %andidate; %ass;

6XZ57 PANVAN3 3W^LYNMLN7 53 JCWW

Preliminar% Lxaminations

0" JLCWN.AY- DWXY

L_PYLJJ/D5^L NWYMA

A..575WNAMA7LMA75CJ

Paper


10/21

REA& THESE INSTRUCTIONS>IRST

)"$6+!

&ate1 !" A030t4 !"#$

&0ration1 ! 5 $" min

Time1 "6 ""?#" $"

@rite yo0r name4 -la an. re3iter n0mer on all t5e Bor yo0 5an. inD @rite in .ar l0e or la- *en on ot5 i.e o; t5e *a*erD

'o0 may 0e a o;t *en-il ;or any .ia3ram or 3ra*5D&o not 0e ta*le4 *a*er -li*4 5i35li35ter4 3l0e or -orre-tion ;l0i.D

AnBer ALL 0etionD

@rite yo0r aner on t5e e*arate Aner Pa*er *roFi.e.DGiFe non7exa-t n0meri-al anBer -orre-t to $ i3ni;i-ant ;i30re4 or # .e-imal *la-e int5e -ae o; an3le in .e3ree4 0nle a .i;;erent leFel o; a--0ra-y i *e-i;ie. in t5e0etionD T5e 0e o; a -ienti;i- -al-0lator i ex*e-te.4 B5ere a**ro*riateD'o0 are remin.e. o; t5e nee. ;or -lear *reentation in yo0r anBerD

At t5e en. o; t5e examination4 ;aten all yo0r Bor e-0rely to3et5erDT5e n0mer o; mar i 3iFen in ra-et at t5e en. o; ea-5 0etion or *art0etionD T5e total n0mer o; mar ;or t5i *a*er i #""D

Setter1 Mr C5i0H@

GT0rn oFerH

This paper consists of printed pages


11/21

2

at!ematical "ormulae

Quadratic!uation

1. éLCEB#L&

Dor !e e1uafon iix: ;,x;c = 0'

Binomiat Expa#ion

(a +b)


12/21

= "t + c — 2bc nos A


13/21

l (H' 0)

3f(8, F

Ans&er A 1uestions+

5 7he line x B% B " 40 intersects the cur2e "I% B %B = 0 at thepoints A and B. Dind

(i) the coordinates of A and B% (!9(ii) the exact 2alue of the distance A6+ 9

2 (a) 5iven that /;'— C'.. 4 — 3 = x)Ax .. *)x — 2) .. 1or a vaues o1 ,1ind the vaues o1 A' B and % (/)

() 7he cuic pol%nomial 5(x) is such that the coecient of I is and thatthe root of the e1uation f(x) 40 are B ' " and 5:+ 3i2en that 5(x) hasa remainder of 0 &hen di2ided % x: 4!' ,nd

(i) the vaue o1 Q(ii) the re7ainder when 1() is divided b + 2

" 3i2en that oand ` are the roots of the e1uation "O:;x B = 0' ,nd

(i) the 2alues of n ;` and o` '

2

(ii) the equation whose rootsare

' and '

4 S,luti,ns t, this uesti,n b* a//uate da8ing 8ill n,t be a//eted.

*he diagra7 shows a rho7bus ABC, in which A is (, 0) and , is on t!e-axis. *he point R(C,) is the 7idpoint o1E% Find

(i) the coordinates o1 C'(ii) the coordinates ot ,'(iii) the coordinates o11, given that "is on , such

that

area o1 triange AC -/ — area o1 triange A,C'

4(iv) the area o1 the rho7bus ABC,.

22

/A

2


14/21

(a) Pro2e(i) (cot F 4tae F) sin F 4 cos F B sec F

(ii) cos’ A - cos6 -- sin(A 44 B& sin(6 B A&

[) 3i2en that =caa ' (sin’ ' ( Fain8caa84 " 'express in !e fom

@ coa (8 0 and 0b < o < U0b+

F 7he depth' h me*es of &ater in a ri2er is i2en % thee1uation

= 0 ; !sin() &here 5 is the numer of hours after

midniht+

(i) %acuate the dep4 o1 the water when 1 .2

(ii) Find 4e 7ai7u7 and 7ini7u7 dep4 o1 water in the i1e.

(iii) J*etch !e aph of h = 0 ; !sin( 4) 1or 0 S;"; 2

!9

!9

!9

(9

7he ri2er ates are closed &hen *e &ater le2el drops to se2en metres and elo&+An alarm

rins &hen !e ates are opened or closed+(i2) Dind !e 2alue of i &hen !e alam

,rst sounds+"9


15/21

% = sin x

BECH4 QUESTION7 ONAFRESHSHEETOFPAPER

H (a) 7he diaram sho&s an open &ater trouh of lenth 5cm &i! a2enical cross4section in the shape of an e1uilateml nianle of side xcm+ 7he &a*r $ouh is made of thin me$l plates and its capacit% is! cm+ 5f the thic*ness of the metal plates is to e ta*en

as negigibe, show 4at 4e area, A rim ' o1 7eta pates used is given b

A-- 4

Dind the 2alue of such !at the area of metal plates used toma*e the trouh is minimum+

(!)

(b) *he diagra7 shows part o1 the curve . sin 2>how that the area o1 the shaded region bounded b the curve, the — ais and theines

—— and 2

3 — —

units'24 4


16/21

8 A particle mo2es alon a sPaiht line such that J seconds after passinthrouh a ,xed

point A' its veoci, v c7 ', is given b

Et the ti7e tAat the partice passes through 4, a second partice Q eaves a 1ised point 4,with

a constant 2elocit% of " cm s + 7he distance et&een point A md point @ is ! cm and !e diaram sho&s !epositions of the

t&o particles at time' t -- ).

A24 c7

B

(i) Find the initia veoci, in c7 s ', o1 the pa7ce S

(ii) 8press in ter7s o1 i, the dispace7ent o1 the pa7ce 0 1ro7 A. 2

(iii) Dind the disonce' in cm' et&een the particles &hen particle isinstantaneousl% at rest+ "9

(i2) 3i2en that particles and Q are Ca2ellin in the same direction from !estart of themotion' ,nd !e time &hen *e t&o particles collide+ 9

K (a) "1 NaO N()‹"r = eO cos/; + c , 1ind () 2

(b) Mi11erentiate ;r 'ii with respect to 2e

ence' or o!er&ise e2aluate Vx/cx A + ea2e %our ans&ers in terms ofa (")

0 Tariabes and ne connected b the equation (2 — ) ( + /) . 3(a) 5iven that the point & ( , 2) ies on the curve (2 — ) ( + /) — 3,

1ind

(i) the vaue o1 — at

(ii) the equation o1 the no7a to the curve at point &

2

/A

(b) 8pain whether a stationar point eists on the curve, soting the reasons cear


17/21

(c) "1 × is increasing at a rate o1 02 units per second, 1ind the rate o1 change o1 atthe instant when . — 4 /)


18/21

5n !e diaam' LD is a tanent to the circle at L+ 3i2en that AB is panllel tof and CC^

produced meets F at *% pro2e that

(i) A//FF is similar to AF*C %

(iii) D4'

FA

EN!

"9

9

"9

7


19/21

" " "

'e9 ina* E+a 20-3 Add $ath 'ae 2 (Enswe7)

(a) (2 , ) , (.,— 0)

2 (a) A .— /, "t ..2, % ..3(b) (i) "; . 2 (ii) — 40

4 (i) % (K,2) (ii) M (0,4) (iii) (,33) (iv) dC units '

U'= 1 cox3+aix* + sinCeosC./

= >eos' + "— coa*+ /Nin2C./

= 4aaa'B.2 + /ain2C

= 2V2caa' */I /siii 2C W

= 2caa 2C+ /ai2C

(i) /4 in (ii) 7a . 4 in , 7in . c7 (iv) t —.KC h

(a) ; . 4 c7


20/21

"

d

C (i) v . c7 s O' (ii)

(in

t:B 5:;t="t;A!

K (a) 2 cos / — / sin/

0 (a) (i)

e:

;

!

3 3

(b) Go stationar point as V2—;)' X0 1or a rea vaues o1 , w 2

:ence (c)

(Y 4C--

(EGHF is si7iar to EFH% )


21/21

4" -- 45.4C

(iii) D> . 45.4C

— 45.4C

(tan.sec thin)

1ro7 (ii)

(iv)'1i . "5."A (tan.sec thin)

(2 M%>)' . "5."A 1vo7 (iii)

U — hi1'

N 4 N%U ' . "5."A

* --+, F.FA!

K

2013 Bukit Panjang SS Copy

Documents

Transcript of 2013 Bukit Panjang SS Copy