[Edu.joshuatly.com] Pahang STPM Trial 2010 Maths T [w Ans] [EC2C8333]

8/12/2019 [Edu.joshuatly.com] Pahang STPM Trial 2010 Maths T [w Ans] [EC2C8333]

1/28

CONFIDENTIAL*

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PEPERIKSAAN PERCUBAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA

NEGERI PAHANG DARUL MAKMUR 2010

Instructions to candidates:

Answer allquestions.

Answers may be written in either English or Bahasa Malaysia.

All 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 in degrees, unless a different level of accuracy

is specified in the question.

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

This question paper consists of 5 printed pages.

950/1, 954/1 STPM 2010

Three hours

MATHEMATICS S

PAPER 1

MATHEMATICS T

PAPER 1

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2/28

CONFIDENTIAL* 2

Mathematical Formulae for Paper 1 Mathematics T / Mathematics S :

Logarithms :

a

xx

b

ba

log

loglog

Series :

)1(2

1

1

nnrn

r

)12)(1(6

1

1

2

nnnrn

r

22

1

3 )1(4

1

nnrn

r

Integration :

dxdxdu

vuvdxdx

dvu

cxfdxxf

xf )(ln)(

)('

ca

x

adx

xa

1

22 tan

11

ca

xdx

xa

1

22sin

1

Series:

Nnwhere

,

21)( 221 nrrnnnnn bba

r

nba

nba

naba

1,!

)1()1(

!2

)1(1)1( 2

xx

r

rnnnx

nnnxx rn where

Coordinate Geometry :

The coordinates of the point which divides the line joining (x1,y1) and (x2,y2) in the

ratio m: nis

nm

myny

nm

mxnx 2121 ,

The distance from ),( 11 yx to 0 cbyax is

22

11

ba

cbyax

Numerical Methods :

Newton-Raphson iteration for 0)( xf :

)('

)(1

n

nnn

xf

xfxx

Trapezium rule :

b

a nn yyyyyhdxxf ])(2[

2

1)( 1210

n

abhrhafyr

andwhere )(

Trigonometry :

BAAABA sincoscossin)sin(

BABABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

AAAAA 2222 sin211cos2sincos2cos AAA 3sin4sin33sin AAA cos3cos43cos 3



3/28

CONFIDENTIAL* 3

1. Given thati

aiz

34

5

where 1, 2 iRa ,

(a) simplify z into the form yix , wherexandyare real numbers, [2 marks]

(b) find the value of a if z is a real number. [2 marks]

2. The real function f is defined by

1,45

1,

1,

)( 2

22

xxk

xk

xxk

xf .

Find the values of k if f is continuous everywhere. [4 marks]

3.(a) Evaluate 2253

5

2

55 loglogloglog yyyy ify= 5. [2 marks]

(b) Find the value of y in surd form if 3)(log)(loglog 352

55 yyy . [3 marks]

4.(a) Solve the equation 14 xx . [2 marks]

(b) On the same axes, sketch the graphs of xy 4 and 1 xy . [2 marks]

(c) Use the information in (a) and (b), find the solution set of the inequality 14

1

x

x.

[3 marks]

5. It is given that matrix

311

21

302

kA , where3

7, kRk .

(a) Show thatAis non-singular. [2 marks]

(b) Find the inverse of A if k=1 . [4 marks]

(c) Use the result in (a), solve the system of linear equations

73 zyx

xzy 2

123 xz . [3 marks]



4/28

CONFIDENTIAL* 4

6. The real functions f and g are defined as follows :

0,:

0,ln2

1:

xxxg

xxxf

(a)Use the sketch graph of function g, explain why the inverse function 1g exists.[3 marks]

(b) Find 1g and state its domain. [2 marks]

(c) Show that f is an increasing function. [2 marks]

(d) State with reason, whether the composite function fg

is defined. [2 marks]

7.(a) Given )34ln(32 2 xey x , find

dx

dy. [2 marks]

(b) The equation of a curve is given by xyxy 4ln2 .

(i) Show that the first derivative of y with respect to x is

)2(2

14

xyx

xy

. [3 marks]

(ii) Find the equation of the normal to the curve at the point (1 , 4). [4 marks]

8. (a) Show that, for all values of p, the point )4,2( 2 ppP lies on the parabola xy 82 .

[2 marks]

(b) Find the equation of the tangent to the parabola xy 82 at the point )4,2( 2 ppP [3 marks]

(c) The tangent to the parabola xy 82 at the point )4,2( 2 ppP meets they-axis at Q.The quadrilateral PQRS is a parallelogram. Given R is the fixed point (-2 , 0),

(i) find the coordinates of S in terms of p, [2 marks](ii) show that as pvaries, the locus of the point S is the parabola )2(22 xy .

[3 marks]



5/28

CONFIDENTIAL* 5

9. (a) By sketching the graph of xy sin3 for 44 x and another suitable graph

on the same axes, show that the equation xecx

cos3 , x in radian , has only one

positive root. [4 marks]

(b) Verify that the positive root of xecx

cos3 lies between 2.2 and 2.3. [2 marks]

(c) Use Newton-Raphson method, find the positive root of xecx

cos3 correct to

three significant figures. [4 marks]

10.(a) Show that

321

16

1

8

1

4

1

2

1

)2( xxxx . [2 marks]

(b) Use the result in (a), expressxx 31)2(

4

in ascending power of x up to and

including the terms in 2x [3 marks]

(c) State the range of values of x for which the expansion in (b) is valid. [2 marks]

(d) By substituting27

1x , find the approximate value of 2 correct to three

decimal places. [3 marks]

11.(a) Determine dxxx

21

2 )2(2 . [2 marks]

Hence, use integration by parts, find dxx

x

22

3

. [3 marks]

(b) Diagram 1 shows the region R, bounded by the curve

xy sec , the x-axis, the y-axis and the line3

x .

(i) Use the trapezium rule with three ordinates to estimatethe area of region R correct to three decimal places.

[3 marks]

(ii) Find the exact volume of the solid generated when theregion R is revolves about the x-axis. [3 marks]

y

xR

3

O

y= sec x

Diagram 1



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CONFIDENTIAL* 6

12. The polynomial nxmxxxf 8)( 23 , where mand nare constants, has a

remainder 2 when divided by x + 2.

Given )(' xf is the first derivative of )(xf with respect to x and 2 is the zero of

)(' xf ,

(a) show that m= 5 and find the value of n, [4 marks]

(b) determine whetherx+ 3 is a factor of )(xf , [2 marks]

(c) show that 0)( xf has only one real root, [3 marks]

(d) express)('

1

xfas a sum of partial fractions. [3 marks]

END OF QUESTION PAPER



7/28

CONFIDENTIAL*




















PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN

JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP

PEPERIKSAAN PERCUBAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA

NEGERI PAHANG DARUL MAKMUR 2010

Instructions to candidates:

Answer allquestions. Answers may be written in either English or Malay.

All 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 in degrees, unless a different level of accuracy is

specified in the question.

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

This question paper consists of printed pages.

954/2 STPM 2010

Three hours

MATHEMATICS T

PAPER 2



8/28


9/28

CONFIDENTIAL* 3

Mathematical Formulae for Paper 2 Mathematics T :

Numerical Methods :

Newton-Raphson iteration for 0)( xf :

)('

)(1

n

nnn

xf

xfxx

Trapezium rule :

b

a nn yyyyyhdxxf ])(2[

2

1)( 1210

n

abhrhafyr

andwhere )(

Correlation and regression :

Pearson correlation coefficient:

22

yyxx

yyxxr

ii

ii

Regression line of yon x:

y= a + bx

where

2i

ii

xx

yyxxb

xbya

Trigonometry

BAAABA sincoscossin)sin( BABABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

AAAAA 2222 sin211cos2sincos2cos AAA 3sin4sin33sin AAA cos3cos43cos 3

2

BAcos

2

BAsin2Bsi nAsin

2

BAsin

2

BAcos2Bsi nAsin

2

BAcos

2

BAcos2BcosAcos

2

BAsin

2

BAsin2BcosAcos



10/28

CONFIDENTIAL* 4

1. (a) Solve the equation

for 0 180 [3]

(b) Express in the form ,where a, band care constants. [4]

2. (a) Express in the form , where R is a

positive constant and an acute angle. [2]

(b) Hence, find the maximum and minimum values for

and their corresponding angles.

Sketch of the graph in the interval 0 360 . [4]

Hence, find the set of values of , with 0 360, which0 . [4]

3. The diagram given shows two intersecting circles ABC and CDEF, XY is a tangent to

the circle ABC at C, while BC is a tangent to CDEF at C.

Prove that , ABC is similar to DCF. [6]

4. The position vectors of three points A , B and C relative to the origin O

are , and respectively.

(a) Find the scalar product of . [2]

(b) Hence , find the ABC and the area of triangle ABC. [4]



11/28

CONFIDENTIAL* 5

5. When a valve is released, water flowed into a large tank that is initially empty.

The volume, v litres , in the tank increase at the rate

where t is measured in hours from the time the valve is released.

(a) What is the initial rate of the water entering the tank ? [1]

(b) Find an expression for v in terms of t. [4]

(c) Determine the value of v when t is 40 minutes. [2]

(d) Find the time taken, to the nearest minute, for the volume of water in the

tank to be 4 litres. [4]

6. Two cyclist, A and B are initially 25 km apart with B on a bearing of N 67E from A.A is moving at 18 km/h in a direction S 20E and B is moving at 12 km/h due south.The velocities of A and B remain unchanged.

(a) Find the velocity of A relative to B. [5]

(b) Find the nearest distance between the two cyclists. [3]

(c) Find the time when the distance between the two cyclists is the nearest. [2]



12/28

1

Marking Scheme

PEPERIKSAAN PERCUBAAN STPM NEGERI PAHANG 2010

Mathematics T Paper 1 (954/1)/ Mathematics S Paper 1 (950/1)

NO. SCHEME MARKS

1(a)

(b)

415

025

415

25

415

25

320

916

341520

)34)(34(

)34)(5(

34

5

a

a

iaa

aaii

ii

iai

i

ai

z

M1use conjugate to

make denominator

real

A1

M1

[A1 [4]

2

14

0)1)(4(

045

)1(45)1(

45limlim

)(lim)(lim

2

22

1

22

1

11

kork

kk

kk

kk

xkxk

xfxf

xx

xx

M1 A1

M1

A1 [4]

3(a)

253

)221)(22(2

1

22...321

5,log...logloglog 2253

5

2

55

yyyyy

M1

A1

(b)

4

4

3

5

5

55

5

5

3

5

2

55

125

5

4

3log

3log4

log33log

3

log1

log

3...)(log)(loglog

y

y

y

y

yyy

y

yyy

M1

A1

A1 [5]

http://edu.joshuatly.com/

http://www.joshuatly.com


13/28

2

NO. SCHEME MARKS

4(a)

3

1

5

1

0)13)(15(

01215

1216

14

2

22

orx

xx

xx

xxx

xx

M1

A1

(b)

B1

B1

(c)

xx

x

x

41

14

1

From the graph, xx 41 when5

1

3

1 xorx

Solution set: }5

1

3

1:{ xorxx

M1 A1

A1 [7]

5(a)

0..

73

73

37

73

)1(30)5(2

11

13

31

20

31

212

Aei

A

kwhen

k

k

kkA

A is non-singular

M1

A1

0

1

1

x

y xy 4

1 xy




14/28


15/28

4

NO. SCHEME MARKS

6(a)

The line y = c, c constant, cuts the graph of xxg )( at only one point

g is one-to-one and onto in 0x 1g exists.

B1

M1

A1

(b)

0,)(

0,

)(

)(

21

2

1

xxxg

xxy

yx

ygx

yxg

B1 B1

(c)

0)('

021

01

0

2

1)('

ln2

1)(

xf

x

xx

xxf

xxf

f is an increasing function.

M1

A1

(d)

gf

gf

DR

DR

),0[),(

),0[),(

function g o f is not defined.

M1

A1 [9]

7(a)

xxe

dx

dy x

34

34 32

2

M1 A1

(b)(i)

)2(2

14

1442

441

2

xyx

xy

dx

dy

x

xy

dx

dyx

dx

dyy

ydx

dyx

xdx

dyy

M1 A1

A1

y

x

xxg )(




16/28

5

NO. SCHEME MARKS

(ii)

17

4

4

17)24)(1(2

1)4)(1(4),4,1(

)2(2

14

normal

T

m

m

dx

dyat

xyx

xy

dx

dy

The equation of normal at (1,4)

072174

446817

)1(17

44

yx

xy

xy

M1

A1

M1

A1 [9]

8(a)

22

222

22

16)2(88

16)4(

)4,2(,8

ppx

ppy

ppPatxy

xy 82 satisfied by P(2p2, 4p)

P lies on the parabola xy 82

M1

A1

(b)

pm

pdx

dym

ppPat

ydx

dy

dx

dyy

xy

T

T

1

4

4

),4,2(

4

82

8

2

2

Equation of tangent

2

22

2

2

24

)2(1

4

pxpy

pxppy

pxp

py

M1

M1

A1




17/28

6

NO. SCHEME MARKS

(c) (i) At Q, x = 0

)2,22(

222

2

4

2

2

2

22

2

2

04,

2

22

2

2'

2

0

)0,2(),2,0(),4,2(),,(

)2,0(

2

2

2

2

2

2

2

2

ppS

pyandpx

ppyand

px

pppyx

MM

RpQppPyxS

pQ

py

ppy

PRQS

M1

A1

(c)(ii) As p varies, locus of P:

)2(2

22

22

2

2222

2

2

2

2

xy

yx

yx

py

pyandpx

B1

M1

A1 [10]

9(a)

xx

xx

ecxx

sin3

sin

13

cos3

y = 3 sin x and y = x

Only one intersection point for 0x

ecxx

cos3 has only one positive root.

B1Correct curve

M1Getting equation of

straight line

A1Correct straight line

seen

A1

-3

3

0

y

4 2 3 4 3 2 x

y = x

y = 3sinx




18/28

7

NO. SCHEME MARKS

(b)

00629.03.2)3.2sin(3)3.2(

02255.02.2)2.2sin(3)2.2(

sin3)(0sin3

sin3

cos3

f

f

xxxfxx

xx

ecxx

Since f(2.2) > 0 and f(2.3) < 0

the positive root lies between 2.2 and 2.3

M1

A1

(c)

28.2

2792.2

8845.2

0842.025.2

25.2('

)25.2(25.2

25.22

3.22.2

1cos3)('

sin3)(

2

1

f

fx

x

xxf

xxxf

28.2

2789.2

9519.2

0010.02792.2

)2792.2('

)2792.2(2792.23

f

fx

The positive root = 2.28 (to 3 s. f.)

M1

A1

A1

A1 [10]

10(a)

...16

1

8

1

4

1

2

1

...8

1

4

1

2

11

2

1

...2!3

)3)(2)(1(

2!2

)2)(1(

21

2

1

212)2(

32

3

32

1

11

xxx

xxx

xxx

xx

M1

A1




19/28

8

NO. SCHEME MARKS

(b)

...4

2322

...2

1

2

3

4

2732

...8

27

2

31...

2

12

...)3(!2

2

3

2

1

)3(2

1

1...16

1

8

1

4

1

2

1

4

)31()2(431)2(

4

2

222

22

232

2

1

1

xx

xxxxx

xxxx

xxxxx

xxxx

M1Expansion of

21

31

x

A1

Expand 21

31 x

correctly

A1

(c) The expansion is valid when

3

1

3

1:

3

1

3

12

1312

xx

x

xandx

xandx

M1

A1

(d)

27

1

x

.).3(414.1

4137.1

81

55

2916

60712

2916

6071

55

2812916

6071

2255

3274

2916

6071

9

8

27

55

4

27

1

4

23

27

122

27

131

27

12

4 2

pd

M1

A1

A1 [10]




20/28

9

NO. SCHEME MARKS

11(a)

cx

cx

dxxx

2

1

2

2

1

2

2

1

2

)2(2

2

1

)2()2(2

cx

xx

dxxxxx

dxxxxx

dxxxx

dxxxdxx

x

2

3

)2(2

)2(22

)()2(2)2(22

1

)2)(2(2

1

)2(2

2

3

222

2

1

222

2

1

22

1

22

2

1

22

2

1

23

2

3

= cxxx 23

222 )2(3

22

M1

A1

M1

A1

A1

(b)(i)

62

03

h

x y = sec x01 x 11y

62

x

2y =1.1547

33

x 3y = 2

Area of R

2

321

3

0

390.1

2)1547.1(2162

1

2

2

1

sec

unit

yyyh

dxx

B1

M1

A1




21/28

10

NO. SCHEME MARKS

(b)(ii) Volume

3

30

3

0

2

2

3

0tan3

tan

tan

sec

unit

x

dxx

dxy

M1

A1

A1 [11]

12(a)

5

08412

08)2(2)2(3)2('

823)('

2

2

m

m

mf

mxxxf

6

216208

2)2(8)2(5)2()2( 23

n

n

nf

M1

A1

M1

A1

(b)

0

6)3(8)3(5)3()3( 23

f

x + 3 is a factor of f(x).

M1

A1

(c) 0)22)(3( 2 xxx

3x or

4

)2)(1(42

4022

2

2

2

acbxx

0222 xx has no real roots

f(x) has only one real root.

M1

M1

A1

(d)

)2(2

1

)43(2

3

)(

1

2

3421,0

2

1)46(1,2

)43()2(1

2431

)2)(43(

1

8103

12

xxxf

ABAx

BBx

xBxA

xB

xA

xxxx

M1

M1

A1 [12]http://edu.joshuatly.com/



22/28

11




23/28

1

Marking Scheme

PEPERIKSAAN PERCUBAAN STPM NEGERI PAHANG 2010

Mathematics T Paper 2 ( 954 / 2 )

Question Scheme Marks

1(a)

***

,

= 90, 180 , =90= 180

= 90 , 180 , 450 , 540=2543 , 5126 , 12834 , 15417

=2543 , 5126 , 90, 12834 , 15417 , 180.

M1

Factor

Formula

M1

A1

1 (b) M1

M1

M1

A1



24/28


25/28

3

2

The solution set is

D1 [ draw

line y=1

and y=0 ]

A1


3.

Given XY is tangent to the ABC at C.

Given BC is a tangent to CDEF at C.

To prove ABC is similar to CDF.

Construction :

Join points F and D.

3ABC = ACX = [ Angles in the alternate segment. ]ACX = DCF = [ Vertically opposite angles ]ABC = DCF =

ACB = OCD = [ Vertically opposite angles ]OCD = CFD = [ Angles in the alternate segment ]ACB = CFD =

CAB = CDF = 180 ( + )[The sum of angles in a triangle is 180. ]Hence , , ABC is similar to DCF. ( AAA )

M1

A1

M1

A1

M1

A1



26/28

4


4 (a)

==

=

=

= +

= 36

M1

A1

4.(b)

or

M1

A1

Area of ABC = AB CB sin ABC

=

= or 13.499 or 13.5

M1

A1



27/28

5


5(a)

Initial time , t = 0 ,

= 5Initial rate is 5 liters per hour.

B1

5 (b)

M1 (separate variables)

M1( correct integration)

M1 ( use limits / find C )

A15 (c) t = 40 minutes , t = , t = hour

v = 3.8164 litres

M1

A1

5 (d) When v= 4 ,

***

,

t = ln 2 hours

t = 42 minutes

M1 ( Quadratic )M1 ( Factorise )

M1 ( value for )

A1


6(a)

D1 [ arrows ]

D1 [ shape ]



28/28

6

= 31 24The velocity of A and B is 7.88 km/h in the direction of

S 5124 E .

M1

[cosine rule]

M1

[sine rule]

A1

6(b)

Shortest distance = 25 sin 6136= 21.9912 km or 21.9 km

D1

M1

A1

6(c)Time =

Time = 1.5095 hour

M1

A1

Documents

[Edu.joshuatly.com] Pahang STPM Trial 2010 Maths T [w Ans] [EC2C8333]