Trial STPM Mathematics T1 PAHANG

CONFIDENTIAL*

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PEPERIKSAAN PERCUBAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA

NEGERI PAHANG DARUL MAKMUR 2010

Instructions to candidates:

Answer all questions.

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

CONFIDENTIAL*

2

Mathematical Formulae for Paper 1 Mathematics T / Mathematics S :

Logarithms :

a

xx

b

b

alog

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 :

dxdx

duvuvdx

dx

dvu

cxfdxxf

xf )(ln

)(

)('

ca

x

adx

xa

1

22tan

11

ca

xdx

xa

1

22sin

1

Series:

N n where

,

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 : n is

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

n

nnxf

xfxx

Trapezium rule :

b

ann yyyyyhdxxf ])(2[

2

1)( 1210

n

abhrhafyr

and where )(

Trigonometry :

BAAABA sincoscossin)sin(

BABABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

AAAAA 2222 sin211cos2sincos2cos

AAA 3sin4sin33sin

AAA cos3cos43cos 3

CONFIDENTIAL*

3

1. Given that i

aiz

34

5

where 1, 2 iRa ,

(a) simplify z into the form yix , where x and y are 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 22

5

3

5

2

55 loglogloglog yyyy if y = 5. [2 marks]

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

5

2

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 , where 3

7, kRk .

(a) Show that A is 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]

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 the y-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 p varies, the locus of the point S is the parabola )2(22 xy .

[3 marks]

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), express xx 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 substituting 27

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

decimal places. [3 marks]

11.(a) Determine dxxx

2

1

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 line 3

x .

(i) Use the trapezium rule with three ordinates to estimate

the area of region R correct to three decimal places.

[3 marks]

(ii) Find the exact volume of the solid generated when the

region R is revolves about the x-axis. [3 marks]

y

x R

3

O

y = sec x

Diagram 1

CONFIDENTIAL*

6

12. The polynomial nxmxxxf 8)( 23 , where m and n are 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 whether x + 3 is a factor of )(xf , [2 marks]

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

(d) express )('

1

xf as a sum of partial fractions. [3 marks]

END OF QUESTION PAPER

Trial STPM Mathematics T1 PAHANG

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