CONFIDENTIAL* 2014-2-KELANTAN-SMJKChungHuaKB_MATHS QA

Section A [45 marks]

Answer all questions in this section.

1. The function g is defined by

g(x) = {(𝑥 − 3)2, 𝑥 ≤ 3

1 −𝑎

𝑥 , 𝑥 > 3

(a) Given that g(x) is continuous at x = 3, find the value of a. [3 marks]

(b) Sketch the graph of y = g(x). [3 marks]

2. The equation of a curve is

x2y + xy2 = 2.

Find the equation of both the tangents to the curve at the point x = 1.

[9 marks]

3. Using an appropriate substitution, evaluate

1

0

3

1

2 .d)1( xxx

[7 marks]

4 Show that the differential equation

0d

d 2 yxyx

yx

may be reduced by the substitution y = vx to the equation

.0d

d 2 xvx

v

Hence, find y in terms of x, given that y = 1 when x = 1. [7 marks]

5. If y2 = 1 + sin x, show that

.01d

d2

d

d2 2

2

2

2

y

x

y

x

yy

Deduce an equation which has the term in 3

3

d

d

x

y. Hence, obtain the expansion

of xsin1 in ascending powers of x up to the term in x3.

[10 marks]

STPM 954/2 [Turn over

* This question paper is CONFIDENTIAL until the examination is over.

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6. Use the trapezium rule with 5 ordinates to find in surd form, an

approximate value for the integral

2

0

.d3 xx

and deduce that ln 3 ).32(4 [6 marks]

Section B [15 marks]

Answer any one questions in this section.

7. State the equations of the asymptotes of the curve

.

)1(

12

2

xy

[2 marks]

Hence sketch the graph of .)1(

12

2

xy By drawing an appropriate line, find

the number of real roots of the equation 2 – 2)1(

1

x= 3x. [7 marks]

Taking x = 0.2 as the first approximation, use the Newton-Raphson method

to find the approximate root of the equation 2 – 2)1(

1

x= 3x correct to 2 decimal

places. [6 marks]

8. In a chemical reaction, substance A is converted to substance B. Throughout

the reacton, the total mass of substance A and the substance B is a constant and

equal to m. The mass of substance B at the time t minutes after the start of the

chemical reaction is x.

At any instant, the rate of increase of the mass B is directly proportional to the

mass of A. Write a differential equation that connects x with t. [2 marks]

Solve this differential equation given that x = 0 when t = 0. [3 marks]

Given that 4

3x m when t = 8 ln 2, show that

t

mx 4

1

e1 . [3 marks]

Hence

(a) find the value of x, in terms of m when t = 8 ln 3. [2 marks]

(b) find the least value of t for which x exceeds 99% of m, [2 marks]

(c) sketch the graph of x against t. [3 marks]

STPM 954/2 [Turn over

* This question paper is CONFIDENTIAL until the examination is over.

CONFIDENTIAL* MARKING SCHEME

Section A [45 marks]

Answer all questions in this section.

1. The function g is defined by

g(x) = {(𝑥 − 3)2, 𝑥 ≤ 3

1 −𝑎

𝑥 , 𝑥 > 3

(a) Given that g(x) is continuous at x = 3, find the value of a. [3 marks]

(b) Sketch the graph of y = g(x). [3 marks]

1 (a)

311lim)(lim

33

a

x

axg

xx

03lim)(lim2

33

xxg

xx = g(3)

B1

g is continuous

31

a = 0 M1

a = 3 A1 [3]

(b)

D1(shape for interval (, 3))

D1(shape for interval (3, ) with

asymptote)

D1(perfect with (0,9) and (3, 0)

and label asymptote, y = 1)

[3]

2. The equation of a curve is

x2y + xy2 = 2.

Find the equation of both the tangents to the curve at the point x = 1.

[9 marks]

2 x2y + xy2 = 2

02)2( 22

dx

dyyxyxy

dx

dyx M1A1

x = 1, y2 + y – 2 = 0 (y – 1)(y + 2) = 0 M1

y = 1, –2 A1

At (1, 1): 0212

dx

dy

dx

dy

dx

dy = –1 B1

At (1, 2):

dx

dy = 0 B1

The equation of tangent at (1, 1): y – 1 = (x – 1)

y = x + 2

The equation of tangent at (1, –2): y = –2

M1

A1

A1 [9]

3. Using an appropriate substitution, evaluate

1

0

3

1

2 .d)1( xxx

[7 marks]

3 Let u = 1 – x du = dx B1

x = 1, u = 0; x = 0, u = 1 B1

0

1

3

1

21

0

3

1

2 d)()1(d)1( uuuxxx M1

0

1

3

7

3

4

3

1

d2 uuuu

A1

0

1

3

10

3

7

3

4

10

3

7

6

4

3

uuu

M1

10

3

7

6

4

30

M1

140

27

A1 [7]

4 Show that the differential equation

0d

d 2 yxyx

yx

may be reduced by the substitution y = vx to the equation

.0d

d 2 xvx

v

Hence, find y in terms of x, given that y = 1 when x = 1. [7 marks]

4 y = vx

dx

dvxv

dx

dy B1

0

d

d 2 yxyx

yx 0)( 2

vxvxx

dx

dvxvx …

0d

d 2 xvx

v

M1

A1

xdxvv

d12

M1

c

x

v

2

1 2

cx

y

x

2

2

A1

When x = 1, y = 1 : 1 =

2

1+ c c =

2

1 M1

2

1

2

2

x

y

x

1

22

x

xy A1 [7]

5. If y2 = 1 + sin x, show that

.01d

d2

d

d2 2

2

2

2

y

x

y

x

yy

Deduce an equation which has the term in 3

3

d

d

x

y. Hence, obtain the expansion

of xsin1 in ascending powers of x up to the term in x3.

[10 marks]

5 y2 = 1 + sin x x

dx

dyy cos2 B1

x

dx

dy

dx

ydy sin22

2

2

2

M1

2

2

2

2

122 ydx

dy

dx

ydy

0122 22

2

2

y

dx

dy

dx

ydy

A1

02422

2

2

2

2

3

3

dx

dyy

dx

yd

dx

dy

dx

yd

dx

dy

dx

ydy

032

2

3

3

dx

dyy

dx

yd

dx

dy

dx

ydy

M1

A1

y(0) = 1, y’(0) =

2

1 B1

y”(0) =

4

1, y”’(0) =

8

1 B1B1

y =

32

!3

8

1

!24

1

2

11sin1 xxxx

M1

32

48

1

8

1

2

11sin1 xxxx A1 [10]

6. Use the trapezium rule with 5 ordinates to find in surd form, an

approximate value for the integral

2

0

.d3 xx

and deduce that ln 3 ).32(4 [6

marks]

6 )3333(2912

1

2

1d3

2

0

xx

M1

324

A1

3ln

83

3ln

1ln3)d3(

3ln

1d3

2

0

2

0

2

0

xxx xx

B1

3ln

8324

M1

ln 3

32

32

32

4

M1

ln 3 )32(4 A1 [6]

Section B [15 marks]

Answer any one questions in this section.

7. State the equations of the asymptotes of the curve

.

)1(

12

2

xy

[2 marks]

Hence sketch the graph of .)1(

12

2

xy By drawing an appropriate line, find

the number of real roots of the equation 2 – 2)1(

1

x= 3x. [7 marks]

Taking x = 0.2 as the first approximation, use the Newton-Raphson method

to find the approximate root of the equation 2 – 2)1(

1

x= 3x correct to 2 decimal

places. [6 marks]

7 Equations of asymptotes: x = 1, y = 2 B1B1 7

The two graphs intersect at only one point,

there is only one real root.

D1(correct shape with

two asymptotes)

D1(label x = 1, y = 2)

D1(any two points:(0, 1),

0,

2

11 ,

0,

2

11 )

D1(perfect)

D1(graph y = 2x)

M1

A1

Let 23

)1(

1)(

2

x

xxf

3)1(

2)('

3

xxf

B1

x0 = 0.2,

1765.09063.6

1625.02.0

)2.0('

)2.0(2.01

f

fx 0.18 M1A1

1759.0

)1765.0('

)1765.0(1765.02

f

fx 0.18 M1A1

x 0.18 A1 [15]

8. In a chemical reaction, substance A is converted to substance B. Throughout

the reacton, the total mass of substance A and the substance B is a constant and

equal to m. The mass of substance B at the time t minutes after the start of the

chemical reaction is x.

At any instant, the rate of increase of the mass B is directly proportional to the

mass of A. Write a differential equation that connects x with t. [2 marks]

Solve this differential equation given that x = 0 when t = 0. [3 marks]

Given that 4

3x m when t = 8 ln 2, show that

t

mx 4

1

e1 . [3 marks]

Hence

(a) find the value of x, in terms of m when t = 8 ln 3. [2 marks]

(b) find the least value of t for which x exceeds 99% of m, [2 marks]

(c) sketch the graph of x against t. [3 marks]

8 Mass of B = x, mass of A = m – x

)( xmdt

dx )( xmk

dt

dx , where k is a constant

M1A1

kdt

xm

dx

)( M1

ln (m – x) = kt + c, where c is a constant

When t = 0, x = 0: ln m = c: ln (m – x) = kt – ln m

A1

kt

m

xm

e )e1( ktmx A1

When t = 8 ln 2, x = ¾ m: )2ln8(e4

14

3

k

m

mm

ln 4 = (8 ln 2)k 4

1k

M1

A1

)e1( 4

1t

mx

A1

(a) When t = 8 ln 3: mmx

9

8)e1(

)3ln8(4

1

M1A1

(b) mx

100

99

100

99)e1( 4

1

t

M1

01.0e 4

1

t

01.0ln4

1 t t > 18.42 t = 19 minutes A1

(c)

D1(shape with asypmtote)

D1(label: x = m)

D1(perfect with point (8ln2, ¾ m)

[15]

CONFIDENTIAL*