Post on 02-May-2017
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) = {(๐ฅ โ 3)2, ๐ฅ โค 3
1 โ๐
๐ฅ , ๐ฅ > 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.
CONFIDENTIAL*
CONFIDENTIAL*
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 โ 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 โ 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) = {(๐ฅ โ 3)2, ๐ฅ โค 3
1 โ๐
๐ฅ , ๐ฅ > 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 โ 2 = 0 (y โ 1)(y + 2) = 0 M1
y = 1, โ2 A1
At (1, 1): 0212
dx
dy
dx
dy
dx
dy = โ1 B1
At (1, 2):
dx
dy = 0 B1
The equation of tangent at (1, 1): y โ 1 = (x โ 1)
y = x + 2
The equation of tangent at (1, โ2): y = โ2
M1
A1
A1 [9]
3. Using an appropriate substitution, evaluate
1
0
3
1
2 .d)1( xxx
[7 marks]
3 Let u = 1 โ 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 โฆ
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, yโ(0) =
2
1 B1
yโ(0) =
4
1, yโโ(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 โ 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 โ 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 โ x
)( xmdt
dx )( xmk
dt
dx , where k is a constant
M1A1
kdt
xm
dx
)( M1
ln (m โ x) = kt + c, where c is a constant
When t = 0, x = 0: ln m = c: ln (m โ x) = kt โ ln m
A1
kt
m
xm
e )e1( ktmx A1
When t = 8 ln 2, x = ยพ 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, ยพ m)
[15]
CONFIDENTIAL*