1
2007
m 1 4
a)
b) h : x x 1
m3
6
f ( x) x 2 2 x 4 2
2
2
2
x 2 2 x ( )2 ( )2 4 ( x 1) 1 4 2
( x 1)2 5 Compare with f ( x) ( x m) 2 n Thus, m 1, n 5
f ( x) x 3
2
7 8b
8b
c log 4 ( ) c log 2 4
When, f ( x) 5 x 3 5
or
x 3 5
x 3 5
x 8
x 2
3
log 2 (
h( x) ax b
)
log 2 8 log2 b log 2 c 2 3 x y 2
9(3n 1 ) 27 n
2
h ( x) h( ax b)
h( ax b) b
a( ax b) b
a 2 x ab b
8
32 3n
1
33 n
2 n 1 3n 1 2n n
1 2
2
h ( x ) 3 6 x 35 a2
36
sin ce, a 0, a 6 ab b 35 6b b 35 7b 35 b 5 thus, a 6, 6 , b 5 4
3 x 2 5 x 2 0
a) (3 x 1)( x 2) 0 x
1 3
16 x,8 x,4 x,... 8 x
9
16 x
or 2
b) hx 2 kx 3 0 For two equal roots, b 2 4ac 0
1 4x , 2 8x
1 2
a) The sequence is a geometric progression
k 2 4h(3) 0 k 2 12h 0 h 5
2 x
2
1
b) The sequence has a common ratio of 1/2
k 2 12
x
2 x 2 x 1 0 (2 x 1)( x 1) 0 The range is
1
10 Arithmetic progression : 5-x, 8, 2x 8-(5-x)=2x-8 3+x=2x-8 x=11 common difference = 3+x =3+11 =14
x 1
2
11 Geometric progression : 27, 18, 12 18 2 a=27 and r= = 27 3 a 27 27 S 81 2 1 1 r 1 3 3
12
y 2 2 x(10 x) ( x) :
y 2 x
20 2 x
At point (p,0), x=p and
y 2 x
q
q 20 2(3)
14 13
x
y
1: 6 h y-intercept = 2 Thus, h=2 y int ercept Gradient = x int ercept For the line
2 6 1
3 For the line y+kx=0 y=-kx The two lines are parallel. 1 Thus, -k= 3 1 1 k k 3 3
15 OD = 3DB 3 OD = OB 4 3 (OA OC ) 4
14 Area of
2
∆ABC=30unit
1 5 4 p 5 2 2 6 2 2
30
30 8 2 p 8 6 p 10 60
24 4 p 60 24 4 p 60
4 p 84 p 21
24 4 p 60 4 p 36 p 9
or 24 4 p 60 4 p 84 p 21 Thus, p=-9 or 21 16 2 1 2a b 2 8 4
5 12
3
Unit vector in the direction of 2a b
3
(9 x 5 y) (9 x 5 y)
4 27 4
x
15 4
y
4 27 4
x
15 4
y
17 cot x + 2 cos x = 0 cos x + 2 cos x = 0 sin x cos x + 2 sin x cos x = 0 cos x (1+2 sin x) = 0 cos x = 0 x= 90 ,270 or 1+2 sin x=0 1 sin x= 2 x= 210 ,330 Thus, x = 90 ,210 ,270 ,330
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
18
5i 12 j
2
5 12 5i 12 j
2
5 i 12 j
52 12 2 5i 12 j
13 13 a) Length of arc BC =
