2009 prelims A-Math 4EX/5NA P1 Solutons 1a
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1b
5 11 3( x 1) 3( x 2) dy 5 11 2 dx 3( x 1) 3( x 2) 2 dy 13 x 1, dx 4 4 AOB 9 BOC 9
2a
y
2b Shaded region =
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+
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( 4) 2 1 4 1 1 4 (4)(4) sin (4)(4) sin (4)(4) sin 2 9 2 9 2 9 2 6.64cm 2
3a
9 13 Mid-pt = , 2 2 m 3 13 9 3 c 2 2 c 20 y 3x 20
3b
20 P ,0 , Q 0,20 3 2
20 PQ 20 21.1units 3 20 10 0 9 0 A= 3 2 5 20 8 0 5 2
3c
4a
1 100 160 180 2 3 3
100units 2 log 8 x p 8p x 23 p x log 2 x 3 p
4b
log 2 y 2 log 2 2 4 log 2 ( y 5) y 2 16( y 5) y 2 16 y 80 0 ( y 4)( y 20) 0 y 4 or y 20
4c
2 3 x 3 2 2 x 8 2 5 x 5 20(2 5 x 1 )
2 5 (5 x ) 72(5 x )
4 9 2 3
5
x 2 y 1, log 2 ( y x) log 2 4 log 2 ( x 4) yx
x4 4 y 5 x 16. x 3, y 1 6a 6b
6c
Gradient = 2
k 4
1 k 2 2 4 k 6 x2 6 y c, 2 x c 4 x2 6 4 2 x A (0,7),
y
7
x 2 4x 7 0 x 3. B (3,4). Area =
+ 3
x 2 4 x 7 dx 0
3
1 4 4 2
x3 2 x 2 7 x 8 3 0 20units 2
8
9a
1 4 2 10 3 1 a 1 4 2 5 b 10 3 1 5 a 3, y 1 dy sin x(sin x cos x ) cos x(cos x sin x) dx (sin x cos x) 2 A 1
sin 2 x sin x cos x cos 2 x sin x cos x (sin x cos x ) 2 1 (sin x cos x) 2 k 1
3 0
3 1 3 dx 3 dx 2 0 (sin x cos x ) (sin x cos x) 2
3 1 3 1 23 0 1 2 3 3 1 3 1 3 1 3 9b
4
1
33 3 2
6
6
f ( x )dx f ( x )dx f ( x )dx 7 1
4
4
4
4
4.5 1.8 f ( x )dx 4.5dx 1.8 f ( x)dx 1
1
1
4.5 x 1.87 4 1
0.9
10a 10b
ds 3t 2 10t 2 dt a 6t 4
v
v adt 3t 2 4t c t 1, v 4 c 3 3t 2 4t 3. s vdt t 3 2t 2 3t 6 10c
t 3 2t 2 3t 6 t 3 5t 2 2t 4 3t 2 t 2 0 2 t s. 3 v P ive, vQ ive. same direction.
11
y
y = 1 3 sin 2 x
y 4
4x
x From graph, 5 solutions.