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3. y = x ln x – x 6. (a) f'(x) = 10ax – 5x –6
4
2
dy = x + 2x ln x – 1 By comparison,
dx 10a = 5
1
2 1 2 2 1 1 a =
x ln x dx = 2x ln x + x – 1 dx – x – dx 2
0 2 0 2 0 2 2 2 –25b = –5
1
4 3
= 1 3 x ln x – x – 1 x – x 4 b = 1
2
2
2 0 4 2 0 5 [3]
1
4
= 1 [(2) ln 2 – 2] – 3 1 (2) – (2)
2
2
4
2 4 2 (b) f'(x) = 5x – 5x
–6
= ln 4 – 1 When f'(x) = 0,
4
–6
[5] 5x = 5x
x
4
2
4. dy = 2x(2 – x) – x (–1) x –6 = 1
dx (2 – x) 2 x = 1
10
2
= 4x – 2x + x 2 x = ±1
4 – 4x + x 2
–x + 4x When x = 1,
2
=
(x – 2) 2 y = 1 + 1
5
= x(4 – x) = 2 1 5
(2 – x) 2
3 3 When x = –1,
x(4 – x) dx = 3 x 2 4
(2 – x) 2 2 – x y = (–1) + 1
5
0 0 (–1) 5
= 9 – 0 = –2
2 – 3 The turning points are (1, 2) and (–1, –2).
= –9 [4]
[4] 7. (a) When x = a, dy = 0.
5. dy = 4x + 4 dx 3 dx
3
dx a – 8 = 0
2a
2
4x
4
+ 4x + c
= 5 Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
a = 8
3
4
dy = 3 when x = –1. a = 2
dx d y 6x – 4x(x – 8) [2]
4
3
2
(–1) + 4(–1) + c = 3 (b) dx 2 = (2x )
4
2 2
–3 + c = 3 = 6x – 4x + 32x
4
4
c = 6 4x 4
dy = x + 4x + 6 = 2x + 32x
4
4
dx 4x 4
y = x + 4x + 6 dx When x = 2,
4
d y
4
2
5
2
= x + 4x + 6x + d dx 2 = 2(2) + 32(2)
4(2)
4
5 2 3
1
When x = –1, y = . = 2
2 3 . 0
(–1) 5 + 2(–1) + 6(–1) + d = 1 2
2
5 2 (2, 3) is minimum point.
d = 47 [4]
10 (c) y = x – 8 dx
3
x
y = + 2x + 6x + 47 2x 2
2
5 10 x 3 8
When x = 3, = 2x 2 – 2x 2 dx
x
y = 3 5 + 2(3) + 6(3) + 47 = – 4 dx
2
5 10 2 x 2
= 89.3 = x 2 + + c
4
[6] 4 x
Cambridge IGCSE
TM
206 Ace Your Additional Mathematics
Answers Add Math.indd 206 14/03/2022 12:29 PM