Page 67 - ACE YR IGCSE A TOP APPR' TO ADD MATH

At (2, 3), Area = 0 0 2.5
1 2.5
2 2 4 2 0 0 4.375 0
+ + c = 3
4 2 1 175
c = 0 = 2 16
4
y = x 2 + 175
2
4 x [4] = 32 unit [7]
2
8. (a) 2x – = 3
x 3 3
2x – 3x – 2 = 0 10. 4x + – 5x dx
2
x
1
5
x = 2 or x = – (rejected) y = x + 3 ln x – x + c
2
4
2 2
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5
When x = 2, y = 1. 2 = 1 + 0 – + c
P(2, 1) c = 3.5 2
[3] y = x + 3 ln x – x + 3.5
5
2
4
2
(b) y = 2x – dx 2
x [4]
= x – 2 ln x + c
2
When x = 2, y = 1 11. 3x + 2x – 7 = 1
2
1 = 2 – 2 ln 2 + c 3x + 2x – 8 = 0
2
2
c = –3 + 2 ln 2 (x + 2)(3x – 4) = 0
y = x – 2 ln x + 2 ln 2 – 3
2
[4] x = –2 (rejected) or x = 4
3
9. (a) dy = 2x + 2 – 2x + 5 2
dx (x + 1) 2 y = 3x + 2x – 7 dx
= 7 = x + x – 7x + c
3
2
(x + 1) 2 4 3 4 2 4
+
The function is not applicable when dy = 0. 0 = 1 2 1 2 – 7 1 2 + c
3
3
3
dx
Therefore, the curve has no turning points. c = 140
[3] 27
(b) When y = 0, y = x + x – 7x + 140
3
2
27
2x – 5 = 0 [5]
x + 1 dy
2x = 5 12. (a) dx = sin x + x cos x
5 [2]
x =
2 (b) sin x dx + x cos x dx = x sin x
2
A 1 5 , 0
2
dy = 7 x cos x dx = x sin x – sin x dx
dx 1 5 + 1 2 2 π = x sin x + cos x
2
3
= 4 4 x cos x dx = x sin x + cos x 4 π 4
7 0 0
m = – 7 = 1.2625 – 1
 4 = 0.263
Equation of normal line: [3]
7
y = – x + 35 13. (a) dy = 8x
2 2
4 8 dx (3 – 4x ) [3]
When x = 0, 8 5x 1
7 35 (b) dx =
2 2
y = – (0) + 5 (3 – 4x ) 3 – 4x 2
4 8
= 35 5x dx = 5 + c
8 (3 – 4x ) 8(3 – 4x )
2 2
2
5
5
5
1
2 4
B 0, 35 2 3 8(3 – 4x ) 1 n = – – 24
8
8
= – 5
6 [3]
Answers 207
Answers Add Math.indd 207 14/03/2022 12:29 PM