Page 65 - ACE YR IGCSE A TOP APPR' TO ADD MATH

4
25. V = πr 2a – 4 = 0
3
3 a 2
dV = 4πr 2 2a = 4
3
dr a = 2
3
dr = dr × dV a = √2
3
dt dV dt [2]
2
= 1 × 18π (b) d y = 2 + 8
4π(0.5) 2 dx 2 x 3
= 18 cm s When x = √2 ,
3
–1
[3] d y 8
2
= 2 +
3
26. (a) 5x h = 500 dx 2 (√2) 3
2
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h = 500 = 2 + 4
5x 2 = 6
= 100 6 . 0
x 2 ( √2 , 2) is a minimum point.
3
A = 5x + 2(xh) + 2(5xh) [3]
2
4
1
1
2 2
2 2
= 5x + 2x 100 + 10x 100 (c) y = 2x – dx
2
x
2
x
x
200 1000 2x 2 4x –1
2
= 5x + + = – + c
x x 2 –1
4
2
= 5x + 1200 = x + + c
x
2
x [3] When x = √2 , y = 2.
3
(b) dA = 10x – 1200x 3 4
–2
2
dx (√2) + 3 + c = 2
1200x = 10x 3 √2
–2
3
x = 120 (√2) + 4 + c = 2
3
3
x = 4.932 √2 6
c = 2 – 3
A = 5(4.932) + 1200 √2
2
4
4.932 y = x + + 2 – 6
2
= 364.93 cm 2 x 3 √2 [4]
[3]
(c) Minimum cost = $20 dy
[1] 2. (a) When x = 1, dx = 0.
2(1) – k
1
27. (a) dy = 2x(2x – 1) + x (3x )(2x – 1) 1 2 √1 = 0
2
3
2
2
3
dx k = 2
= 2x√2x – 1 + 3x 4 [1]
3
√2x – 1 (b) y = 2x – 2 dx
3
3
= 2x(2x – 1) + 3x 4 √x
√2x – 1 = 2x – 2 dx
3
4
= 7x – 2x √x √x
3
√2x – 1 3 1
A = 7, B = –2 = 2x 2 – 2x 2 + c
[4] 3 1
2 2
2 x(7x – 2) 3
3
4
(b) = x – 4√x + c
2
1 √2x – 1 3
3
3
4
= x √2x – 1 4 2 = √x – 4√x + c
2
3
3
1 3
= 14.49
[2] When x = 1, y = 2.
4 √1 – 4√1 + c = 2
3
Integration 3 14
c = 3
1. (a) At (a, 2), dy = 0 4 14
dx y = √x – 4√x +
3
3 3 [4]
Answers 205
Answers Add Math.indd 205 14/03/2022 12:29 PM