Page 53 - ACE YR IGCSE A TOP APPR' TO ADD MATH

19. (a) 1 = 3 tan x = – 1
sin (x – 25°) √3
1
sin (x – 25°) = x = 5π rad or 11π rad
3 6 6
x − 25° = 19.471°, 160.529° [4]
x = 44.47°, 185.53°
[4] 10 Permutations and Combinations
(b) 1 = 1
1
tan 2x + π 2 1. (a) 4 P 4 = 1
3
1
tan 2x + π 2 = 1 5 3 P 5 5 1 [1]
3
P
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0 < x < 2π (b) 5 P 3 5 = 20 [1]
π
π < 2x + < 13π
3 3 3 2. (a) 4! × 5! = 2880
π 5π 9π 13π 17π [2]
2x + = , , ,
3 4 4 4 4 (b) (6! × P ) + (6! × P ) = 60480
5
4
3
3
x = 11π , 23π , 35π , 47π [3]
24 24 24 24 (c) 6! × 4! = 17280
x = 1.44 rad, 3.01 rad, 4.58 rad, 6.15 rad [2]
[4] 5 5 5 5 5
1 3. ( C × C ) + ( C × C ) + C = 155 [3]
1
2
3
4
2
20. sin q – cos q
cos q sin q 4. (a) 9! = 362880
= 1 – cos q (b) 8! × 4! = 967680 [1]
sin q cos q sin q
[2]
2
= 1 – cos q (c) 10! = 3628800
sin q cos q [2]
2
= sin q 8
sin q cos q 5. (a) P = 6720
5
= sin q (b) P = 840 [1]
7
cos q 4
= tan q 6720 – 840 = 5880
[4] [2]
3
6
(c) ( C × P ) × 2 + ( C × P ) × 2 = 1680
4
6
21. A = 4 1 3 1 3 [4]
B = 1.8
10
C = –3 6. (a) C = 120
7
[3] [2]
9
(b) C = 84
22. (a) 3 [1] 6 [2]
(b) 180° [1]
10
7. (a) C = 120
3
2
23. 2(cosec q − 1) + 3 + 3 cosec q = 0 [2]
9
2
2 cosec q − 2 + 3 + 3 cosec q = 0 (b) C = 36
2
2
2 cosec q + 3 cosec q + 1 = 0 [2]
(2 cosec q +1 )(cosec q + 1) = 0 8. ( C × C ) + C = 95
6
4
6
1
3
4
1 = – 1 or 1 = –1 [3]
sin q 2 sin q 9. 3! × 5! × 3! × 3! = 25920
sin q = –2 (invalid) or sin q = –1 [2]
q = 3π rad
2 10. C = 56
8
[4] 3 [2]
1
24. √3 e cos x = –e sin x 11. (a) 8 × 2! × 8! = 645120
x
2
x
2
x
2
e = 0 (invalid) [2]
sin x = – 1 (b) 10! – 645120 = 2983680
cos x √3 [2]
Answers 193
Answers Add Math.indd 193 14/03/2022 12:29 PM