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4. (a) 2x + 7x + 3 (b) 3x + 1
2
= (2x + 1)(x + 3) x – 5 3x + x – 15x – 5
3
2
2
2
3
1
1
When x = − , f – 1 2 = 0 3x + x – 15x
2
3
2 2 3x + x – 15x – 5
3
2
1
1
1
2 – 1 2 2 3 + A – 1 2 2 + B – 1 2 – 6 = 0 3x + x – 15x – 5
2
2
2
3
2
= 1 (3x + x − 15x − 5) ÷ (x − 5)
2 25 − A + 2B = 0 = 3x + 1
2
= √3 When x = −3, f(−3) = 0. f(x) = (3x + 1)(x − 5)
2(−3) + A(−3) + B(−3) − 6 = 0 [4]
3
2
= √(7) – 20 − 3A + B = 0 8. (a) f(−2) = (−2) − 3(−2) + 2
2
3
Solve the simultaneous equations to get: = 0 (shown)
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2
2
= 1 8 + 3 A = 3, B = −11 [6] (b) x + 0x – 2x + 1 [1]
2
2
3
1 (b) x + 2 x + 0x – 3x + 2
2
3
= 2 2 x – 2 x + 2x
3
2
2
3
2
= 2x + 7x + 3 2x + 3x – 11x – 6 3 2
2x + 7x + 3x x – 2x – 3x
2
3
3
2
– 4x – 14x – 6 x – 2x – 4x
2
3
2
– 4x – 14x – 6 x – 2x – 3x + 2
2
x – 2x – 3x + 2
3
2
f(x) = (x − 2)(2x + 1)(x + 3) f(x) = (x + 2)(x − 2x + 1)
2
[4] = (x + 2)(x − 1)
2
5. 3(1) + (1) − A(1) + B(1) − 6 = 0 [4]
2
3
4
3 + 1 − A + B − 6 = 0 9. (a) When x = 1, f(1) = 0
−A + B = 2 1 + a − b + 10 = 0
3(2) + (2) − A(2) + B(2) − 6 = 20 a − b = −11
3
4
2
48 + 8 − 4A + 2B − 6 = 20 When x = −1, f(−1) = 12
−2A + B = −15 −1 + a + b + 10 = 12
Solve the simultaneous equations to get: a + b = 3
A = 17, B = 19 Solve the simultaneous equations to get:
[5] a = −4, b = 7
[5]
6. When x − 3 = 0 (b) f(x) = 0
2
x = ±√3 (x − 1)(x − 3x − 10) = 0
2
When x = √3, (x − 1)(x − 5)(x + 2) = 0
Remainder x = 1 or x = 5 or x = −2
= 2(√3) + 5(√3) − 7(√3) + 1 [4]
4
6
2
= 79 10. f(2) = 25
When x = −√3, a(2) − 4(2) − b(2) + 3 = 25
2
3
Remainder 4a − b = 19
= 2(−√3) + 5(−√3) − 7(−√3) + 1 f(1) = 3
6
4
2
= 79 a(1) − 4(1) − b(1) + 3 = 3
3
2
[3] a − b = 4
7. (a) x − 5 = 0 Solve the simultaneous equations to get:
2
x = √5 or x = −√5 a = 5, b = 1
When x = √5 , f(√5 ) = 0. [5]
3(√5 ) + a(√5 ) + b(√5 ) − 5 = 0 11. When x = 2
2
3
2
3
When x = −√5 , f(−√5 ) = 0. (2) − (2) − 14(2) + 24 = 0
3
2
3(−√5 ) + a(−√5 ) + b(−√5 ) − 5 = 0 x + x + x – 12
3
2
3
x – 2 x – x – 14x + 24
2
Solve the simultaneous equations to get: x – 2x
3
2
a = 1, b = −15 x – x – 14x
2
3
[5]
x – 2x – 2x
2
3
x – 2x– 12x + 24
3
3
x – 2x– 12x + 24
Cambridge IGCSE
TM
176 Ace Your Additional Mathematics
Answers Add Math.indd 176 14/03/2022 12:29 PM