FURTHER MATHEMATICS EXAMINATION LIKELY QUESTIONS FOR K-12 SCHOOL STUDENTS

FURTHER MATHEMATICS
EXAMINATION LIKELY QUESTIONS FOR K-12 SCHOOL STUDENTS
1. The root of the quadratic equation 3𝑥 2 − 𝑥 − 1 = 0 are 𝛼 𝑎𝑛𝑑 𝛽. Find the value
2 1 2
of 𝛼 2 + 𝛽2 + . A. − B. 0 C. D. 1
9 2 9

2. A binary operation ∆ is defined on the set of real numbers by 𝑎∆𝑏 = 𝑎𝑏 . Find
1 1
the value of 3∆ − 2. A. -9 B. − C. D. 1
9 9
2𝑥−1
3. 𝑔: 𝑥 → , where x is real and 𝑥 ≠ 1. Find the inverse, 𝑔−1 .
𝑥+1
𝑥 𝑥+1 1
A. 𝑔−1 : 𝑥 → ,𝑥≠2 B. 𝑔−1 : 𝑥 → ,𝑥≠
2−𝑥 2𝑥−1 2
𝑥+1 1−𝑥
C. 𝑔−1 : 𝑥 → ,𝑥≠2 D. 𝑔−1 : 𝑥 → , 𝑥 ≠ −2
2−𝑥 2+𝑥

4. Which of the following binary operations is not commutative?
A. 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏 B. 𝑎 ∗ 𝑏 = 2𝑎 + 2𝑏 − 𝑎𝑏
1 1
C. 𝑎 ∗ 𝑏 = + D. 𝑎 ∗ 𝑏 = 𝑎 − 𝑏 − 𝑎𝑏
𝑎 𝑏

5. If (3 + 4√3)(2 − 𝑎√3) = 18 + 2√3, find the value of a.
A. -4 B. -2 C. 2 D. 3
6. Given that 2 log 𝑦 𝑥 = 3, find the relation involving x and y.
A. 𝑦 3 = 𝑥 2 B. 𝑦 3 = 𝑥 C. 𝑦 3 = 𝑥 3 D. 𝑦 = 𝑥 3
1
7. Find the value of x in 82𝑥+1 = A. -5 B. -2 C. -1 D. 4
512

8. Find the equation of the line which passes through the point (1,3) and which is
perpendicular to the line 2𝑦 − 3𝑥 + 1 = 0.
A. 2𝑦 + 3𝑥 − 9 = 0 B. 2𝑦 − 3𝑥 − 9 = 0 C. 3𝑦 + 2𝑥 − 11 = 0 D. 2𝑦 +
3𝑥 − 3 = 0
9. Two functions 𝑓and 𝑔 are defined by𝑓: → 𝑥 2 − 1. Find𝑔𝑜𝑓.
A. 𝑥 2 + 2 B. 𝑥 2 + 3 C. 𝑥 2 + 3𝑥 + 8 D. 𝑥 2 + 6𝑥 + 8

, 10. If 𝛼 𝑎𝑛𝑑 𝛽 are the roots of 4𝑥 2 − 9𝑥 − 16 = 0, which of the following
statements is/are correct?
9 9
I. 𝛼 + 𝛽 = II. 𝛼𝛽 = −4 III. 𝛼 + 𝛽 = −
4 4

A. I only B. II only C. I and II only D. II and III only
11. A binary operation is defined by 𝑎 ∗ 𝑏 = 𝑎2 − 𝑏 2 + 𝑎𝑏, where a and b are real
numbers. Evaluate √2 ∗ √3 A. 17 B. √6 − 1 C. 1 + √6
D. 1 − √6
3𝑥+1 𝐴 𝐵
12. If ≡ + , find the value of A + B.
𝑥 2 −3𝑥+2 (𝑥−1) (𝑥−1)

A. -3 B. 2 C. 3 D. 4
13. Find the value of m for which the equation 𝑥 2 − 𝑚𝑥 + 4 = 0 has equal roots.
A. ±2 B. ±3 C. ±4 D. ±5
14. 𝑥 − 2 is a factor of the polynomial𝑓(𝑥) = 2𝑥 3 − 𝑥 2 − 𝑛𝑥 − 2, where n is a
constant. Find n. A. -10 B. -5 C. 5 D. 11
1 3 3
15. If 3(32𝑥 ) = . Find the value of x. A. 2 B. C. − D. -2
27 2 2

16. If 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 + 3𝑥 + 2, find 𝑓(−1). A. 14 B. 2 C. 0 D. -6
𝑥 √𝑦
17. Express log ( 1 ) in terms of log 𝑥 , log 𝑦 𝑎𝑛𝑑 log 𝑧.
𝑧3
1 1 1
A. log 𝑧 − log 𝑥 − log 𝑦 B. 3 log 𝑧 − log 𝑥 − log 𝑦
3 2 2
1 1 1
C. log 𝑥 − log 𝑦 − 3 log 𝑧 D. log 𝑥 + log 𝑦 − log 𝑧
2 2 3

18. Find the coefficient of 𝑥 5 in the binomial expression of (2 + 𝑥)7 .
A. 21 B. 28 C. 84 D. 140
3
4 −2 8 2 3 27
19. Evaluate (9) A. 27 B.
3
C.
2
D.
8

20. Evaluate 8𝐶2 - 5𝐶3 A. 4 B. 6 C. 18 D. 33
2𝑥−1
21. 𝑔: 𝑥 → , 𝑥 ≠ −1. Find the inverse, 𝑔−1 : 𝑥
𝑥+1