Further Mathematics Past Questions and Answers

If the polynomial \(f(x) = 3x^{3} - 2x^{2} + 7x + 5\) is divided by (x - 1), find the remainder.

A -17
B -7
C 5
D 13

Question 32:

\(P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}\) are subsets of \(U = {1, 2, 3, ... , 12}\). Which of the following statements is true?

A \(Q \cap R = \varnothing\)
B \(R \subset P\)
C \((R \cap P) \subset (R \cap U)\)
D \(n(P' \cap R) = 2\)

Question 33:

If \(\log_{3}a - 2 = 3\log_{3}b\), express a in terms of b.

A \(a = b^{3} - 3\)
B \(a = b^{3} - 9\)
C \(a = 9b^{3}\)
D \(a = \frac{b^{3}}{9}\)

Question 34:

If \(\alpha\) and \(\beta\) are the roots of \(2x^{2} - 5x + 6 = 0\), find the equation whose roots are \((\alpha + 1)\) and \((\beta + 1)\).

A \(2x^{2} - 9x + 15 = 0\)
B \(2x^{2} - 9x + 13 = 0\)
C \(2x^{2} - 9x - 13 = 0\)
D \(2x^{2} - 9x - 15 = 0\)

Question 35:

Resolve \(\frac{3x - 1}{(x - 2)^{2}}, x \neq 2\) into partial fractions.

A \(\frac{x}{2(x - 2)} - \frac{5}{(x - 2)^{2}}\)
B \(\frac{5}{(x - 2)} + \frac{x}{2(x - 2)^{2}}\)
C \(\frac{1}{2(x - 2)} + \frac{5x}{2(x- 2)^{2}}\)
D \(\frac{-1}{2(x - 2)} + \frac{8x}{2(x - 2)^{2}}\)