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Further Mathematics an advanced secondary school mathematics subject. The subject usually picks up where the regular maths class leaves off and covers topics like mathematical induction, complex numbers, polar curves, and conic sections, differential equations, matrices, and statistical inference.

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Simplify \(\frac{\sqrt{3}}{\sqrt{3} -1} + \frac{\sqrt{3}}{\sqrt{3} + 1}\)

A)

\(\frac{1}{2}\)

B)

3

C)

\(2\sqrt{3}\)

D)

6

Find the domain of \(g(x) = \frac{4x^{2} - 1}{\sqrt{9x^{2} + 1}}\)

A)

\({x : x \in R, x = \frac{1}{2}}\)

B)

\(x: x \in R, x\neq \frac{1}{3}\)

C)

\(x : x \in R, x = \frac{1}{3}\)

D)

\(x: x \in R\)

Given that \(f(x) = 3x^{2} -Â 12x + 12\) and \(f(x) = 3\), find the values of x.

A)

1, 3

B)

-1, -3

C)

1, -3

D)

-1, 3

A binary operation * is defined on the set of real numbers, by \(a * b = \frac{a}{b} + \frac{b}{a}\). If \((\sqrt{x} + 1) * (\sqrt{x} - 1) = 4\), find the value of x.

A)

6

B)

5

C)

4

D)

3

If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.

A)

\(\frac{5\sqrt{10}}{4}\)

B)

\(4\sqrt{10}\)

C)

\(5\sqrt{10}\)

D)

\(\frac{4\sqrt{10}}{5}\)

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

\(P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}\) 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)

\(n(P' \cap R) = 2\)

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)

\(a = \frac{b^{3}}{9}\)

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)

\(2x^{2} - 9x - 15 = 0\)

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)

\(\frac{-1}{2(x - 2)} + \frac{8x}{2(x - 2)^{2}}\)

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