Simplify each of the following expressions: (i) (3+√3)(2+√2) (ii) (3+√3)(3-√3) (iii) (√5+√2)^2 (iv) (√5-√2)(√5+√2)

Find the maximum and minimum values of each of the following trigonometric expressions: (i) $12 \sin \theta - 5 \cos \theta$ (ii) $12 \cos \theta + 5 \sin \theta + 4$ (iii) $5 \cos \theta + 3 \sin \left(\frac{\pi}{6} - \theta\right) + 4$ (iv) $\sin \theta - \cos \theta + 1$

Express each of the following as a rational number in the form $\frac{p}{q}$:

(i) $6^{-1}$
(ii) $(-7)^{-1}$
(iii) $\left(\frac{1}{4}\right)^{-1}$
(iv) $(-4)^{-1} \times \left(\frac{-3}{2}\right)^{-1}$
(v) $\left(\frac{3}{5}\right)^{-1} \times \left(\frac{5}{2}\right)^{-1}$

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form: (i) $1+i$ (ii) $\sqrt{3}+i$ (iii) $1-\mathrm{i}$ (iv) $\frac{1-i}{1+i}$ (v) $\frac{1}{1+i}$ (vi) $\frac{1+2i}{1-3i}$ (vii) $\sin 120^{\circ}-\mathrm{i}\cos 120^{\circ}$ (viii) $\frac{-16}{1+i\sqrt{3}}$

Determine the intervals in which the following function is strictly increasing or strictly decreasing:

$$ f(x) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3x^2 + \frac{36}{5} x + 11 $$

• It is increasing in $(\infty, -2) \cap (1, 3)$ and decreasing in $(-2, 1) \cup (3, \infty)$.

• It is increasing in $(-2, 1) \cup (3, \infty)$ and decreasing in $(\infty, -2) \cup (1, 3)$.

• It is increasing in $(-2, 1) \cap (3, \infty)$ and decreasing in $(-\infty, -2) \cap (1, 3)$.

• It is increasing in $(-\infty, -2) \cup (1, 3)$ and decreasing in $(-2, 1) \cap (3, \infty)$.

Which of the following fractions are equivalent to $\frac{1}{4}$? Choose all the correct options.

(A) $\frac{2}{4}$

(B) $\frac{1}{8}$

(C) $\frac{2}{8}$

(D) $\frac{5}{12}$