(i) $a=-3.594, b = -5.187, c=7.303$.
(ii) $x=-0.589$.
(iii) $y=-5.19x+7.30$.
(ii) $-1.73 < x < 0.414$ or $x > 1.73$.
(i) $\frac{1}{n} \left \{ f\left( \frac{1}{n} \right) + f\left( \frac{2}{n} \right) + \cdots + f\left( \frac{n}{n} \right) \right \}$ is the total area of $n$ rectangles. These rectangles approximate the area under the curve, which is given by $\displaystyle \int_0^1 f(x) \; \mathrm{d}x$.
As $n \to \infty$, the approximation becomes exact and $\lim_{n \to \infty} \frac{1}{n} \left \{ f\left( \frac{1}{n} \right) + f\left( \frac{2}{n} \right) + \cdots + f\left( \frac{n}{n} \right) \right \} = \displaystyle \int_0^1 f(x) \; \mathrm{d}x$.
(ii) $\frac{3}{4}$.
$k=\frac{1}{32}$.
(i) A scaling of scale factor $\frac{1}{4}$ parallel to the $y$-axis. A translation of 3 units along the positive $x$-axis direction.
(i) $y= 2x-2x^2 + \frac{8}{3}x^3 + \ldots$
(ii) $a=2, b=\frac{5}{3}, c=-\frac{3}{5}$.
Coefficient of $x^4 = -\frac{104}{27}$.
(i) Maximum $h=32$ cm.
(ii) $t=160-40\sqrt{16-\frac{h}{2}}$, 46.9 years.
(i) $73.4^\circ$.
(ii) $(3,1,-10),(\frac{13}{7},-\frac{5}{7},-\frac{46}{7})$.
$(\frac{17}{7},\frac{1}{7},-\frac{58}{7})$.
(iii) $36x-2y+11z=-4$.
(a)(i) Since every horizontal line $y=k, k\in \mathbb{R}$ intersects the graph of $y=f(x)$ at most once, $f$ is one-one and it has an inverse.
(a)(ii) $f^{-1}(x) = \sqrt{1-\frac{1}{x}}$, $D_{f^{-1}} = (-\infty,0)$.
(b) $\left (-\infty, 1-\frac{\sqrt{3}}{2} \right ] \; \cup \; \left [ 1 + \frac{\sqrt{3}}{2}, \infty \right )$.
(i) $A=1,B=-1$.
(ii) $S_n = \frac{1}{3} - \frac{1}{2n+3}$.
(iii) $499$.
(i) 0.224.
(ii) 0.150.
(iii) 0.824.
(i) 0.4.
(ii) 0.185.
(iii) $0.165 \leq P(A' \cap B' \cap C') \leq 0.365$.