We are asked to solve a series of calculus problems including finding roots of equations, finding range and domain of functions, applying trigonometric identities, finding solutions to trigonometric equations, determining if functions are odd or even, finding derivatives.

1. Problem Description

We are asked to solve a series of calculus problems including finding roots of equations, finding range and domain of functions, applying trigonometric identities, finding solutions to trigonometric equations, determining if functions are odd or even, finding derivatives.

2. Solution Steps

3. a) (i) sin(2$\theta$) = tan($\theta$)

We know that sin(2$\theta$) = 2sin($\theta$)cos($\theta$). So,

2sin($\theta$)cos($\theta$) = sin($\theta$)/cos($\theta$)

2sin($\theta$)cos$^2$($\theta$) - sin($\theta$) = 0

sin($\theta$)(2cos$^2$($\theta$) - 1) = 0

sin($\theta$) = 0 or 2cos$^2$($\theta$) - 1 = 0

If sin($\theta$) = 0, then $\theta$ = n$\pi$ where n is an integer.

If 2cos$^2$($\theta$) - 1 = 0, then cos$^2$($\theta$) = 1/2, so cos($\theta$) = $\pm \frac{1}{\sqrt{2}}$ = $\pm \frac{\sqrt{2}}{2}$.

This implies $\theta$ = $\frac{\pi}{4}$ + $\frac{n\pi}{2}$ where n is an integer.

3. a) (ii) 4sin$^2$(3t) - 3sin(3t) = 1

Let u = sin(3t). The equation becomes 4u$^2$ - 3u - 1 =

0.

(4u + 1)(u - 1) = 0

u = -1/4 or u = 1

sin(3t) = -1/4 or sin(3t) = 1

If sin(3t) = 1, then 3t = $\frac{\pi}{2}$ + 2n$\pi$

t = $\frac{\pi}{6}$ + $\frac{2n\pi}{3}$

If sin(3t) = -1/4, then 3t = arcsin(-1/4) + 2n$\pi$ or 3t = $\pi$ - arcsin(-1/4) + 2n$\pi$ = $\pi$ + arcsin(1/4) + 2n$\pi$

t = $\frac{1}{3}$arcsin(-1/4) + $\frac{2n\pi}{3}$ or t = $\frac{\pi}{3}$ + $\frac{1}{3}$arcsin(1/4) + $\frac{2n\pi}{3}$

3. b) (i) g(x) = -|$x$| + 2

g(-x) = -|$-x$| + 2 = -|$x$| + 2 = g(x). Therefore g(x) is an even function.

4. b) (ii) f(x) = $\frac{x^2}{x^3-x}$ = $\frac{x^2}{x(x^2-1)}$ = $\frac{x}{x^2-1}$

f(-x) = $\frac{-x}{(-x)^2 - 1}$ = $\frac{-x}{x^2-1}$ = -f(x). Therefore f(x) is an odd function.

3. Final Answer

3 a) (i) $\theta$ = n$\pi$ and $\theta$ = $\frac{\pi}{4}$ + $\frac{n\pi}{2}$

3 a) (ii) t = $\frac{\pi}{6}$ + $\frac{2n\pi}{3}$, t = $\frac{1}{3}$arcsin(-1/4) + $\frac{2n\pi}{3}$, and t = $\frac{\pi}{3}$ + $\frac{1}{3}$arcsin(1/4) + $\frac{2n\pi}{3}$

3 b) (i) g(x) is even.

3 b) (ii) f(x) is odd.