Suggested solutions - please let me know of any errors:
UPDATED: worked solutions attached to this post.
Question 1a
\(\frac{4g-10\sqrt{3}-5}{2}\)
Question 1b
\(\frac{10-5\sqrt{3}}{4}\)
Question 1c
\(20-10\sqrt{3}\)
Question 2
\(-\frac{10}{3}+\frac{8\sqrt{2}}{3}\)
Question 3
cis\(\left(\frac{7\pi}{12}\right)\), cis\(\left(-\frac{\pi}{12}\right)\), cis\(\left(-\frac{3\pi}{4}\right)\)
Question 4
\(x \in \left(-\infty, \frac{7-\sqrt{5}}{2}\right)\)
Question 5a
\(m=4\)
Question 5b
\(\frac{1}{18}\left(47\vec{i}-10\vec{j}+7\vec{k}\right)\)
Question 6a
By chain rule, \(f'(x)=3\times \frac{1}{(3x-6)^2+1}=\frac{3}{9x^2-36x+37}\) as required.
Question 6b
\(f''(x)=\frac{-3(18x-36)}{\left(9x^2-36x+37\right)^2}\)
\(f''(x)=0 \rightarrow 18x-36=0 \rightarrow x=2\).
Since \(\left(9x^2-36x+37\right)^2 > 0\) for all \(x\), \(f''(x) > 0\) when \(x<2\)and \(f''(x) < 0\) when \(x>2\), so \(f''(x)\) changes sign at \(x=2\).
Question 6c
Graph of arctan function with horizontal asymptotes \(y=\frac{\pi}{2},\frac{3\pi}{2}\), and point of inflection at \((2,\pi)\)
Question 7a
\(f(x)\) is continuous everywhere if \(m+n=\frac{4}{1+1^2} \rightarrow m+n=2\).
\(f'(x)\) is continuous everywhere if \(m=\frac{-8\times 1}{(1+1^2)^2} \rightarrow m=-2\).
Hence, \(m=-2,n=4\)
Question 7b
\(3+\frac{\pi}{3}\)
Question 8
\(2\pi\left(\log_e\left(2\sqrt{3}+2\right)+\frac{\pi}{3}\right)\)
Question 9a
\(\begin{aligned}[t]\left(\frac{dy}{dt}\right)^2 &= \left(\frac{1}{1+t}-\frac{1}{4(1-t)}\right)^2 \\
&= \frac{1}{(1+t)^2}-\frac{1}{2(1-t)(1+t)}+\frac{1}{16(1-t)^2} \\
&= \frac{1}{(1+t)^2}-\frac{1}{2(1-t^2)}+\frac{1}{16(1-t)^2}\end{aligned}\)
So \(a=1, b=-2, c=16\) as required.
Question 9b
\(\log_e\left(\frac{3}{2}\right)-\frac{1}{4}\log_e\left(\frac{1}{2}\right)\)