NSAA 2017 Section 1, Part E (Mathematics questions)

Which one of the following is a simplification of$$1-{\left(\frac{3+\sqrt{3}}{6-2\sqrt{3}}\right)}^2$$

$k$ is the smallest positive value of $x$ which is a solution to both the equations $2\sin x+1=0$ and $2\cos 2x=1$

How many values of $x$ in the range $0\le x\le k$ are solutions to at least one of these equations?

Which of the following is a solution to the equation $3^{(2x+1)}-6(3^x)=0$?

The graph of the function $y=x^3+p{x}^2+qx+6$, where $p$ and $q$ are real constants, has a local maximum when $x=2$ and a local minimum when $x=4$.

What are the values of $p$ and $q$?

Given that $y=(2+3x{)}^6$, what is the coefficient of $x^3$ in $\frac{\mathrm{d}y}{\mathrm{d}x}$?

A geometric progression has first term equal to 1 and common ratio $\frac{1}{2}\sin 2x$

The sum to infinity of the series is $\frac{4}{3}$

Find the possible values of $x$ in the range $\pi \le x\le 2\pi$

The sequence of numbers $u_1,u_2,u_3,\dots ,u_n,\dots$ is given by$$u_1=2$$$$u_{n+1}=p{u}_n+3$$where $p$ is an integer.

The fourth term, $u_4$, is equal to $-7$

What is the value of $u_1+u_2+u_3+u_4$?

Find the complete set of values of $x$ for which$$\frac{x^3-6x^2+9x-4}{x}>0$$

The equations of two straight lines are $y=3+(2p^2-p)x$ and $y=7+(p-2)x$, where $p$ is a real constant.

For certain values of $p$, the two lines are perpendicular.

Which of the following numbers is closest to the greatest such value of $p$?