NSAA 2018 Section 1, Part E (Mathematics questions)

Curve $C$ has equation $y=9-x^2$

Line $L$ has equation $y=5$

What is the area enclosed between $C$ and $L$?

How many solutions of the equation $2{\sin }^3\theta =\sin \theta$ lie in the interval $-\frac{\pi }{2}\le \theta \le \pi$?

The line $y=x+k$, where $k$ is a constant, is a tangent to the curve $y=3x^2-2x+1$

What is the value of $k$?

In a particular arithmetic progression:

the $13^{\text{th}}$ term is six times the $1^{\text{st}}$ term

the $11^{\text{th}}$ term is 1 less than twice the $5^{\text{th}}$ term

What is the $3^{\text{rd}}$ term of the progression?

Evaluate$${\log }_2\left(\frac{5}{4}\right)+{\log }_2\left(\frac{6}{5}\right)+{\log }_2\left(\frac{7}{6}\right)+\cdots +{\log }_2\left(\frac{64}{63}\right)$$

Circle $C$ has equation $(x+3{)}^2+(y-2{)}^2=5$

The length of the tangent from the circle $C$ to the point $P$ is $5\sqrt{3}$

What is the shortest distance from $P$ to $C$?

What is the coefficient of $x^3$ in the expansion of $(1-2x{)}^5(1+2x{)}^5$?

Given that$${\int }_0^2{x}^{m}dx=\frac{16\sqrt{2}}{7}$$and$${\int }_0^2{x}^{m+1}dx=\frac{32\sqrt{2}}{9}$$what is the value of $m$?

The dimensions of a solid cuboid, in cm, are $x$, $2x$ and $y$

The volume of the cuboid is $576{\text{ cm}}^3$.

At this volume, the surface area of the cuboid has its maximum value.

What is the area, in ${\text{cm}}^2$, of the face that has the largest area?