ENGAA 2021 Section 1, Part B (Mathematics questions)

Given that$$y={\left(2\sqrt{x}-\frac{1}{2\sqrt{x}}\right)}^2$$find the value of $\frac{\mathrm{d}y}{\mathrm{d}x}$ when $x=\frac{1}{2}$

A particular arithmetic series has first term $a$ and common difference $d$.

The sum of the first $k$ terms of this series is denoted by $S_k$

Which of the following is a simplification of $S_{n+1}-S_{n-1}$?

Find how many distinct real solutions there are to the equation$$(x^2+4x+3{)}^2=1$$

The line $x=1$ divides the circle $x^2+y^2=4$ into two segments.

What is the area of the smaller segment?

What is the mean of ${\log }_{10}27$, ${\log }_{10}64$, and ${\log }_{10}216$?

Which of the following is the largest in value?

(All angles are in radians.)

A geometric progression has first term $u_1=a$ and common ratio $r$.

The sum to infinity of the geometric progression is $\frac{8}{5}$

The sum to infinity of the even-numbered terms $(u_2+u_4+u_6+\cdots )$ is $\frac{3}{5}$

What is the value of $a+r$?

At how many distinct points do the following two curves meet?$$y=(x-4)(x^2-2x-8)$$$$y=-x^2+8x-16$$

Evaluate$$\frac{3}{\sqrt{27}+\sqrt{21}}+\frac{3}{\sqrt{24}+\sqrt{18}}+\frac{3}{\sqrt{21}+\sqrt{15}}+\cdots +\frac{3}{\sqrt{9}+\sqrt{3}}$$

$PQRS$ is a rectangle.

$P$ and $Q$ lie on the $x$-axis.

$Q$ and $R$ lie on the line $x=15$

$S$ lies on the curve $y=\sqrt{x}$

What is the maximum possible area of the rectangle?