ENGAA 2020 Section 1, Part B (Mathematics questions)

$(x-1)$ and $(x-2)$ are both factors of $x^4+a{x}^3+b{x}^2-12x+4$

What are the values of $a$ and $b$?

What is the area of the region enclosed between the curve $y=\frac{1}{2}{x}^2$, the line $y=-x$, and the lines $x=1$ and $x=3$?

A line with non-zero gradient $m$ is reflected in the line $y=x$

What is the gradient of the reflected line?

The sum of the first 20 terms of an arithmetic progression is 50.

The sum of the next 20 terms of the arithmetic progression is $-50$.

What is the sum of the first 100 terms of the arithmetic progression?

The line L with equation $y=mx+c$, where $m>0$ and $c\ge 0$, passes through the point $(2,4)$.

A line is drawn through the point $(2,4)$ perpendicular to L.

The triangle enclosed between the two lines and the $y$-axis has area 5 square units.

What is the larger of the two possible values of $m$?

$P$ and $Q$ are two different geometric progressions.

The 3rd term of each geometric progression is 4.

The 5th term of each geometric progression is 2.

What is the modulus of the difference between the sums to infinity of $P$ and $Q$?

The curve$$y=x^3+3\sqrt{5}p{x}^2+3px+13$$has two distinct turning points.

What are all the possible values of $p$?

Find the number of solutions of the equation$$14{\cos }^{3}x+10{\sin }^{2}x\cos x=13\cos x$$in the range $-2\pi \le x\le 2\pi$

Find the product of the real roots of the equation$${\left({\log }_{10}{x}^2\right)}^2+{\log }_{10}x=3$$

Find the maximum value of the gradient of the curve with equation$$y=2-4x+4x^{\frac{3}{2}}-x^2$$where $x>0$