ENGAA 2022 Section 1, Part B (Mathematics questions)

Find the value of$${\int }_1^4\frac{2x^2-3}{x\sqrt{x}}\mathrm{d}x$$

The diagram shows a semicircle of radius 5 units and a triangle.

The triangle has vertices at $(0,0)$, $(10,0)$ and $(x,y)$.

$(x,y)$ is a point on the arc of the semicircle.

Which of the following is an expression in terms of $x$ for the area of this triangle?

Four mathematically similar solids, W, X, Y and Z, have the following properties:

- The ratio of the lengths of W to the lengths of X is $1:2$
- The ratio of the surface area of X to the surface area of Y is $2:1$
- The ratio of the volume of Y to the volume of Z is $1:2$

What is the order of the solids when arranged in increasing volume?

Which one of the following expressions is equal to$$\frac{{\left(2+\sqrt{20}\right)}^2}{{\left(1+\sqrt{5}\right)}^3}$$

Evaluate$$\sum _{r=2}^{\infty }\left(\frac{1}{2^r}+\frac{1}{3^r}+\frac{1}{4^r}\right)$$

What is the complete set of real values of $x$ for which$$x^2\left(x^2+4\right)<21$$

A sector of a circle has perimeter 24.

For what value of the radius does the sector have the maximum possible area?

The curve $y=x^2-x-6$ intersects the $x$-axis at the points $A$ and $B$, and has a minimum at the point $C$.

The rectangle $ABDE$ has two of its vertices at $A$ and $B$.

The point $C$ lies on the edge $DE$, between $D$ and $E$.

What is the area of the rectangle $ABDE$?

A straight line passes through the points $(0,2a)$ and $(a,0)$, where $a$ is a positive constant.

What is the perpendicular distance of the point $P(a,2a)$ from this line?

Find the sum of the real solutions to the equation$$2^x-{\left(\sqrt{2}\right)}^{x+6}+12=0$$