NSAA 2019 Section 1, Part E (Mathematics questions)

Find the area of the shape bounded by the four lines:$$2y+x=4$$$$x=-6$$$$x=0$$$$y=0$$

The curve$$y=x^3+p{x}^2+qx+r$$has a local maximum when $x=-1$ and a local minimum when $x=3$.

What is the value of $p$?

It is given that$$7\cos x+\tan x\sin x=5$$where $0^{\circ }<x<90^{\circ }$

What are the possible values of $\tan x$?

An equilateral triangle of side $8$ cm is drawn so that its vertices lie on the circumference of a circle, as shown in the diagram.

What is the total of the three areas shaded in the diagram, in ${\text{cm}}^2$?

Which one of the following is the real solution of the equation$$3\times 5^{2x+1}-5^x-2=0$$

For a particular function $f(x)$, it is given that:$${\int }_{-2}^{2}2f(x)dx+{\int }_2^{4}f(x)dx=4$$and also:$${\int }_{-2}^{2}5f(x)dx-{\int }_{-2}^{4}f(x)dx=7$$Find the value of ${\int }_2^{4}f(x)dx$

Given that$$f(x)={\int }_0^x(3+2t{)}^{7}dt$$what is the coefficient of $x^4$ in the expansion of $f(x)$ in powers of $x$?

The three internal angles in a triangle are $\alpha$, $\beta$ and $\theta$, and$$3\tan \alpha -2\sin \beta =2$$$$5\tan \alpha +6\sin \beta =8$$What is the value of $\theta$ in degrees?

Find the complete set of values of $x$ for which$$x^3-2x^2-7x-4>0$$