NSAA 2016 Section 1, Part E (Mathematics questions)

When $x=2$ is substituted in the expression $x^3+p{x}^2+qx+p^2$ the result is 0.

When $x=1$ is substituted into the same expression, the result is $-3.5$.

Find all possible value(s) of $p$.

A square $PQRS$ is drawn above the $x$-axis with the side $PQ$ on the $x$-axis.

$P$ is the point $(-5,0)$ and $Q$ is the point $(1,0)$.

A circle is drawn inside the square with diameter equal in length to the side of the square.

Which one of the following is an equation of the circle?

The first term of a convergent geometric series is 8.

The fifth term is 2.

The sixth term is real and positive.

What is the sum to infinity of this series?

(The sum to infinity of a convergent geometric series is given by $\frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio.)

Tangents are drawn from a point $P$ to a circle of radius 10 cm.

The centre of the circle is $C$ and the distance $PC$ is 20 cm.

Which one of the following is an expression for the shaded area in square centimetres?

Given that $7\cos \theta -3\tan \theta \sin \theta =1$, which one of the following is true?

The complete set of values of $a$ for which the equation $3x^2=(a+2)x-3$ has two real distinct roots is

The straight line with equation $y=mx+3$, where $m>0$, $m\ne 1$, is perpendicular to the line with equation $y=px+2$

The lines cut the $x$-axis at the points $L$ and $M$ respectively. The length of $LM$ is 5 units.

What is the value of $m+p$ given that $m>1$?

$f(x)=x^3-a^{2}x$ where $a$ is a positive constant.

Find the complete set of values of $x$ for which $f(x)$ is an increasing function.

The curve $y=x^2$ is translated by the vector $\left(\begin{array}{c}4\\ 3\end{array}\right)$ and then reflected in the line $y=-1$

Which one of the following is an equation of the resulting curve?