NSAA 2022 Section 1, Part A (Mathematics questions)

Which one of the following is a simplification of$$y{\left(\frac{3x^{\frac{1}{2}}z}{y^3}\right)}^2$$

Triangle $PQR$ has a right angle at $Q$.

The point $T$ lies on $QR$ such that $QT=\frac{1}{4}QR$

$PT=6$ cm

$PR=12$ cm

What is the length of $QT$, in cm?

Find the complete set of values of $x$ that satisfy the inequality$$\frac{3}{4}(5-x)-\frac{1}{2}(6-x)-x<0$$

I have two fair dice, X and Y, each of which has six sides.

The faces on X are labelled 1, 1, 2, 3, 4, 5.

The faces on Y are labelled 2, 3, 4, 5, 6, 6.

I roll the dice together and calculate my total score by adding the number rolled on X to the number rolled on Y.

What is the probability that my total score is greater than 9?

Rob keeps a record of what he earns each day.

On Monday, he earned 50% less than he earned on Sunday. On Tuesday, he earned 20% more than he earned on Monday. On Wednesday, he earned 30% less than he earned on Tuesday.

On Wednesday, he earned £84.

How much did Rob earn on Sunday?

The $n$th term of a sequence T is $(n-3{)}^2$, where $n$ is a positive integer.

The $n$th term of another sequence V is $3n+p$, where $p$ is a constant and $n$ is a positive integer.

The 10th term in T is equal to twice the 5th term in V.

What is the 4th term in V?

Which one of the following is a simplification of$$\frac{5x^2-17x-12}{25x^2-9}÷\frac{x^2+x-12}{x^2-x-6}$$

S is a list of six numbers:$$1,2,x,x+1,x+1,15$$where $2\le x\le 14$

The mean of S is one more than the median of S.

What is the value of $x$?

A rectangle $PQRS$ has length $(2x-1)$ cm and width $(x+1)$ cm as shown on the diagram.

A larger rectangle is made by adding 3 cm to both the length and the width of $PQRS$, as shown.

The larger rectangle has an area of $360{\text{ cm}}^2$.

What is the ratio of $PQ$ to $PS$?

$t$ is inversely proportional to the square of $w$.

$t$ and $w$ are positive numbers.

$t=36$ when $w=2\times 10^{-2}$

What is the value of $w$ when $t=100$?

$PQRS$ is a trapezium as shown.

$\tan RSQ=\frac{5}{8}$

What is the length of $PS$, in metres?

A cyclist rides along a track to the top of a hill then immediately turns around and descends along the same track to her starting point.

She takes 40 minutes at an average speed of $12{\text{ km h}}^{-1}$ to reach the top.

Her average speed for the whole journey is $15{\text{ km h}}^{-1}$.

What is the average speed of her descent?

A solid cylinder has radius $r$ cm and height $h$ cm.

A cube has side length $3r$ cm.

The total surface area of the cylinder is equal to four times the total surface area of the cube.

Which of the following is an expression for $h$ in terms of $r$?

Consider the equation $2x^2+4x+c=0$, where $c$ is a constant.

The positive difference between the roots of this equation is $\sqrt{10}$.

What is the value of $c$?

The variables $x$ and $y$ are related by the equation:$$x=5-\frac{2y^3+1}{1-2y^3}$$Which of the following is a rearrangement to make $y$ the subject?

$PQR$ is a triangle as shown.

$S$ and $T$ are points on the sides $PQ$ and $PR$.

$ST$ is parallel to $QR$.

$PS=3$ cm $ST=x$ cm $QS=(x+2)$ cm $QR=(2x+1)$ cm

What is the length, in cm, of $QR$?

Three different numbers are chosen at random from $\sqrt{1}$, $\sqrt{2}$, $\sqrt{3}$, $\sqrt{4}$, $\sqrt{5}$.

What is the probability that the three numbers form the three sides of a right-angled triangle?

P, Q and R are regular polygons.

Q has three times as many sides as P.

An interior angle of Q is $10^{\circ }$ larger than an interior angle of P.

R has twice as many sides as Q.

How much larger is an interior angle of R than an interior angle of Q, in degrees?

The point $(-1,5)$ is translated to the point $(3,2)$ by two successive translations.

The first translation is by the vector $\left(\begin{array}{c}3p\\ -4p\end{array}\right)$

The second translation is by the vector $\left(\begin{array}{c}q\\ -2q\end{array}\right)$

What is the value of $p+q$?

Consider the graphs of the form$$y=x^2+2ax+a$$What is the complete range of values of $a$ for which the minimum point of the graph lies above the $x$-axis?