NSAA 2017 Section 1, Part A (Mathematics questions)

Evaluate$$\frac{{\left(\sqrt{12}+\sqrt{3}\right)}^2}{{\left(\sqrt{12}-\sqrt{3}\right)}^2}$$

Solve fully the inequality$$2x^2\ge 15-x$$

The equation gives $y$ in terms of $x$:$$y=3{\left(\frac{x}{2}-1\right)}^2-5$$Which one of the following is a rearrangement for $x$ in terms of $y$?

A fruit stall sells apples costing £$x$ each, and pears costing £$y$ each.

Sam bought 2 apples and 5 pears, and the total cost of these was £$P$.

Lesley bought 3 apples and 2 pears, and the total cost of these was £$Q$.

Which of the following is an expression for the cost, in pounds (£), of a pear?

$P$ is directly proportional to $Q$ squared.

When $P$ is 2, $Q$ is 4.

$Q$ is inversely proportional to $R$.

When $Q$ is 2, $R$ is 5.

What is $P$ in terms of $R$?

Two sequences are defined by the following rules:

In sequence $S$ the $n^{\text{th}}$ term is $7n+1$

In sequence $T$ the $n^{\text{th}}$ term is $99-n^2$

What is the smallest value of $n$ for which the $n^{\text{th}}$ term of sequence $S$ is greater than the $n^{\text{th}}$ term of sequence $T$?

Which one of the following is a simplification of$$2-\frac{x^2\left(9x^2-4\right)}{x^3(2-3x)}$$

The parallelogram $OPQR$, labelled clockwise, is in the first quadrant $(x\ge 0,y\ge 0)$ with $O$ at the origin.

The point $R$ has coordinates $\left(\frac{3a}{2},0\right)$ and the point $Q$ has coordinates $(2a,a+1)$.

The area of $OPQR$ is 9 square units.

What are the coordinates of point $P$?

What is the value of $x$ that makes the following expression correct?$$2^{3+2x}{4}^x{8}^{-x}=4\sqrt{2}$$

There are 100 students in Year 10.

Each student studies exactly one of French, German, and Spanish.

$X$ girls study French and there are $3X$ girls in total.

$2Y$ boys study German.

There are 35 students studying Spanish of which $Y$ are boys.

Which of the following is an expression for the total number of students studying German?

An exterior angle of a regular polygon with $n$ sides is $4^{\circ }$ larger than an exterior angle of a regular polygon with $(n+3)$ sides.

What is the value of $n$?

The bearing of a ship $R$ from a lighthouse $L$ is $220^{\circ }$

A canoe $C$ is due North of $R$.

$C$ is the same distance from the ship and the lighthouse.

What is the bearing of $L$ from $C$?

The hands of a 12-hour analogue clock move continuously. When the time on the clock is 4:00, the angle between the minute hand and the hour hand is $120^{\circ }$.

What is the angle between the two hands at 4:40?

The cost of manufacturing a cake is directly proportional to the volume of the cake.

The baker makes a 70% profit when he sells a large rectangular cake.

The baker sells a large rectangular cake for £6.80

The baker decides to sell smaller rectangular cakes. The length, width, and height of the smaller cakes are all half of those of the large rectangular cake.

He sells a pack of 6 of the smaller cakes for £6.50

How much profit does he make on the pack of smaller cakes?

A pet shop has 4 female rabbits and $x$ male rabbits for sale.

A customer buys 2 of the rabbits, chosen at random, and each rabbit is equally likely to be chosen.

The probability that both the chosen rabbits are male is $\frac{1}{3}$.

What is the value of $x$?

The diagram shows a square with side of length $x$ cm. A circle is drawn with centre O which lies at the mid-point of one of the sides of the square. This side forms part of a diameter of the circle. The circle passes through two corners of the square as shown.

What is the area, in ${\text{cm}}^2$, of the shaded part of the semi-circle?

A cylindrical hollow metal pipe is 16 cm long.

It has an external diameter of 10 cm and an internal diameter of 8 cm.

The density of the metal from which the pipe is made is 8 grams per ${\text{cm}}^3$.

What is the mass of the pipe in grams?

The shortest distance between two opposite sides of a regular hexagon is 12 cm.

Find the area, in ${\text{cm}}^2$, of the regular hexagon.