NSAA 2023 Section 1, Part A (Mathematics questions)

The surface area of a solid sphere of radius $R$ is equal to the total surface area of 10 solid closed cylinders of radius $r$ and height $4r$.

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

(The surface area of a sphere of radius $R$ is $4\pi R^2$.)

Which of the following statements is/are correct?

1. $\frac{47}{93}÷\frac{33}{71}=\frac{71}{93}÷\frac{33}{47}$

2. 53% of 84 = 84% of 53
3. ${\left(2\sqrt{3}\right)}^2={\left(3\sqrt{2}\right)}^2$

Which of the following is a correct rearrangement of$$y=p-\frac{q-r}{s-x}$$to make $x$ the subject?

Six numbers in increasing order are:$$2,x,x,y,23.5,27.5$$The mean of the six numbers is 12.

The median of the six numbers is 7.25.

What is the mode of the six numbers?

$WXYZ$ is a square of side length 1.

$WM:MX=1:2$

$XN:NY=3:1$

$YP:PZ=4:1$

What is the area of triangle $MNP$?

Which of the following is a simplification of$$\frac{4x-8}{x^2+4x-12}÷\frac{x-6}{x^3-36x}$$

Given that$$\frac{27^{2(x-2)}}{9^{(2x-3)}}=(81{)}^{\frac{3}{2}}$$what is the value of $x$?

The solutions of the equation $3x^2-6x-2=0$ are $p$ and $q$, where $p>q$.

What is the value of $3p+2q$?

Last year, the salary of the coach of a football club was 80% of the salary of the star player.

At the start of the new year, the coach received a 15% increase in salary and the star player received a 38% increase in salary.

What percentage of the star player's new salary is the coach's new salary?

Two of the angles of a triangle are $90^{\circ }$ and $\theta$, where$$\tan \theta =\sqrt{2}$$The length of the hypotenuse of the triangle is $3\sqrt{6}$ cm.

What is the area of the triangle, in ${\text{cm}}^2$?

An athlete's training session consists of several complete repetitions of a three-part programme:

1. Walk 100 m at an average speed of $6{\text{ km h}}^{-1}$

2. Jog 200 m at an average speed of $10{\text{ km h}}^{-1}$

3. Run 100 m at an average speed of $20{\text{ km h}}^{-1}$

What is the athlete's average speed for the complete training session, in ${\text{km h}}^{-1}$?

The line joining the points with coordinates $(q,2-q)$ and $(2q+2,q-4)$ is perpendicular to the line with equation $3x-4y+5=0$

What is the value of $q$?

Two objects X and Y are similar.

The surface area of object Y is double the surface area of object X.

The volume of object Y is $7\sqrt{2}{\text{ cm}}^3$ more than the volume of object X.

What is the volume of object X, in ${\text{cm}}^3$?

$x$, $y$ and $z$ are positive variables.

$y$ is inversely proportional to the square of $x$.

When $y=20$, $x=3$

$z$ is directly proportional to $x$.

When $z=1.2$, $x=6$

What is $z$ when $y=80$?

The equation$${\left(\frac{a\times 10^4+2a\times 10^3}{3\times 10^{-1}}\right)}^2=8\times 10^9$$has two solutions for $a$.

What is the positive difference between these two solutions?

For any real number $x$, the function $V(x)$ is defined as:$$V(x)=10^{x^2+6x+7}$$What is the smallest value of $V(x)$?

$X$ and $Y$ are the end-points of a line segment.

Point $P$ has coordinates $(-8,5)$.

$P$ lies on the line segment $XY$ such that $XP:PY$ is $1:2$ and $\overrightarrow{XP}=\left(\begin{array}{c}4\\ -3\end{array}\right)$

A point $Q$ is such that $\overrightarrow{QY}=\left(\begin{array}{c}7\\ 6\end{array}\right)$

What are the coordinates of point $Q$?

The $2^{\text{nd}}$ term of a linear (arithmetic) sequence is equal to twice the $8^{\text{th}}$ term.

The $5^{\text{th}}$ term is $\frac{9}{4}$

What is the $20^{\text{th}}$ term of the sequence?

Find the maximum value of$$2^{\sin x}\times 3^{-\sin x}$$where $0^{\circ }\le x\le 360^{\circ }$

Two bags contain counters which are identical except for colour.

The first bag contains 3 blue and 2 yellow counters.

The second bag contains $x$ red, 1 blue and 2 yellow counters.

A counter is picked at random from the first bag and placed in the second bag.

A counter is then picked at random from the second bag. The probability that this counter is yellow is 0.2. What is the probability that this counter is red?