May/June 2024

Write $3y^2 - 12y - 15$ in the form $3(y + a)^2 + b$, with $a$ and $b$ as constants.

Quadratics

The circle is given by $(x - 3)^2 + y^2 = 18$. A line of the form $y = mx + c$ passes through $(0, -9)$ and is tangent to the circle.

Coordinate geometry

The function has rule $f(x) = \frac{4}{x^3} - \frac{3}{x} + 2$ for $x \neq 0$. Its graph, $y = f(x)$, is displayed in the diagram.

Differentiation

The diagram contains two curves. One has equation $y = \sin x$, while the other has equation $y = f(x)$.

Functions

Within the expansion of $(3 + ax)^6$, the coefficient of $x^3$ is $160$.

Series

The curve is defined by $y=f(x)$, where $f(x)=(2x-1)\sqrt{3x-2}-2$. The points below are on the curve. Values that are not exact have been rounded to $5$ decimal places. $A(2,4)$, $B(2.0001,k)$, $C(2.001,4.00625)$, $D(2.01,4.06261)$, $E(2.1,4.63566)$, $F(3,11.22876)$. The table gives the gradients of chords $AB$, $AC$, $AD$ and $AF$.

Differentiation

Prove the identity $\dfrac{\sin^2 x - \cos x - 1}{1 + \cos x} = -\cos x$.

Trigonometry

The function $f$ is given by $f(x)=\dfrac{2}{x^2}+4$ for $x<0$, and the diagram displays the graph of $y=f(x)$.

Functions

The figure shows $AOD$ and $BC$ as a pair of parallel straight lines. Arc $AB$ lies on a circle with centre $O$ and radius $15\text{ cm}$. The angle $BOA$ is $\theta$ radians. Arc $CD$ lies on a circle with centre $O$ and radius $10\text{ cm}$. The angle $COD$ is $\frac{1}{2}\pi$ radians.

Circular measure

The initial three terms of an arithmetic progression are $25$, $4p - 1$ and $13 - p$, with $p$ as a constant. Determine the value of the tenth term of the progression.

Series

The diagram presents part of the curve whose equation is $y = \frac{1}{(5x - 4)^3}$, together with the lines $x = 2.4$ and $y = 1$. The curve meets the line $y = 1$ at the point $(1, 1)$.

Integration

For the expansion of $(1 - 4x)^6$, the coefficient of $x^2$ is 12 times the coefficient of $x^2$ in the expansion of $(2 + ax)^5$.

Series

A curve is given by $y = (5 - 2x)^{\tfrac{3}{2}} + 5$ for $x < \tfrac{5}{2}$.

Differentiation

The curve $y = x^2$ is mapped onto the curve $y = 4(x - 3)^2 - 8$.

Functions

Show that $\frac{7\tan\theta}{\cos\theta} + 12 = 0$ may be rewritten as $12\sin^2\theta - 7\sin\theta - 12 = 0$.

Trigonometry

The function $f$ is specified by $f(x) = \sqrt{x} - 1$ for $x > 1$.

Functions

The first two terms of an arithmetic progression are $\tan \theta$ and $\sin \theta$, respectively, where $0 < \theta < \frac{1}{2}\pi$.

Series

The curve given by $y = 2x - 8x^{\frac{1}{2}}$ reaches a minimum at $A$ and cuts the positive $x$-axis at $B$.

Integration

A circle is given by $(x - 6)^2 + (y + a)^2 = 18$. The line $y = 2a - x$ touches the circle, so it is a tangent.

Coordinate geometry

The diagram depicts a symmetric plate $ABCDEF$. The line $ABCD$ is straight, and $BC$ measures $2\text{ cm}$. The two sectors $ABF$ and $DCE$ each have radius $r\text{ cm}$, and each angle $ABF$ and $DCE$ is $\frac{1}{3}\pi$ radians.

Circular measure

The function $f$ has derivative $f'(x) = 6(2x - 3)^2 - 6x$ for $x \in \mathbb{R}$.

Differentiation

Find the coefficient of $x^2$ when $(2 - 5x)(1 + 3x)^{10}$ is expanded.

Series

For the geometric progression $a_1, a_2, a_3, \ldots$, the first term is $2$ and the common ratio is $r$, where $r > 0$. It is also given that $\frac{2}{9} a_5 + 7a_3 = 8$.

Series

The function $f$ is given by $f(x) = 10 + 6x - x^2$ for all $x \in \mathbb{R}$.

Functions

The diagram depicts the curve $y = k\cos(x - \frac{1}{6}\pi)$, where $k$ is a positive constant and $x$ is measured in radians. The curve meets the $x$-axis at point $A$, and $B$ is a minimum point.

Trigonometry

The diagram depicts a sector of a circle with centre $C$. The radii $CA$ and $CB$ each measure $r$ cm, and the reflex angle $ACB$ has size $\theta$ radians. The shaded sector has perimeter $65$ cm and area $225\text{ cm}^2$.

Circular measure

Show that $\cos\theta(7\tan\theta - 5\cos\theta) = 1$ can be transformed into $a\sin^2\theta + b\sin\theta + c = 0$, with integer values for $a$, $b$ and $c$ to be determined.

Trigonometry

The curve is defined by $y = 2x^2 - \frac{1}{2x} + 3$.

Differentiation

A curve goes through the point $\left(\frac{4}{5}, -3\right)$ and has $\frac{dy}{dx} = \frac{-20}{(5x - 3)^2}$.

Functions

An arithmetic progression has first term $1.5$, and the total of its first ten terms is $127.5$.

Series

The circle with equation $x^2 + y^2 - 6x + 2y - 15 = 0$ crosses the $y$-axis at the points $A$ and $B$. The tangents to the circle at $A$ and $B$ intersect at the point $P$.

Coordinate geometry

The diagram presents the curve with equation $y = \sqrt{2x^3 + 10}$.

Integration

The curve is defined by $y = 2\tan x - 5\sin x$ for $0 \leq x < \frac{1}{2}\pi$.

Differentiation

The curve is defined by $x^2 \ln y + y^2 + 4x = 9$.

Differentiation

On a single diagram, sketch the graphs of $y = |3x - 8|$ and $y = 5 - x$.

Algebra

Show that the expression $3\tan 2\theta + \tan(\theta + 45^\circ)$ is equivalent to $\dfrac{\tan^2\theta + 8\tan\theta + 1}{1 - \tan^2\theta}$.

Trigonometry

A curve is defined by $y = \frac{1 + e^{2x}}{1 + 3x}$. It has one and only one stationary point $P$.

Numerical solution of equations

The sketch displays the curve given by $y = \sqrt{\sin 2x + \sin^2 2x}$ for $0 \leq x \leq \frac{1}{6}\pi$. The shaded area lies between the curve, the vertical line $x = \frac{1}{6}\pi$ and the horizontal line $y = 0$.

Integration

The polynomial $p(x)$ has been defined as $p(x) = 9x^3 + 6x^2 + 12x + k$, with $k$ a constant.

Integration

Solve for the values of $x$ in $|5x + 7| > |2x - 3|$.

Algebra

Apply logarithms to solve the equation $6^{2x - 1} = 5e^{3x + 2}$. State your answer to $4$ significant figures.

Logarithmic and exponential functions

The diagram depicts the curve with equation $y = 8e^{-x} - e^{2x}$. The curve meets the y-axis at point $A$ and the x-axis at point $B$. The shaded area is enclosed by the curve and the two axes.

Integration

The curve has the parametric equations $x = 4\cos^2 t$, $y = \sqrt{3}\sin 2t$, for values of $t$ satisfying $0 < t < \frac{1}{2}\pi$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = 9x^3 + 18x^2 + 5x + 4$.

Integration

The diagram represents the curve given by the equation $y = \frac{\ln(2x + 1)}{x + 3}$. This curve has a maximum point $M$.

Numerical solution of equations

Prove that the identity $2\sin\theta \cosec 2\theta \equiv \sec\theta$ is true.

Trigonometry

Solve the inequality given by $|5x + 7| > |2x - 3|$.

Algebra

Solve $6^{2x - 1} = 5\mathrm{e}^{3x + 2}$ by using logarithms. Give your answer correct to $4$ significant figures.

Logarithmic and exponential functions

The diagram depicts the curve with equation $y = 8\mathrm{e}^{-x} - \mathrm{e}^{2x}$. This curve meets the $y$-axis at $A$ and the $x$-axis at $B$. The shaded area lies between the curve and the two coordinate axes.

Integration

A curve is given by the parametric equations $x = 4\cos^2 t$, $y = \sqrt{3}\sin 2t$, for t-values satisfying $0 < t < \frac{1}{2}\pi$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = 9x^3 + 18x^2 + 5x + 4$.

Integration

The graph depicts the curve with equation $y = \frac{\ln(2x + 1)}{x + 3}$. This curve includes a maximum point $M$.

Numerical solution of equations

Prove that $2\sin\theta \cosec 2\theta \equiv \sec\theta$.

Trigonometry

Expand $(3 + x)\sqrt{1 - 2x}$ as a series in ascending powers of $x$, including terms up to and including $x^2$, and simplify the coefficients.

Algebra

If $2x = \tan y$, prove that $\frac{dy}{dx} = \frac{2}{1 + 4x^2}$.

Integration

A field contains 300 plants of one species, and every one of them may catch a particular disease. Let x be the number infected at time t after the first plant becomes infected. The rate at which x changes is proportional to the product of the number already infected and the number not yet infected. The variables x and t are regarded as continuous, and it is given that $\frac{dx}{dt} = 0.2$ and $x = 1$ when $t = 0$.

Differential equations

Solve the equation $\ln(x - 5) = 7 - \ln x$. Give your answer correct to $2$ decimal places.

Logarithmic and exponential functions

The variables $x$ and $y$ are linked by $a^y = bx$, where $a$ and $b$ are constants. The graph of $y$ plotted against $\ln x$ is a straight line that passes through $(0.336, 1.00)$ and $(1.31, 1.50)$, as shown in the diagram.

Numerical solution of equations

The complex number $u$ is defined by $u = -1 - i\sqrt{3}$.

Complex numbers

The curve is given by $y = \frac{e^{\sin x}}{\cos^2 x}$ for $0 \leq x \leq 2\pi$.

Differentiation

By sketching an appropriate pair of graphs, demonstrate that the equation $\cosec\,\frac{1}{2}x = e^{x} - 3$ has exactly one root, called $\alpha$, in the interval $0 < x < \pi$.

Numerical solution of equations

On a single Argand diagram, sketch the loci described by the equations $|z - 3 + 2i| = 2$ and $|w - 3 + 2i| = |w + 3 - 4i|$ where $z$ and $w$ are complex numbers.

Complex numbers

Take the substitution $u = 1 - \sin x$.

Integration

Two straight lines, $l_1$ and $l_2$, have the equations $l_1: \mathbf{r} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + a\mathbf{k})$ and $l_2: \mathbf{r} = -\mathbf{i} - \mathbf{j} - \mathbf{k} + \mu(3\mathbf{i} - 2\mathbf{j} - 2\mathbf{k})$, where $a$ is constant. These lines, $l_1$ and $l_2$, are perpendicular.

Vectors

For this question, $a$ is a positive constant.

Algebra

If $y = \sec^3\theta$ is rewritten as $\frac{1}{\cos^3\theta}$, demonstrate that $\frac{dy}{d\theta} = 3\sin\theta\sec^4\theta$.

Differential equations

Write $\frac{6x^2 - 9x - 16}{2x^2 - 5x - 12}$ as partial fractions.

Algebra

The variables $x$ and $y$ are related by $a^{2y-1} = b^{x-y}$, with $a$ and $b$ constant.

Algebra

A curve is given by the equation $ye^{2x} + y^2 e^x = 6$.

Differentiation

You are told that $e^{2x} = 5 + \cos 3x$ has just one root. By calculation, show that this root is within $0.7 \le x \le 0.8$.

Numerical solution of equations

The diagram represents the curve $y = x e^{-ax}$, with $a$ as a positive constant, together with its maximum point $M$.

Integration

Show that, after simplification, $\cos^4 \theta - \sin^4 \theta = \cos 2\theta$.

Integration

Points $A$, $B$ and $C$ are given by position vectors $\overrightarrow{OA} = -2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$, $\overrightarrow{OB} = 5\mathbf{i} + 2\mathbf{j}$ and $\overrightarrow{OC} = 8\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}$, with $O$ as the origin. The line $l_1$ goes through $B$ and $C$.

Vectors

The complex numbers $z$ and $\omega$ are given by $z = 1 - i$ and $\omega = -3 + 3\sqrt{3}i$.

Complex numbers

Solve the equation $8^{3-6x} = 4 \times 5^{-2x}$. Present your answer correct to $3$ decimal places.

Logarithmic and exponential functions

The equations for two straight lines are $\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k})$ and $\mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$, with $a$ as a constant.

Vectors

Apply the substitution $2x = \tan\theta$ to determine the exact value of the integral.

Integration

Determine the exact coordinates of the stationary point on the curve $y = e^{2x} \sin 2x$ for $0 \leq x \leq \frac{1}{2}\pi$.

Differentiation

The square roots of $24 - 7i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ both real and exact.

Complex numbers

The variables $x$ and $y$ obey the equation $ky = e^{cx}$, with $k$ and $c$ as constants. The plot of $ y$ against $x$ is a straight line that goes through the points $(2.80, 0.372)$ and $(5.10, 2.21)$, as the diagram shows.

Differential equations

Write $\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}$ as a sum of partial fractions.

Algebra

On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both $|z - 4 - 3i| \leq 2$ and $\arg(z - 2 - i) \geq \frac{1}{3}\pi$.

Complex numbers

Consider the function $f(x) = 8x^3 + 54x^2 - 17x - 21$.

Trigonometry

Express $3\cos 2x - \sqrt{3}\sin 2x$ in the form $R\cos(2x + \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. Give the exact values of $R$ and $\alpha$.

Trigonometry

A container shaped as a cuboid has a square base with side $x$ and height $(10-x)$. It is stated that $x$ changes with time, $t$, where $t>0$. The container's volume is falling at a rate inversely proportional to $t$. When $t=\tfrac{1}{10}$, $x=\tfrac{1}{2}$ and the rate at which $x$ decreases is $\tfrac{20}{37}$.

Differential equations

A car is initially at rest, then accelerates at $2\,\text{m}\,\text{s}^{-2}$ for $10\,\text{s}$. It then continues at a steady speed for $30\,\text{s}$. After that, the car decelerates uniformly until it comes to rest over $20\,\text{s}$.

Kinematics of motion in a straight line

The forces, with magnitudes $20\,\text{N}$ and $F\,\text{N}$, act at point $P$ in the directions indicated in the diagram.

Forces and equilibrium

A train with mass $180\,000\,\text{kg}$ climbs a straight hill of length $1.5\,\text{km}$, which is tilted at an angle of $1.5^\circ$ to the horizontal. During the climb, the total work needed to overcome the resistance to motion is $12\,000\,\text{kJ}$, and the train's speed drops from $45\,\text{m s}^{-1}$ to $40\,\text{m s}^{-1}$.

Energy, work and power

A car with mass $1700\,\text{kg}$ is towing a trailer with mass $300\,\text{kg}$ on a level straight road. A light inextensible cable, parallel to the road, joins the car to the trailer. The constant resistive forces opposing motion are $400\,\text{N}$ on the car and $150\,\text{N}$ on the trailer. The car’s engine has power $14\,000\,\text{W}$.

Newton's laws of motion

A straight slope with length $60\,\text{m}$ is inclined at an angle of $12^\circ$ to the horizontal. A bobsled begins at the top of the slope with a speed of $5\,\text{m s}^{-1}$. It travels directly down the slope.

Energy, work and power

A particle travels along a straight line, beginning at point $O$. At time $t\,\text{s}$ after leaving $O$, its velocity is $v\,\text{m s}^{-1}$. It is given that $v = kt^2 - 2t - 8$, where $k$ is a positive constant. The greatest velocity of the particle is $4.5\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

Particle $P$, with mass $0.2\,\text{kg}$, is launched vertically upwards from level ground at $25\,\text{m s}^{-1}$.

Momentum

The cyclist together with the bicycle has a combined mass of $72\,\text{kg}$. While moving on a horizontal road, the cyclist experiences a total resistive force of $28\,\text{N}$.

Energy, work and power

A particle $P$ travels along a straight line. At time $t\,\text{s}$ after departing from point $O$ on the line, $P$ has velocity $v\,\text{m s}^{-1}$, where $v = 44t - 6t^2 - 36$.

Kinematics of motion in a straight line

Four coplanar forces with magnitudes $P\,\text{N}$, $10\,\text{N}$, $16\,\text{N}$ and $2\,\text{N}$ act at a point in the directions indicated in the diagram. The forces are in equilibrium.

Forces and equilibrium

A car of mass $1400\,\text{kg}$ has a resistance-to-motion magnitude of $kv^2\,\text{N}$ when its speed is $v\,\text{m s}^{-1}$, with $k$ as a constant.

Energy, work and power

A particle with mass $0.8\,\text{kg}$ rests on a rough plane inclined at an angle of $28^\circ$ to the horizontal. It is maintained in equilibrium by a force of magnitude $T\,\text{N}$. This force acts at an angle of $35^\circ$ above the line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is $0.2$.

Forces and equilibrium

Particles $A$, $B$ and $C$, with masses $5\text{ kg}$, $1\text{ kg}$ and $2\text{ kg}$ respectively, are initially at rest in that sequence on the straight smooth horizontal track $XYZ$. At the start, $A$ is at $X$, $B$ is at $Y$ and $C$ is at $Z$. $A$ is launched towards $B$ with speed $6\,\text{m s}^{-1}$, and at the same moment $C$ is launched towards $B$ with speed $v\,\text{m s}^{-1}$. In the later motion, $A$ collides with $B$ and they coalesce to form particle $D$. Then $D$ collides with $C$ and they coalesce to form particle $E$, which then moves towards $Z$.

Momentum

Particles $P$ and $Q$, with masses $2.5\text{ kg}$ and $0.5\text{ kg}$ respectively, are joined by a light inextensible string that runs over a small smooth pulley fixed at the top of a plane inclined at an angle of $30^\circ$ to the horizontal. $P$ rests on the plane while $Q$ hangs below the pulley, with $Q$ initially $2\,\text{m}$ lower than $P$ in level (see diagram). $P$ is let go from rest with the string taut and moves down the plane. The plane is rough, and the coefficient of friction between the plane and $P$ is $0.2$.

Energy, work and power

Particles $P$ and $Q$, with masses $0.2\,\mathrm{kg}$ and $0.5\,\mathrm{kg}$, are initially stationary on a smooth horizontal plane. $P$ is then sent towards $Q$ at a speed of $6\,\mathrm{m\,s^{-1}}$. Once $P$ and $Q$ have collided, the speed of $P$ is $1\,\mathrm{m\,s^{-1}}$.

Momentum

A particle of mass $0.2\,\mathrm{kg}$ is fastened to one end of a light inextensible string. The opposite end of the string is fixed at a point on a vertical wall. The particle is held in equilibrium by a force of magnitude $X\,\mathrm{N}$, acting at right angles to the string, with the string taut and inclined at an angle of $30^\circ$ to the wall (see the diagram).

Forces and equilibrium

A car moves on a straight road with constant acceleration $a\,\mathrm{m\,s^{-2}}$, where $a>0$. The car goes through the points $A$, $B$ and $C$ in that order. At $A$, the car has speed $u\,\mathrm{m\,s^{-1}}$ in the direction $AB$. The distance $BC$ is twice the distance $AB$. The car takes $8$ seconds to go from $A$ to $B$ and $10$ seconds to go from $B$ to $C$.

Kinematics of motion in a straight line

A particle moves along a straight line. At time $t\,\text{s}$ after departing from a point $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = kt^2 - 4t + 3$. The particle covers $6\,\text{m}$ during the first $2\,\text{s}$ of its motion. You may take $v > 0$ throughout the first $2\,\text{s}$.

Kinematics of motion in a straight line

A van with mass $4500\,\text{kg}$ is pulling a trailer with mass $750\,\text{kg}$ along a straight hill that slopes at angle $\theta$ to the horizontal, where $\sin \theta = 0.05$. The van and trailer are joined by a light rigid tow-bar parallel to the road. Constant resistive forces act on the van and the trailer, with $2500\,\text{N}$ on the van and $300\,\text{N}$ on the trailer.

Newton's laws of motion

A cyclist is moving on a straight, level road. The combined mass of the cyclist and her bicycle is $80\,\text{kg}$. A constant resistive force of magnitude $32\,\text{N}$ opposes the cyclist’s motion. At a moment when she is moving at $7\,\text{m s}^{-1}$, her acceleration is $0.1\,\text{m s}^{-2}$.

Energy, work and power

The track $ABCD$ is shown lying in a vertical plane. $AB$ is a straight smooth slope at an angle of $30^\circ$ to the horizontal, $BC$ is a horizontal rough straight section, and $CD$ is another rough straight slope at an angle of $30^\circ$ to the horizontal. Each of $AB$, $BC$ and $CD$ has length $2\,\text{m}$. A particle is let go from rest at $A$. The coefficient of friction between the particle and both $BC$ and $CD$ is $\mu$. The particle’s speed is unchanged as it goes through $B$ and also as it goes through $C$.

Energy, work and power

For 20 values of $x$, the totals are $\sum (x - 30) = 439$ and $\sum (x - 30)^2 = 12\,405$. For a further 25 values of $x$, the totals are $\sum (x - 30) = 470$ and $\sum (x - 30)^2 = 11\,346$.

Representation of data

Adult raccoons of one particular species have tail lengths that are normally distributed, with mean $28\text{ cm}$ and standard deviation $3.3\text{ cm}$.

The normal distribution

The table below gives the heights, in cm, of 200 adults in Barimba: height intervals $130 \le h < 150$, $150 \le h < 160$, $160 \le h < 170$, $170 \le h < 175$, $175 \le h < 195$ with matching frequencies $16$, $32$, $76$, $64$, $12$.

Representation of data

A game between two players uses a fair 4-sided dice with faces numbered $1$, $2$, $3$ and $4$. In a turn, the dice is thrown repeatedly, with no more than three throws allowed. As soon as a $4$ appears, the turn ends and no further throws are taken in that turn. Any player who throws a $4$ in a turn earns $1$ point.

Probability

For a particular region of the Arctic, the chance of snow on any one day is $0.7$, and this does not depend on any other day.

The normal distribution

Harry has three coins. One of the coins is biased, with the chance of getting a head on a throw equal to $\frac{1}{3}$. The second coin is biased, with the chance of getting a head on a throw equal to $\frac{1}{4}$. The third coin is biased, with the chance of getting a head on a throw equal to $\frac{1}{5}$. Harry throws all three coins. The random variable $X$ counts the number of heads he gets.

Discrete random variables

The eight digits $1, 2, 2, 3, 4, 4, 4, 5$ are placed in one row.

Permutations and combinations

Each year, Rajesh enters once for a ticket to a music festival. His chance of success in any given year is $0.3$, and the outcomes in different years are independent.

Discrete random variables

Seva owns a coin that is biased so that, whenever it is tossed, the chance of getting a head is $\frac{1}{3}$. He also has a bag with $4$ red marbles and $5$ blue marbles. Seva throws the coin. If a head is obtained, he chooses one marble from the bag at random. If a tail is obtained, he chooses two marbles from the bag at random without replacement.

Probability

A normal distribution with mean $131$ grams and standard deviation $54$ grams can be used to represent orange weights. Oranges are put into the categories small, medium or large. A large orange has a weight of at least $184$ grams, and $20\%$ of oranges are in the small category.

The normal distribution

The paired back-to-back stem-and-leaf diagram gives the yearly salaries of 19 workers in each of the two firms, Petral and Ravon. Key: $2\,|\,31\,|\,5$ represents a Petral worker earning $\$31\,200$ and a Ravon worker earning $\$31\,500$.

Representation of data

Jasmine owns one $5 coin, two $2 coins and two $1 coins. She picks two of these coins at random. The random variable $X$ gives the total value, in dollars, of the two chosen coins.

Discrete random variables

Residents of Mahjing were surveyed and asked to sort their local bus service: 25% of residents described the service as good. 60% of residents described the service as satisfactory. 15% of residents described the service as poor.

Discrete random variables

What is the number of distinct arrangements of the $10$ letters in REGENERATE?

Permutations and combinations