May/June 2022

Write $x^2 - 8x + 11$ in the form $(x + p)^2 + q$, with $p$ and $q$ both constants.

Quadratics

The curve is given by $\frac{d^2 y}{dx^2} = 6x^2 - \frac{4}{x^3}$. Its stationary point is at $(-1, \frac{9}{2})$.

Differentiation

In this arithmetic progression, the 13th term is $12$ and the total of the first $30$ terms is $-15$.

Series

In the expansion of $(2x^2 + \frac{k^2}{x})^5$, let the coefficient of $x^4$ be $a$. In the expansion of $(2kx - 1)^4$, let the coefficient of $x^2$ be $b$.

Quadratics

Prove the identity $$\frac{\sin^3 \theta}{\sin \theta - 1} - \frac{\sin^2 \theta}{1 + \sin \theta} = -\tan^2 \theta (1 + \sin^2 \theta).$$

Trigonometry

The diagram depicts sector $ABC$ from a circle whose centre is $A$ and whose radius is $r$. The line $BD$ is at right angles to $AC$. The angle $CAB$ is $\theta$ radians.

Circular measure

The function $f$ is given by $f(x) = \dfrac{x^2 - 4}{x^2 + 4}$, with the condition $x > 2$.

Functions

The curve is defined by $y = \sqrt{3x - 2}$, while the straight line is $y = \frac{1}{2}x + 1$. These two graphs meet at points $A$ and $B$.

Integration

The graph of $y = \sin x$ is mapped onto the graph $y = 4\sin\left(\tfrac{1}{2}x - 30^\circ\right)$.

Trigonometry

The circle is given by the equation $x^2 + y^2 + 6x - 2y - 26 = 0$.

Coordinate geometry

In $(3 + x)^5$, the coefficient of $x^4$ is the same as the coefficient of $x^2$ in $(2x + \frac{a}{x})^6$.

Series

The functions $f$ and $g$ are given by $f(x) = \frac{2x + 1}{2x - 1}$ for $x \neq \frac{1}{2}$, $g(x) = x^2 + 4$ for $x \in \mathbb{R}$. The diagram shows a section of the graph of $y = f(x)$.

Functions

The function $f$ is defined by $f(x) = 4\cos^4 x + \cos^2 x - k$ for $0 \leq x \leq 2\pi$, where $k$ is a constant.

Quadratics

In a geometric progression, the second term is $10$ and the third term is $8$.

Series

For the curve, $\frac{dy}{dx} = 3\sqrt{4x - 7} - 4x^{-\frac{1}{2}}$, and it is known to pass through the point $(4, \frac{5}{2})$.

Integration

In an arithmetic progression, the first three terms are $k$, $6k$ and $k + 6$ in that order.

Series

The curve is given by $y = 4x^2 - kx + \frac{1}{2}k^2$ and the line is given by $y = x - a$, where $k$ and $a$ are constants.

Differentiation

The diagram shows the curve given by equation $y = 5x^{\frac{1}{2}}$ and the straight line given by equation $y = 2x + 2$.

Integration

The diagram depicts the sector $OBAC$ of a circle that has centre $O$ and radius $10\,\text{cm}$. Point $P$ is on $OC$, and $BP$ is at right angles to $OC$. The angle $AOC = \frac{\pi}{6}$, while the arc length $AB$ is $2\,\text{cm}$.

Circular measure

The circle is given by the equation $x^2 + y^2 + ax + by - 12 = 0$. The points $A(1, 1)$ and $B(2, -6)$ are located on the circle.

Coordinate geometry

The curve is given by $y = 3x + 1 - 4\sqrt{3x + 1}$ for $x > -\frac{1}{3}$.

Differentiation

In the expansion of $(p + \frac{1}{p}x)^4$, the coefficient of $x^3$ is 144.

Quadratics

The function $f$ is given by $f(x) = (4x + 2)^{-2}$ for $x > -\tfrac{1}{2}$.

Integration

The point $P$ is on the line whose equation is $y = mx + c$, where $m$ and $c$ are positive constants. A curve is given by $y = -\tfrac{m}{x}$. There is one point $P$ on the curve for which the straight line is tangent to the curve at $P$.

Differentiation

The diagram displays a section of the curve whose equation is $y = p \sin(q\theta) + r$, with $p$, $q$ and $r$ as constants.

Trigonometry

An arithmetic progression starts with term $4$ and has common difference $d$. The total of its first $n$ terms is $5863$.

Series

The curve given by $y = x^2 + 2x - 5$ is translated by $\begin{pmatrix}-1\\3\end{pmatrix}$. Find the equation of the translated curve, and give your answer in the form $y = ax^2 + bx + c$.

Functions

Solve $6\sqrt{y} + \frac{2}{\sqrt{y}} - 7 = 0$.

Trigonometry

The function $f$ is given by $f(x) = 2x^2 - 16x + 23$ for $x < 3$.

Functions

The diagram illustrates the circle whose equation is $(x - 2)^2 + (y + 4)^2 = 20$, with centre $C$. Point $B$ is at $(0, 2)$, and the line segment $BC$ cuts the circle at $P$.

Coordinate geometry

The diagram displays the curve whose equation is $y = x^{\frac{1}{2}} + 4x^{-\frac{1}{2}}$. The line $y = 5$ cuts the curve at the points $A (1, 5)$ and $B(16, 5)$.

Integration

In the diagram, triangle $ABC$ has $AB = BC = 6\,\text{cm}$ and $\angle ABC = 1.8$ radians. The arc $CD$ is part of a circle with centre $A$, while $ABD$ is a straight line.

Circular measure

The variables $x$ and $y$ obey the equation $y = 4^{2x-a}$, where $a$ is an integer. As the diagram indicates, the graph of $\ln y$ against $x$ forms a straight line that passes through $(0, -20.8)$, with the second coordinate stated correct to $3$ significant figures.

Logarithmic and exponential functions

Express the equation $7\tan \theta + 4\cot \theta - 13\sec \theta = 0$ using $\sin \theta$ only.

Trigonometry

The diagram displays the curve whose equation is $y = 3\sin x - 3\sin 2x$ for $0 \le x \le \pi$. This curve crosses the $x$-axis at the origin and again at the points whose $x$-coordinates are $a$ and $\pi$.

Integration

The curve is given by the equation $x^2 y + 2y^3 = 48$.

Differentiation

Using sketches of $y = |5 - 2x|$ and $y = 3 \ln x$ on one diagram, show that the equation $|5 - 2x| = 3 \ln x$ has exactly two roots.

Numerical solution of equations

The curve is given by $y = \frac{9e^{2x} + 16}{e^x - 1}$.

Differentiation

The polynomial $p(x)$ has definition $p(x) = 2x^3 + 5x^2 + ax + 2a$, and $a$ is an integer.

Integration

Suppose that $y = \frac{\ln x}{x^2}$,

Differentiation

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 5x - 3$.

Algebra

A curve is given by $e^{2x}\cos 2y + \sin y = 1$.

Differentiation

Use the trapezium rule with three intervals to show that the value of $\int_{1}^{4} \ln x\,dx$ comes out as approximately $\ln 12$.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = 2x^3 + ax^2 - 3x - 4$, with $a$ a constant. It is stated that $(x - 4)$ is a factor of $p(x)$.

Logarithmic and exponential functions

The diagram illustrates the curve $y = 3e^{2x-1}$. The shaded region is enclosed by the curve and the lines $x = a$, $x = a + 1$ and $y = 0$, with $a$ a constant. The area of the shaded region is stated to be $120$ square units.

Numerical solution of equations

The diagram plots the curves $y = \sqrt{2\pi - 2x}$ and $y = \sin^2 x$ over $0 \leq x \leq \pi$. The shaded area is enclosed by the two curves and the line $x = 0$.

Integration

Express $3\sin 2\theta \sec \theta + 10\cos(\theta - 30^\circ)$ in the form $R \sin(\theta + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the value of $\alpha$ correct to 2 decimal places.

Trigonometry

Suppose that $y = \frac{\ln x}{x^2}$.

Differentiation

Sketch the graphs of $y = |2x - 9|$ and $y = 5x - 3$ on the same diagram.

Algebra

The curve is given by the equation $e^{2x}\cos 2y + \sin y = 1$.

Differentiation

Apply the trapezium rule using three intervals to demonstrate that $\int_{1}^{4} \ln x\,dx$ is approximately $\ln 12$.

Integration

The polynomial $p(x)$ is given by $p(x) = 2x^3 + ax^2 - 3x - 4$, with $a$ a constant. It is stated that $(x - 4)$ is a factor of $p(x)$.

Logarithmic and exponential functions

The figure displays the curve $y = 3e^{2x-1}$. The shaded area is enclosed by the curve together with the lines $x = a$, $x = a + 1$ and $y = 0$, where $a$ is a constant. The area of this shaded region is given as $120$ square units.

Numerical solution of equations

The diagram depicts the curves $y = \sqrt{2\pi - 2x}$ and $y = \sin^2 x$ for $0 \leq x \leq \pi$. The shaded area lies between the two curves and the line $x = 0$.

Integration

Express $3\sin 2\theta \sec \theta + 10\cos(\theta - 30^\circ)$ in the form $R\sin(\theta + \alpha)$ where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the value of $\alpha$ correct to 2 decimal places.

Trigonometry

Solve the equation $2(3^{2x-1}) = 4^{x+1}$, and give your answer rounded to 2 decimal places.

Logarithmic and exponential functions

The graph of $y = x\sqrt{\sin x}$ has a single stationary point for $0 < x < \pi$, and this occurs when $x = a$ (see diagram).

Numerical solution of equations

Expand $(2-x^2)^{-2}$ as a series in ascending powers of $x$, up to and including the term in $x^4$, and simplify the coefficients.

Algebra

Solve $2\cot 2x + 3\cot x = 5$ for $0^\circ < x < 180^\circ$.

Trigonometry

The differential equation $\frac{dy}{dx} = \frac{xy}{1 + x^2}$ is satisfied by the variables $x$ and $y$, and when $x = 0$, $y = 2$.

Differential equations

Let the polynomial $ax^3 - 10x^2 + bx + 8$, where $a$ and $b$ are constants, be represented by $p(x)$. It is stated that $(x - 2)$ is a factor of both $p(x)$ and $p'(x)$.

Algebra

Let $I = \int_0^3 \frac{27}{\left(9 + x^2\right)^2}\,dx$.

Integration

The complex number $u$ is defined as $u = \dfrac{\sqrt{2} - a\sqrt{2} i}{1 + 2i}$, where $a$ is a positive integer.

Complex numbers

The curve is given by $x^3 + y^3 + 2xy + 8 = 0$.

Differentiation

In the diagram, $OABCDEFG$ forms a cuboid, with $OA = 2$ units, $OC = 4$ units and $OG = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OG$ respectively. Point $M$ is the midpoint of $DF$. Point $N$ lies on $AB$ so that $AN = 3NB$.

Vectors

Solve the equation $\ln(\mathrm{e}^{2x} + 3) = 2x + \ln 3$, and give your answer correct to $3$ decimal places.

Logarithmic and exponential functions

Let $\mu$ stand for the complex number $-1 + \sqrt{7}i$. It is stated that $\mu$ is a root of $2x^3 + 3x^2 + 14x + k = 0$, where $k$ is a real constant.

Complex numbers

Solve $3\cos 2\theta = 3\cos \theta + 2$, for $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

Let the polynomial $ax^3 + x^2 + bx + 3$ be represented by $p(x)$. You are told that $p(x)$ is divisible by $(2x - 1)$ and that, when $p(x)$ is divided by $(x + 2)$, the remainder is $5$.

Algebra

The curve has equation $y = \cos^3 x\sqrt{\sin x}$. It is stated that there is exactly one stationary point for the curve in the interval $0 < x < \frac{1}{2}\pi$.

Differentiation

By sketching an appropriate pair of graphs, show that the equation $\ln x = 3x - x^2$ has one real root.

Numerical solution of equations

The variables $x$ and $y$ are related by the differential equation $\frac{dy}{dx} = xe^{y-x}$, and $y=0$ when $x=0$.

Differential equations

A curve is given by $x^3 + 3x^2y - y^3 = 3$.

Differentiation

Define $f(x)$ by $f(x) = \dfrac{x^2 + 9x}{(3x - 1)(x^2 + 3)}$.

Integration

The vector equations for the lines $l$ and $m$ are $\mathbf{r} = -\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 5\mathbf{i} + 4\mathbf{j} + 3\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} + \mathbf{k})$ respectively, where $a$ and $b$ are constants.

Vectors

Find the set of x-values, in terms of $a$, that satisfy the inequality, with $a$ being a positive constant.

Algebra

The constant $a$ satisfies $\int_{1}^{a} x^{2} \ln x \, dx = 4$.

Numerical solution of equations

Solve the equation $\cos(\theta - 60^\circ) = 3\sin\theta$, where $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

Show that the equation $\log_3(2x + 1) = 1 + 2\log_3(x - 1)$ can be rearranged into a quadratic equation in $x$.

Logarithmic and exponential functions

The curve $y = e^{-4x}\tan x$ has two stationary points for $0 \leq x < \tfrac{1}{2}\pi$.

Differentiation

Let $u$ stand for the complex number $3 - i$.

Complex numbers

A curve is described parametrically by $x = \frac{1}{\cos t}$ and $y = \ln \tan t$, where $0 < t < \frac{1}{2}\pi$.

Differentiation

Define $f(x)$ by $f(x) = \dfrac{5x^2 + 8x - 3}{(x - 2)(2x^2 + 3)}$.

Algebra

Let $N$ represent the number of insects in the population at $t$ days after observations begin. The change in the insect count is described by the differential equation $\frac{dN}{dt} = k N^{\frac{3}{2}} \cos 0.02t$, where $k$ is a constant and $N$ is regarded as a continuous variable. It is also given that $N = 100$ when $t = 0$.

Differential equations

With respect to origin $O$, point $A$ has position vector $overrightarrow{OA} = \mathbf{i} + 5\mathbf{j} + 6\mathbf{k}$. The line $l$ is represented by $\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

Vectors

A car begins from rest and travels in a straight line with constant acceleration over a distance of $200\,\text{m}$, arriving at a speed of $25\,\text{m s}^{-1}$. It then continues at this same speed for $400\,\text{m}$, before slowing down uniformly to rest in $5\,\text{s}$.

Kinematics of motion in a straight line

Particles $P$ and $Q$, with masses $0.5\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are joined by a light inextensible string. The string is taut, with $P$ positioned vertically above $Q$. A force of magnitude $10\,\text{N}$ acts on $P$ vertically upwards.

Newton's laws of motion

A $300\,\text{kg}$ crate is motionless on rough horizontal ground. The coefficient of friction between the crate and the ground is $0.5$. A force of size $X\,\text{N}$, making an angle $\alpha$ above the horizontal, acts on the crate, with $\sin \alpha = 0.28$.

Forces and equilibrium

Three coplanar forces with magnitudes $20\,\text{N}$, $100\,\text{N}$ and $F\,\text{N}$ act at a point. Their directions are indicated in the diagram.

Forces and equilibrium

At point $O$ on a straight horizontal test track, racing cars $A$ and $B$ are stationary side by side. The mass of $A$ is $1200\text{ kg}$. A’s engine exerts a constant driving force of $4500\text{ N}$. By the time $A$ reaches point $P$, its speed is $25\text{ m s}^{-1}$. The distance $OP$ is $d\text{ m}$. From $O$ to $P$, the work done against the resistance force acting on $A$ is $75\,000\text{ J}$.

Newton's laws of motion

A particle leaves the point O and then travels along a straight line. The velocity v\text{ m s}^{-1} of the particle at time t\text{ s} after it has left O is given by v = k(3t^2 - 2t^3), where k is a constant.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, travel down the same line of greatest slope on a smooth plane. The plane is inclined at $30^\circ$ to the horizontal, and $A$ is situated above $B$ on the plane. At the moment of collision, the speeds of $A$ and $B$ are $3\,\text{m s}^{-1}$ and $2\,\text{m s}^{-1}$ respectively. During the collision, the speed of $A$ is lowered to $2.5\,\text{m s}^{-1}$.

Momentum

Two small smooth spheres \(A\) and \(B\), each with the same radius and with masses of \(5\,\text{kg}\) and \(3\,\text{kg}\) respectively, rest on a smooth horizontal plane. At the start, \(B\) is at rest, whereas \(A\) moves towards \(B\) at a speed of \(8.5\,\text{m s}^{-1}\). The spheres collide, and afterwards \(A\) continues in the same direction, but at one quarter of \(B\)'s speed.

Momentum

Forces with magnitudes $60\,\text{N}$, $20\,\text{N}$, $16\,\text{N}$ and $14\,\text{N}$ lie in the same plane and act at a point in the directions indicated in the diagram.

Forces and equilibrium

Particles $A$ and $B$, whose masses are $2.4\,\text{kg}$ and $1.2\,\text{kg}$ respectively, are joined by a light inextensible string that passes over a fixed smooth pulley. $A$ is initially at a height of $2.1\,\text{m}$ above a horizontal plane and $B$ is $1.5\,\text{m}$ above the plane. The particles hang vertically and are released from rest. In the motion that follows, $A$ arrives at the plane and does not rebound, and $B$ does not reach the pulley.

Newton's laws of motion

A particle $A$, travelling along a straight horizontal track at a constant speed of $8\text{ m s}^{-1}$, goes past a fixed point $O$. Four seconds later, a second particle $B$ goes past $O$, moving along a parallel track in the same direction as $A$. Particle $B$ is moving at speed $20\text{ m s}^{-1}$ as it passes $O$ and has a constant deceleration of $2\text{ m s}^{-2}$. $B$ comes to rest when it returns to $O$.

Kinematics of motion in a straight line

A block with mass $12\,\text{kg}$ rests on a plane inclined at $24^{\circ}$ to the horizontal. It is connected to a light string that lies at an angle of $36^{\circ}$ above the line of greatest slope. The string tension is $65\,\text{N}$ (see diagram). The coefficient of friction between the block and the plane is $\mu$. The block is in limiting equilibrium and is just about to move up the plane.

Forces and equilibrium

A car with mass $900\,\text{kg}$ is travelling uphill on a slope of angle $\sin^{-1}(0.12)$ above the horizontal. Its starting speed is $11\,\text{m s}^{-1}$. After $12\,\text{s}$, it has covered $150\,\text{m}$ up the slope and is moving at $16\,\text{m s}^{-1}$. The car’s engine is outputting power at a steady rate of $24\,\text{kW}$.

Energy, work and power

Particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t$ seconds is defined by $v = 0.5t$ for $0 \leq t \leq 10$, and by $v = 0.25t^2 - 8t + 60$ for $10 \leq t \leq 20$.

Kinematics of motion in a straight line

Particles $P$ and $Q$, having masses $0.3\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are initially at rest on a smooth horizontal plane. $P$ is sent off at a speed of $4\,\text{m s}^{-1}$ directly towards $Q$. Once $P$ and $Q$ collide, $Q$ starts to move with a speed of $3\,\text{m s}^{-1}$.

Momentum

Particle $P$ is projected vertically upwards from level ground. $P$ attains a maximum height of $45\,\text{m}$. After returning to the ground, $P$ comes to rest and does not rebound.

Kinematics of motion in a straight line

A particle travels along a straight line, with displacement $s$ metres at time $t$ seconds after departing from a fixed point $O$. It starts from rest and is at points $P$, $Q$ and $R$ at times $t = 5$, $t = 10$ and $t = 15$ respectively, then returns to $O$ at time $t = 20$. The distances $OP$, $OQ$ and $OR$ are $50\,\text{m}$, $150\,\text{m}$ and $200\,\text{m}$ respectively. The diagram displays a displacement-time graph representing the particle’s motion from $t = 0$ to $t = 20$. The graph is formed by two curved sections $AB$ and $CD$ and two straight sections $BC$ and $DE$.

Kinematics of motion in a straight line

The diagram depicts a block of mass $10\,\text{kg}$ hanging under a horizontal ceiling from two strings, $AC$ and $BC$, whose lengths are $0.8\,\text{m}$ and $0.6\,\text{m}$ respectively; these strings are fixed to points on the ceiling. The angle $ACB = 90^\circ$. A horizontal force of magnitude $F\,\text{N}$ acts on the block, and the block is in equilibrium.

Forces and equilibrium

A cyclist travels on a straight, level road. The combined mass of the cyclist and her bicycle is $70\text{ kg}$. At the moment when her speed is $4\text{ m s}^{-1}$, her acceleration is $0.3\text{ m s}^{-2}$. A steady resistive force of magnitude $30\text{ N}$ acts.

Energy, work and power

Particles $P$ and $Q$, with masses $0.3\text{ kg}$ and $0.2\text{ kg}$ respectively, are joined by a light inextensible string. The string goes over a fixed smooth pulley at $B$, which is fitted between two inclined planes. $P$ is on the smooth plane $AB$, inclined at $60^\circ$ to the horizontal. $Q$ is on the plane $BC$, inclined at $30^\circ$ to the horizontal. The string is taut, and the particles are able to move along the lines of greatest slope of the two planes (see diagram).

Newton's laws of motion

A particle $P$ travels along a straight line and passes through a point $O$. Its velocity $v\ \text{m s}^{-1}$ at time $t$ seconds after passing $O$ is given by $v = \frac{9}{4} + \frac{b}{(t + 1)^2} - ct^2,$ where $b$ and $c$ are positive constants. When $t = 5$, the velocity of $P$ is zero and the acceleration is $-\frac{13}{12}\ \text{m s}^{-2}$.

Kinematics of motion in a straight line

Determine the number of distinct arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.

Permutations and combinations

A Book Club contains 6 men and 8 women. The club’s committee is formed from five members. Mr Lan and Mrs Lan are both members of the club.

Permutations and combinations

The table gives a summary of the journey times to college for 2500 students. The variable is time taken $t$ minutes, with classes $0 \leq t < 20$, $20 \leq t < 30$, $30 \leq t < 40$, $40 \leq t < 60$, $60 \leq t < 90$ and the corresponding frequencies $440$, $720$, $920$, $300$, $120$.

Representation of data

Jacob has four coins. One of the coins is biased so that, when it is tossed, the chance of getting a head is $\frac{7}{10}$. The other three coins are fair. Jacob tosses all four coins once. Let the random variable $X$ be the number of heads obtained. A probability table for $X$ is shown, with $x = 0,1,2,3,4$ and probabilities $P(X=0)=\frac{3}{80}$, $P(X=1)=a$, $P(X=2)=b$, $P(X=3)=c$, $P(X=4)=\frac{7}{80}$.

Probability

For one type of leaf, lengths measured in cm follow the distribution $N(5.2,\,1.5^2)$. A second leaf type is also represented by a normal distribution. A scientist records the lengths of a random sample of 500 leaves of this type and discovers that 46 are shorter than $3\text{ cm}$ and 95 are longer than $8\text{ cm}$.

The normal distribution

Janice is taking part in a computer game. She has to clear level 1 and then level 2 in order to finish the game. At each level, she is allowed no more than two attempts. • For level 1, the chance that Janice clears it on her first attempt is $0.6$. If her first attempt is unsuccessful, the probability that she clears it on her second attempt is $0.3$. • Once Janice clears level 1, she goes straight on to level 2. • For level 2, the chance that Janice clears it on her first attempt is $0.4$. If her first attempt is unsuccessful, the probability that she clears it on her second attempt is $0.2$.

Probability

It is given that, for $n$ values of the variable $x$, $\sum (x - 200) = 446$ and $\sum x = 6846$.

Representation of data

A balanced 6-sided die has the faces marked $1, 2, 2, 3, 3, 3$. It is thrown twice. The random variable $X$ stands for the sum of the two numbers shown.

Discrete random variables

The back-to-back stem-and-leaf diagram gives the diameters, in cm, of 19 cylindrical pipes made by each of the two companies, A and B. Key: $1\,|\,35\,|\,3$ shows that the pipe diameter from company A is $0.351\text{ cm}$ and from company B is $0.353\text{ cm}$.

Representation of data

The masses, in kg, of bags of rice made by Anders are modelled by the distribution $N(2.02, 0.03^2)$.

The normal distribution

In a large college, $28\%$ of the students play no musical instrument, $52\%$ play exactly one musical instrument, and the rest play two or more musical instruments. A random sample of $12$ students is selected from the college.

The normal distribution

Find the number of distinct arrangements of the $9$ letters in CROCODILE.

Permutations and combinations

Hanna has 12 hollow chocolate eggs, and each one contains a sweet. The eggs are identical in appearance, but Hanna knows that 3 hold a red sweet, 4 hold an orange sweet and 5 hold a yellow sweet. One after another, Hanna’s three children each selects one egg at random and eats it, keeping the sweet inside.

Probability

For each of the 150 students, the time taken, $t$ minutes, to finish a puzzle was recorded. The results are summarised in the table.

Representation of data