May/June 2023

Solve the equation $4\sin\theta + \tan\theta = 0$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

The diagram depicts a section of the curve with equation $y = \frac{4}{(2x - 1)^2}$ together with sections of the lines $x = 1$ and $y = 1$. The curve goes through the points $A(1, 4)$ and $B\left(\frac{3}{2}, 1\right)$.

Integration

The curve is defined by $\frac{dy}{dx} = 6x^2 - 30x + 6a$, with $a$ being a positive constant. There is a stationary point on the curve at $(a, -15)$.

Differentiation

The diagram displays circle $P$, whose centre is $(0, 2)$ and whose radius is $10$, together with the tangent to the circle at $A(6, 10)$. It also displays a second circle $Q$, centred at the point where this tangent crosses the $y$-axis, with radius $\frac{5}{2}\sqrt{5}$.

Coordinate geometry

Find the first three terms in the expansion, in ascending powers of $x$, of $(2 + 3x)^4$.

Series

The graphs of $y = f(x)$ and $y = g(x)$ are shown in the diagram.

Functions

The diagram depicts a sector $ABC$ of a circle whose centre is $A$ and whose radius is $8\text{ cm}$. The sector has area $\frac{16}{3}\pi\text{ cm}^2$. Point $D$ is located on arc $BC$.

Coordinate geometry

For positive constant $k$, the line $y = kx - k$ is just tangent to the curve $y = -\frac{1}{2x}$.

Differentiation

The initial three members of an arithmetic progression are $\frac{p^2}{6}$, $2p - 6$ and $p$.

Series

The curve is given by $y = 2 + 3\sin \frac{1}{2}x$ for $0 \leq x \leq 4\pi$.

Trigonometry

The functions $f$ and $g$ are specified below, with $a$ and $b$ taken as constants. $f(x) = 1 + \frac{2a}{x - a}$ for $x > a$ and $g(x) = bx - 2$ for $x \in \mathbb{R}$.

Functions

Water enters a tank at a steady rate of $500\,\text{cm}^3$ per second. The water depth in the tank is $h$ cm, measured $t$ seconds after filling begins. When the water depth is $h$ cm, the volume, $V\,\text{cm}^3$, of water in the tank is given by the formula $V = \dfrac{4}{3}(25 + h)^3 - \dfrac{62500}{3}$.

Differentiation

The curve is given by $\frac{dy}{dx} = \frac{4}{(x - 3)^3}$ for $x > 3$, and it passes through $(4, 5)$.

Integration

A circle has equation $(x-a)^2+(y-3)^2=20$. The straight line $y=\frac{1}{2}x+6$ is tangent to the circle at point $P$.

Coordinate geometry

The curve is given by $y=k\sqrt{4x+1}-x+5$, where $k$ is a positive constant.

Differentiation

In the expansion of $(x + a)^6$, the coefficient of $x^4$ is $p$, and in the expansion of $(ax + 3)^4$, the coefficient of $x^2$ is $q$. You are told that $p + q = 276$.

Series

Write $4x^2 - 24x + p$ in the form $a(x + b)^2 + c$, with $a$ and $b$ integers and $c$ expressed using the constant $p$.

Quadratics

Solve $8x^6 + 215x^3 - 27 = 0$.

Quadratics

The diagram illustrates the curve given by $y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}$ for $x > 0$. The curve crosses the $x$-axis at the points $(0, 0)$ and $(4, 0)$.

Integration

The diagram depicts sector $OAB$ of a circle whose centre is $O$. The angle $AOB = \theta$ radians, and $OP = AP = x$.

Circular measure

By expanding $(\cos\theta + \sin\theta)^2$ first, find the three solutions of $(\cos\theta + \sin\theta)^2 = 1$ when $0 \leq \theta \leq \pi$.

Trigonometry

The diagram presents the graph of $y = f(x)$, where $f$ is defined by $f(x) = 3 + 2\sin\left(\frac{1}{4}x\right)$ for $0 \leq x \leq 2\pi$.

Functions

The second term in a geometric progression is $16$, and the sum to infinity is $100$.

Series

The diagram displays the graph of $y = f(x)$, made up of the two straight lines $AB$ and $BC$. The lines $A'B'$ and $B'C'$ give the graph of $y = g(x)$, obtained by carrying out a sequence of two transformations, in either order, on $y = f(x)$.

Functions

The diagram illustrates the points $A\,(1\tfrac{1}{2}, 5\tfrac{1}{2})$ and $B\,(7\tfrac{1}{2}, 3\tfrac{1}{2})$ on the curve with equation $y = 9x - (2x + 1)^{\tfrac{3}{2}}$.

Integration

The function $f$ is given on $x \in \mathbb{R}$ by $f(x) = x^2 - 6x + c$, where $c$ is a constant. It is known that $f(x) > 2$ for every value of $x$.

Quadratics

Write the full expansion of $(x + \frac{2}{x})^5$.

Series

Demonstrate that the equation $3\tan^2 x - 3\sin^2 x - 4 = 0$ can be rewritten in the form $a\cos^4 x + b\cos^2 x + c = 0$, with $a$, $b$ and $c$ as the constants to determine.

Trigonometry

The circle is given by $(x - 1)^2 + (y + 4)^2 = 40$, and the line $y = x - 9$ cuts it at $A$ and $B$.

Coordinate geometry

The diagram depicts sector $OAB$ from a circle with centre $O$ and radius $r$ cm. The angle $AOB = \theta$ radians. You are told that arc length $AB$ is $9.6$ cm and that sector area $OAB$ is $76.8$ cm$^2$.

Coordinate geometry

The function $f$ is given by $f(x) = 2 - \frac{5}{x + 2}$, with $x > -2$.

Functions

In a progression, the first term is $a$ and the second term is $\frac{a^2}{a + 2}$, with $a$ being a positive constant.

Series

The curve, which goes through $(0, 3)$, is described by $y = f(x)$. Also given is $f'(x) = 1 - \frac{2}{(x-1)^3}$.

Coordinate geometry

Use logarithms to determine the solution of the equation $12^x = 3^{2x+1}$. Give your answer correct to 3 significant figures.

Logarithmic and exponential functions

The curve is described by $y = \frac{2 + 3 \ln x}{1 + 2x}$.

Differentiation

You are told that $\int_0^a (3e^{2x} - 1)\,dx = 12$, with $a$ a positive constant.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = 2x^3 + 3x^2 + kx - 30$, with $k$ as a constant. You are told that $(x - 3)$ is a factor of $p(x)$.

Algebra

The diagram depicts the curve given by the parametric equations $x = 4e^{2t}$ and $y = 5e^{-t}\cos 2t$, for $-\frac{1}{4}\pi \le t \le \frac{1}{4}\pi$. The curve includes a maximum point $M$.

Differentiation

Show that, by evaluating the integral, $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \left(4\cos^2 2x + \frac{1}{\cos^2 x}\right) \, dx = 4\sqrt{3} + \frac{\pi}{6} - 1.$$

Integration

Write $7\cos\theta + 24\sin\theta$ in the form $R\cos(\theta - \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give $\alpha$ correct to 2 decimal places.

Trigonometry

Solve the equation $\sec^2\theta + 5\tan^2\theta = 9 + 17\sec\theta$ for $0^\circ < \theta < 360^\circ$.

Trigonometry

The variables $x$ and $y$ are linked by the equation $y = Ae^{(A-B)x}$, where $A$ and $B$ are constants. The plot of $\ln y$ against $x$ is a straight line and passes through the points $(0.4, 3.6)$ and $(2.9, 14.1)$, as shown in the diagram.

Algebra

The diagram depicts part of the curve $y = \frac{6}{2x + 3}$. The shaded area is enclosed by the curve together with the lines $x = 6$ and $y = 2$.

Integration

The diagram depicts the graph of $y = 3 - e^{-\frac{1}{2}x}$. On the same diagram, sketch $y = |5x - 4|$, and demonstrate that $3 - e^{-\frac{1}{2}x} = |5x - 4|$ has precisely two real roots. The two roots of $3 - e^{-\frac{1}{2}x} = |5x - 4|$ are labelled $\alpha$ and $\beta$, with $\alpha < \beta$.

Numerical solution of equations

The diagram depicts the curve with equation $y = e^{-\frac{1}{2}x}(x^2 - 5x + 4)$. It meets the $x$-axis at the points $A$ and $B$, and reaches a maximum at $C$.

Differentiation

Show that the identity $4\sin(\theta + \frac{1}{3}\pi)\cos(\theta - \frac{1}{3}\pi) = \sqrt{3} + 2\sin 2\theta$ is true.

Trigonometry

The curve is described by the parametric equations $x = \frac{2t + 3}{t + 2}$ and $y = t^2 + at + 1$, where $a$ is a constant. At the point $P$ on the curve, the gradient is given as $1$.

Differentiation

Find the solutions of the equation $\sec^2 \theta + 5\tan^2 \theta = 9 + 17\sec \theta$ for $0^{\circ} < \theta < 360^{\circ}$.

Trigonometry

The variables $x$ and $y$ obey the equation $y = Ae^{(A-B)x}$, where $A$ and $B$ are constants. The graph of $\ln y$ against $x$ is a straight line that passes through $(0.4, 3.6)$ and $(2.9, 14.1)$, as illustrated in the diagram.

Logarithmic and exponential functions

The diagram illustrates a section of the curve $y = \frac{6}{2x + 3}$. The shaded area is enclosed by the curve together with the lines $x = 6$ and $y = 2$.

Integration

The diagram gives the graph of $y = 3 - e^{-\frac{1}{2}x}$.

Numerical solution of equations

The diagram depicts the curve whose equation is $y = e^{-\frac{1}{2}x}(x^2 - 5x + 4)$. The curve meets the $x$-axis at the points $A$ and $B$, and reaches a maximum at the point $C$.

Differentiation

Show that, by using trigonometric identities, $4\sin\left(\theta + \frac{1}{3}\pi\right)\cos\left(\theta - \frac{1}{3}\pi\right) = \sqrt{3} + 2\sin 2\theta$.

Trigonometry

A curve is given in parametric form by $x = \frac{2t + 3}{t + 2}$ and $y = t^2 + at + 1$, where $a$ is constant. At the point $P$ on the curve, the gradient is 1.

Differentiation

Find the solution of $3e^{2x} - 4e^{-2x} = 5$. State the answer correct to $3$ decimal places.

Logarithmic and exponential functions

The polynomial $x^3 + 5x^2 + 31x + 75$ is represented by $p(x)$.

Complex numbers

Sketch the graph for $y = \lvert 2x + 3 \rvert$.

Algebra

Calculate the coefficient of $x^3$ in the binomial expansion of $(3 + x)\sqrt{1 + 4x}$.

Algebra

Show that the equation $\sin 2\theta + \cos 2\theta = 2\sin^2 \theta$ may be rewritten in the form $\cos^2 \theta + 2\sin \theta \cos \theta - 3\sin^2 \theta = 0$.

Trigonometry

The curve is given by $x^2y - ay^2 = 4a^3$, where $a$ is a constant that is not zero.

Differentiation

Measured from origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\vec{OA} = \begin{pmatrix}2\\1\\3\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}4\\3\\2\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3\\-2\\-4\end{pmatrix}$. The quadrilateral $ABCD$ is a parallelogram.

Vectors

The variables $x$ and $y$ obey the differential equation $\cos 2x\,\frac{dy}{dx} = \frac{4\tan 2x}{\sin^2 3y}$, with $0 \le x < \frac{1}{4}\pi$. Also, $y = 0$ when $x = \frac{1}{6}\pi$.

Differential equations

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

Integration

The constant $a$ is defined by $\int_{0}^{a} x e^{-2x}\, dx = \frac{1}{8}$.

Numerical solution of equations

Solve $|5x - 3| < 2|3x - 7|$.

Algebra

The diagram illustrates the curve $y = (x + 5)\sqrt{3 - 2x}$ together with its maximum point $M$.

Integration

The position vectors of points $A$ and $B$ are $\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$ and $2\mathbf{i} - \mathbf{j} + \mathbf{k}$ respectively. The equation of the line $l$ is $\mathbf{r} = \mathbf{i} - \mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})$.

Vectors

Solve the equation $\ln(2x^2 - 3) = 2\ln x - \ln 2$, and give your answer in exact form.

Logarithmic and exponential functions

Using an Argand diagram, sketch the locus of points corresponding to complex numbers $z$ for which $|z + 3 - 2i| = 2$.

Complex numbers

Solve the equation $2\cos x - \cos\left(\frac{1}{2}x\right) = 1$ for $0 \leq x \leq 2\pi$.

Trigonometry

The complex number $2 + yi$ is represented by $a$, with $y$ being a real number and $y < 0$. It is stated that $f(a) = a^3 - a^2 - 2a$.

Complex numbers

The equation $\cot\left(\frac{1}{2}x\right) = 3x$ has a single root in the interval $0 < x < \pi$, and this root is called $\alpha$.

Numerical solution of equations

The curve is defined by $3x^2 + 4xy + 3y^2 = 5$.

Differentiation

The variables $x$ and $y$ obey the differential equation $\frac{dy}{dx} = \frac{4 + 9y^2}{e^{2x+1}}$. It is also given that $y = 0$ when $x = 1$.

Differential equations

Define $f(x) = \dfrac{2x^2 + 17x - 17}{(1 + 2x)(2 - x)^2}$.

Integration

Solve the equation $\ln(x + 5) = 5 + \ln x$. Give your answer correct to 3 decimal places.

Logarithmic and exponential functions

Consider $f(x) = \dfrac{21 - 8x - 2x^2}{(1 + 2x)(3 - x)^2}$.

Algebra

The complex number $z$ is specified by $z = \dfrac{5a - 2i}{3 + ai}$, where $a$ is an integer. It is also given that $\arg z = -\dfrac{\pi}{4}$.

Complex numbers

Find the quotient together with the remainder when $2x^4 - 27$ is divided by $x^2 + x + 3$.

Algebra

On an Argand diagram sketch, shade the set of points corresponding to complex numbers $z$ that satisfy the stated inequalities.

Complex numbers

A curve has parametric equations $x = \frac{\cos \theta}{2 - \sin \theta}$, $y = \theta + 2\cos \theta$.

Differentiation

The diagram displays the section of the curve $y = x^2 \cos 3x$ for $0 \leq x \leq \frac{\pi}{6}$, together with its maximum point $M$, where $x = a$.

Numerical solution of equations

Express $3\\cos x + 2\\cos(x - 60^\\circ)$ as $R\\cos(x - \\alpha)$, with $R > 0$ and $0^\\circ < \\alpha < 90^\\circ$. State $R$ exactly, and give $\\alpha$ correct to $2$ decimal places.

Trigonometry

By using the substitution $u = \cos x$, show that $\int_{0}^{\pi} \sin 2x\, e^{2\cos x}\, dx = \int_{-1}^{1} 2u e^{2u}\, du$.

Integration

The variables $x$ and $y$ are linked by the differential equation $\frac{dy}{dx} = \frac{y^2 + 4}{x(y + 4)}$ for $x > 0$. It is also stated that $x = 4$ when $y = 2\sqrt{3}$.

Differential equations

The equations of the lines $l$ and $m$ are $l: \; \mathbf{r} = a\mathbf{i} + 3\mathbf{j} + b\mathbf{k} + \lambda(c\mathbf{i} - 2\mathbf{j} + 4\mathbf{k})$, $m: \; \mathbf{r} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + \mathbf{k})$. Relative to origin $O$, the position vector of $P$ is $4\mathbf{i} + 7\mathbf{j} - 2\mathbf{k}$.

Vectors

Particles $P$ and $Q$, with masses $m\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are initially stationary on a smooth horizontal plane. $P$ is given a speed of $5\,\text{m s}^{-1}$ straight towards $Q$. After $P$ and $Q$ collide, $P$ continues in the same direction as before, but with speed $2\,\text{m s}^{-1}$.

Momentum

Particle $P$, with mass $0.4\,\text{kg}$, is fired straight up from level ground at a speed of $10\,\text{m s}^{-1}$.

Energy, work and power

A particle travels in a straight line, beginning from rest. The displacement $s\,\text{m}$ from a fixed point $O$ on the line after $t\,\text{s}$ is $s = t^{\frac{5}{2}} - \frac{15}{4}t^{\frac{3}{2}} + 6$.

Kinematics of motion in a straight line

At time $t\,\text{s}$ after moving away from the fixed point $O$, the particle has velocity $v\,\text{m s}^{-1}$. The diagram presents a velocity-time graph that represents the particle's motion. It is made up of 5 straight line segments. The particle increases its speed to $0.9\,\text{m s}^{-1}$ over a period of $3\,\text{s}$, then continues at this constant speed for $6\,\text{s}$, and then stops instantly $1\,\text{s}$ later. The particle then reverses and comes to rest at $O$ again at time $T\,\text{s}$.

Kinematics of motion in a straight line

Four coplanar forces act at a point, with magnitudes of $F\,\text{N}$, $10\,\text{N}$, $50\,\text{N}$ and $40\,\text{N}$. Their directions are indicated in the diagram.

Forces and equilibrium

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.1\,\text{kg}$ respectively, are connected to the two ends of a light inextensible string. This string goes over a fixed smooth pulley $B$ that is attached to two inclined planes. Particle $P$ is situated on a smooth plane $AB$ inclined at $60^{\circ}$ to the horizontal. Particle $Q$ is situated on a plane $BC$ inclined at an angle of $\theta^{\circ}$ to the horizontal. The string is taut, and the particles can travel along the lines of greatest slope of the planes (see diagram).

Newton's laws of motion

A car with mass $1200\,\text{kg}$ is moving along a straight horizontal road. The car’s engine has constant power equal to $16\,\text{kW}$. A constant resistive force of magnitude $500\,\text{N}$ acts on the car.

Energy, work and power

A particle with mass $1.6\text{ kg}$ is released from rest at a point $9\text{ m}$ above the horizontal ground. Immediately before it strikes the ground, the particle’s speed is $12\text{ m s}^{-1}$.

Energy, work and power

Particles $A$ and $B$, whose masses are $3.2\text{ kg}$ and $2.4\text{ kg}$ respectively, rest on a smooth horizontal table. $A$ travels towards $B$ at a speed of $v\text{ m s}^{-1}$ and then strikes $B$, which is travelling towards $A$ at a speed of $6\text{ m s}^{-1}$. After the collision, both particles are at rest.

Momentum

Forces of magnitudes $30\text{ N}$, $15\text{ N}$, $33\text{ N}$ and $P\text{ N}$ act together at a point in the directions shown in the diagram, where $\tan \alpha = \frac{4}{3}$. The forces are in equilibrium.

Forces and equilibrium

An athlete with mass $84\,\text{kg}$ is moving along a straight road.

Energy, work and power

A particle of mass $0.6\,\text{kg}$ rests on a rough plane inclined at $35^\circ$ to the horizontal. It is held in equilibrium by a horizontal force of magnitude $P\,\text{N}$ acting in a vertical plane that contains a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is $0.4$.

Forces and equilibrium

A particle $P$ is initially at rest and then travels in a straight line away from point $O$. At time $t\,\text{s}$ after leaving $O$, the velocity of $P$, $v\,\text{m s}^{-1}$, is $v = bt + ct^{\frac{3}{2}}$, where $b$ and $c$ are constants. The velocity of $P$ is $8\,\text{m s}^{-1}$ when $t = 4$ and $13.5\,\text{m s}^{-1}$ when $t = 9$.

Kinematics of motion in a straight line

Particles $P$ and $Q$, with masses $2\,\text{kg}$ and $0.25\,\text{kg}$ respectively, are joined by a light inextensible string passing over a fixed smooth pulley. Particle $P$ rests on an inclined plane set at an angle of $30^\circ$ to the horizontal. Particle $Q$ is suspended beneath the pulley. Points $A$, $B$ and $C$ are on the line of greatest slope of the plane, with $AB = 0.8\,\text{m}$ and $BC = 1.2\,\text{m}$ (see diagram). Particle $P$ is let go from rest at $A$ with the string taut and moves down the plane. While $P$ travels from $A$ to $C$, $Q$ does not get to the pulley. The section of the plane from $A$ to $B$ is rough, and the coefficient of friction between the plane and $P$ is $0.3$. The section from $B$ to $C$ is smooth.

Kinematics of motion in a straight line

Particles $P$ and $Q$, with masses $0.1\,\text{kg}$ and $0.4\,\text{kg}$ respectively, can move freely on a smooth horizontal plane. $P$ is launched at a speed of $4\,\text{m s}^{-1}$ towards stationary $Q$. After $P$ and $Q$ collide, they have equal speeds.

Momentum

A car with mass $1500\,\text{kg}$ is pulling a trailer of mass $m\,\text{kg}$ on a level straight road. The car and trailer are joined by a tow-bar that is horizontal, light and rigid. A resistance force of $F\,\text{N}$ acts on the car and a resistance force of $200\,\text{N}$ acts on the trailer. The engine of the car provides a driving force of $3200\,\text{N}$, the car accelerates at $1.25\,\text{m s}^{-2}$ and the tension in the tow-bar is $300\,\text{N}$.

Forces and equilibrium

A smooth ring $R$ of mass $0.2\,\text{kg}$ is placed on a light string $ARB$. The two ends of the string are fixed at points $A$ and $B$, with $A$ vertically above $B$. The string is taut, and $ABR = 90^\circ$. The angle between the segment $AR$ of the string and the vertical is $60^\circ$. The ring is kept in equilibrium by a force of magnitude $X\,\text{N}$, acting on the ring in a direction perpendicular to $AR$ (see diagram).

Forces and equilibrium

A lorry with mass $15\,000\,\text{kg}$ travels along a straight horizontal road from $A$ to $B$. Its speed is $20\,\text{m s}^{-1}$ at $A$ and $25\,\text{m s}^{-1}$ at $B$. The engine power of the lorry is constant, and the resistance to motion has constant magnitude $6000\,\text{N}$. The acceleration of the lorry at $B$ is $0.5$ times the acceleration of the lorry at $A$.

Energy, work and power

A particle begins at rest at point $O$ and travels along a straight line. At time $t$ after leaving $O$, its acceleration is $a\,\text{m s}^{-2}$, with $a = kt^{\frac{1}{2}}$ for $0 \leq t \leq 9$, where $k$ is constant. When $t = 9$, the particle's velocity is $1.8\,\text{m s}^{-1}$. For $t > 9$, the velocity $v\,\text{m s}^{-1}$ is $v = 0.2(t - 9)^2 + 1.8$.

Kinematics of motion in a straight line

An elevator is drawn upward by a cable. It speeds up vertically at $0.4\,\text{m s}^{-2}$ for $5\,\text{s}$, then continues at a steady speed for $25\,\text{s}$. After that, it slows down at $0.2\,\text{m s}^{-2}$ until it is stationary.

Newton's laws of motion

The diagram presents the vertical cross-section $XYZ$ of a rough slide. $YZ$ is a straight segment of length $2\,\text{m}$ that makes an angle of $\alpha$ with the horizontal, where $\sin\alpha = 0.28$. At $Y$, $YZ$ touches the curved part $XY$ tangentially, and $X$ is $1.8\,\text{m}$ above level of $Y$. A child of mass $25\,\text{kg}$ slides down the slide, beginning from rest at $X$. The work done by the child against the resistance force in moving from $X$ to $Y$ is $50\,\text{J}$.

Energy, work and power

For 50 values of $x$, the totals are $\sum (x - q) = 700$ and $\sum (x - q)^2 = 14\,235$, with $q$ as a constant.

Representation of data

Determine how many different committees of 6 people can be formed from 6 men and 8 women when the committee has to contain 3 men and 3 women.

Permutations and combinations

Determine how many distinct arrangements can be made from the 8 letters in the word COCOONED.

Permutations and combinations

A mathematical puzzle is presented to a large group of students. The completion times are normally distributed, with mean $14.6$ minutes and standard deviation $5.2$ minutes.

The normal distribution

The table summarises the populations of $150$ villages in the UK, rounded to the nearest hundred.

Representation of data

Eli has four fair $4$-sided dice, each face labelled $1, 2, 3, 4$. He rolls all four together. The random variable $X$ stands for the number of $2$s that appear.

Discrete random variables

A wildlife magazine for children is issued every Monday. During the next 12 weeks, each edition will come with a model animal as a free gift. The five possible models are tiger, leopard, rhinoceros, elephant and buffalo, and each has the same chance of being placed in the magazine. Sahim purchases one copy of the magazine every Monday.

Discrete random variables

The random variable $X$ may take the values $-2$, $2$ and $3$. You are told that $P(X = x) = k(x^2 - 1)$, where $k$ is a constant.

Discrete random variables

A sports event lasts for $4$ days and begins on Sunday. The chance that rain falls on Sunday is $0.4$. For each later day, the chance of rain is $0.7$ if the previous day was rainy and $0.2$ if the previous day was dry.

Probability

The back-to-back stem-and-leaf diagram below shows the monthly salaries, in dollars, of $27$ employees in each of companies A and B. The key indicates that $1\,|\,27\,|\,6$ stands for $\$2710$ for company A and $\$2760$ for company B.

Representation of data

A fair 5-sided spinner is numbered $1, 2, 3, 4, 5$ on its faces. It is spun again and again until the side showing $2$ appears on the face where the spinner lands. The random variable $X$ represents how many spins are needed.

Discrete random variables

Western bluebird lengths are normally distributed, with mean $16.5\text{ cm}$ and standard deviation $0.6\text{ cm}$. One random sample of $150$ of these birds is chosen.

The normal distribution

Among a set of $25$ people, $6$ are swimmers, $8$ are cyclists and $11$ are runners. Every person takes part in just one of these sports. A team of $7$ people is chosen from these $25$ people to enter a competition.

Permutations and combinations