Oct/Nov 2022

Solve for $x$ in $3x + 2 = \frac{2}{x - 1}$.

Quadratics

The curves $y = 2x^{\frac{1}{2}} + 1$ and $y = \frac{1}{2}x^2 - x + 1$ meet at $A(0, 1)$ and $B(4, 5)$, as the diagram shows.

Integration

The diagram represents the circle whose equation is $x^2 + y^2 = 20$. The tangents that touch the circle at $B$ and $C$ go through $A(0, 10)$.

Coordinate geometry

A curve has derivative $\frac{dy}{dx} = 12\left(\frac{1}{2}x - 1\right)^{-4}$. The curve is known to pass through $P(6, 4)$.

Differentiation

The curve is given by $y = ax^{\frac{1}{2}} - 2x$, for $x > 0$, with $a$ as a constant. It has a stationary point at $P$, whose $x$-coordinate is $9$.

Coordinate geometry

In the expansion of $(1 + \frac{2}{p}x)^5 + (1 + px)^6$, the coefficient of $x^2$ is 70.

Quadratics

The diagram displays a sector $OAB$ from a circle centred at $O$. The arc $AB$ measures 8 cm. The perimeter of the sector is stated to be 20 cm.

Circular measure

Show that the equation $\frac{1}{\sin \theta + \cos \theta} + \frac{1}{\sin \theta - \cos \theta} = 1$ can be rewritten in the form $a\sin^2 \theta + b\sin \theta + c = 0$, where the constants $a$, $b$ and $c$ are to be determined.

Trigonometry

A tool used to drive fence posts into the ground is known as a ‘post-rammer’. The distances, in millimetres, that the post moves into the ground with each impact of the post-rammer form a geometric progression. The first three impacts make the post sink into the ground by $50\text{ mm}$, $40\text{ mm}$ and $32\text{ mm}$ respectively.

Series

The function $f$ is specified by $f(x) = 2 - \dfrac{3}{4x - p}$, with domain $x > \dfrac{p}{4}$ and $p$ a constant.

Differentiation

The functions $f$ and $g$ are each defined for every $x \in \mathbb{R}$, and are given by $f(x) = x^2 - 4x + 9$ and $g(x) = 2x^2 + 4x + 12$.

Quadratics

The coordinates of points $A$ and $B$ are $(5, 2)$ and $(10, -1)$, respectively.

Coordinate geometry

The diagram presents a cross-section $RASB$ of an aircraft body. It is made up of a sector $OARB$ of a circle with radius $2.5\,\text{m}$ and centre $O$, a sector $RASB$ of a second circle with radius $2.24\,\text{m}$ and centre $P$, and a quadrilateral $OAPB$. Angle $AOB = \frac{2\pi}{3}$ and angle $APB = \frac{5\pi}{6}$.

Coordinate geometry

Determine the coordinates of the curve's minimum point for $y = \frac{9}{4}x^2 - 12x + 18$.

Integration

The first three terms in an arithmetic progression are $a$, $2a$ and $a^2$, with $a$ being a positive constant.

Series

Find the values of $k$ for which the equation $8x^2 + kx + 2 = 0$ has no real roots.

Trigonometry

A geometric progression has third term $1764$ and the sum of its second and third terms is $3444$.

Series

The curve given by $y = f(x)$ is carried over to the curve given by $y = g(x)$ by a stretch in the $x$-direction with factor $0.5$, then a translation of $\begin{pmatrix}0\\1\end{pmatrix}$.

Functions

The curve is defined by the equation $y = 4x^2 + 20x + 6$.

Quadratics

Prove that $\frac{\sin \theta}{\sin \theta + \cos \theta} + \frac{\cos \theta}{\sin \theta - \cos \theta} = \frac{\tan^2 \theta + 1}{\tan^2 \theta - 1}$.

Trigonometry

The curve is defined by $\frac{dy}{dx} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}$. It passes through the point $(3, 5)$.

Differentiation

The functions $f$ and $g$ are given by $f(x) = x + \frac{1}{x}$ for $x > 0$, and $g(x) = ax + 1$ for $x \in \mathbb{R}$, with $a$ a constant.

Functions

Solve $8\sin^2\theta + 6\cos\theta + 1 = 0$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

The diagram depicts the circle $x^2 + y^2 = 2$ and the line $y = 2x - 1$ meeting at the points $A$ and $B$. The point $D$ on the $x$-axis is positioned so that $AD$ is perpendicular to the $x$-axis.

Integration

The coordinates of points $A$, $B$ and $C$ are $A(5,-2)$, $B(10,3)$ and $C(2p,p)$, with $p$ as a constant.

Coordinate geometry

The function $f$ has the rule $f(x) = -2x^2 - 8x - 13$ for $x < -3$.

Quadratics

Determine the first three terms, in ascending powers of $x$, of the expansion of $(1 + 2x)^5$.

Series

In a large industrial water tank, if the water depth is x metres, then the water volume V \text{ m}^3 in the tank is given by $V = 243 - \frac{1}{3}(9 - x)^3$. The tank is being filled at a steady rate of $3.6\text{ m}^3$ per hour.

Differentiation

The diagram depicts a curve with a maximum point at $(8, 12)$ and a minimum point at $(8, 0)$. This curve is obtained by applying two transformations to a circle. The first transformation is a translation of $\begin{pmatrix} 7 \\ -3 \end{pmatrix}$. The second transformation is a stretch in the $y$-direction.

Coordinate geometry

It is given that $\alpha = \cos^{-1}\left(\frac{8}{17}\right)$.

Trigonometry

For the curve $y = f(x)$, $f'(x) = \frac{-3}{(x + 2)^4}$.

Differentiation

The diagram depicts two congruent circles that overlap at points $A$ and $B$, with centres at $P$ and $Q$. Each circle has radius $r$, and the separation $PQ$ is $\frac{5}{3}r$.

Circular measure

A geometric progression has first term $216$ and fourth term $64$.

Series

Solve the inequality $|2x - 5| > x$.

Algebra

Use logarithms to find the solution of $14e^{-2x} = 5^{x+1}$, giving your answer correct to $3$ significant figures.

Logarithmic and exponential functions

It is given that $\sec\theta = \sqrt{17}$, with $0 < \theta < \frac{1}{2}\pi$.

Trigonometry

By sketching an appropriate pair of graphs on the same diagram, show that the equation $e^{-\frac{1}{2}x} = x^5$ has exactly one real root.

Numerical solution of equations

The curve is given by $4e^{2x}y + y^2 = 21$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = 12x^3 - 9x^2 + 8x - 4$.

Integration

The graph depicts the curve given by $y = \frac{2\ln x}{3x + 1}$. It meets the $x$-axis at $A$ and reaches a maximum at $B$. The shaded area is enclosed by the curve together with the lines $x = 3$ and $y = 0$.

Integration

The function $f(\theta)$ is given by $f(\theta) = 12\sin\theta\cos\theta + 16\cos^2\theta$.

Trigonometry

Find the solutions of the equation $\sec \theta = 5 \cosec \theta$ for $0^\circ < \theta < 360^\circ$.

Trigonometry

The equation $|4x - 1| = |x + 3|$ has solutions $x = p$ and $x = q$, with $p < q$.

Algebra

The variables $x$ and $y$ are related by $y = A x^k$, with $A$ and $k$ as constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line that passes through the points $(0.56,\, 2.87)$ and $(0.81,\, 3.47)$, as shown in the diagram.

Integration

The polynomial $p(x)$ is defined by $p(x) = ax^3 + 23x^2 - ax - 8$, where $a$ is a constant. You are told that $(2x + 1)$ is one factor of $p(x)$.

Logarithmic and exponential functions

The graph given by $y = x\ln(4x + 1) - 3x$ has a single stationary point, $P$.

Numerical solution of equations

The diagram depicts the curves $y = \dfrac{6}{3x + 2}$ and $y = 3e^{-x} - 3$ for x-values from $0$ to $4$. The shaded area lies between the two curves and the lines $x = 0$ and $x = 4$.

Integration

The curve is given parametrically by $x = 3\cos 2\theta$, $y = 4\sin\theta$, with $\pi \leq \theta \leq \dfrac{3\pi}{2}$. The points $P$ and $Q$ are on the curve. At $P$, the gradient of the curve is $2$. The line $3x + y = 0$ intersects the curve at $Q$.

Differentiation

Solve $|2x - 5| > x$ as an inequality.

Algebra

Use logarithms to solve the equation $14e^{-2x} = 5^{x + 1}$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

You are told that $\sec \theta = \sqrt{17}$ for $0 < \theta < \frac{1}{2}\pi$.

Trigonometry

Sketch a suitable pair of graphs on the same diagram to show that the equation $e^{-\frac{1}{2}x} = x^5$ has only one real root.

Numerical solution of equations

The curve satisfies the equation $4e^{2x}y + y^2 = 21$.

Differentiation

The polynomial $p(x)$ has the expression $p(x) = 12x^3 - 9x^2 + 8x - 4$.

Integration

The diagram displays the curve with equation $y = \frac{2\ln x}{3x + 1}$. It cuts the $x$-axis at point $A$ and reaches a maximum at point $B$. The shaded area is enclosed by the curve and the lines $x = 3$ and $y = 0$.

Integration

The function $f(\theta)$ is given by $f(\theta) = 12 \sin \theta \cos \theta + 16 \cos^2 \theta$.

Trigonometry

Sketch a graph of $y = |2x + 1|$.

Algebra

Suppose $f(x) = \dfrac{2x^2 + 7x + 8}{(1 + x)(2 + x)^2}$.

Algebra

From the diagram, $OABCD$ is a solid shape with $OA = OB = 4$ units and $OD = 3$ units. The edge $OD$ is vertical, $DC$ runs parallel to $OB$, and $DC = 1$ unit. The base $OAB$ is horizontal, while $ngle AOB = 90^07$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OD$ respectively. $M$ is the midpoint of $AB$, and $N$ lies on $BC$ so that $CN = 2NB$.

Vectors

On a sketch of an Argand diagram, indicate by shading the set of points that correspond to complex numbers $z$ meeting the stated inequalities.

Complex numbers

Solve the equation $2^{3x-1} = 5(3^{-x})$. Give your answer in the form $\frac{\ln a}{\ln b}$, with $a$ and $b$ integers.

Logarithmic and exponential functions

For $0^{\circ} < x < 180^{\circ}$, solve $\tan(x + 45^{\circ}) = 2\cot x$.

Trigonometry

Let the complex numbers $u$ and $w$ be given by $u = 2e^{\frac{1}{4}\pi i}$ and $w = 3e^{\frac{1}{3}\pi i}$.

Complex numbers

Show that $\cos 4\theta + 4\cos 2\theta + 3 = 8\cos^4 \theta$.

Trigonometry

A curve is described by $y = \dfrac{x}{\cos^2 x}$, for $0 < x < \tfrac{1}{2}\pi$. At the point where $x = a$, the tangent to the curve has gradient $12$.

Numerical solution of equations

In one chemical reaction, the quantity, $x$ grams, of a substance is rising. The differential equation linking $x$ and $t$, where $t$ is the time in seconds from the start of the reaction, is $\frac{dx}{dt} = kx e^{-0.1t}$, with $k$ a positive constant. It is also stated that $x = 20$ at the beginning of the reaction.

Differential equations

The diagram illustrates a section of the curve $y = (3 - x)e^{-\frac{x}{3}}$ for $x \geq 0$, together with its minimum point $M$.

Integration

Find $x$ from the equation $2^{3x-1} = 5(3^{1-x})$. Write your answer in the form $\ln a \over \ln b$ where $a$ and $b$ are integers.

Logarithmic and exponential functions

Define $f(x)$ by $f(x) = \dfrac{4 - x + x^2}{(1 + x)(2 + x^2)}$.

Integration

Let $p(x)$ denote the polynomial $2x^3 - x^2 + a$, with $a$ as a constant. It is known that $(2x + 3)$ is a factor of $p(x)$.

Algebra

A curve is given by $y = \sin x \sin 2x$. There is a stationary point on the curve for $0 < x < \tfrac{1}{2}\pi$.

Numerical solution of equations

Express $4 \cos x - \\sin x$ in the form $R \cos(x + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and state $\alpha$ correct to 2 decimal places.

Trigonometry

Solve $z^2 - 6iz - 12 = 0$, with answers expressed as $x + iy$, where $x$ and $y$ are exact real numbers.

Complex numbers

Measured from the origin $O$, the position vectors of $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}$ and $\vec{OC} = \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}$.

Vectors

The variables $x$ and $\theta$ satisfy the differential equation $x \sin^2 \theta \, \frac{dx}{d\theta} = \tan^2 \theta - 2 \cot \theta$, for $0 < \theta < \frac{1}{2}\pi$ and $x > 0$. It is also given that $x = 2$ when $\theta = \frac{1}{4}\pi$.

Differential equations

The diagram depicts a section of the curve $y = \sin \sqrt{x}$. This section crosses the $x$-axis at the point where $x = a$.

Integration

The diagram depicts a semicircle with diameter $AB$, centre $O$ and radius $r$. The shaded part is the minor segment bounded by the chord $AC$, and its area is one third of the area of the semicircle. The angle $CAB$ is $\theta$ radians.

Numerical solution of equations

Solve the equation $\ln(2x - 1) = 2\ln(x + 1) - \ln x$. Write your answer to $3$ decimal places.

Logarithmic and exponential functions

A gardener is topping up an ornamental pool with water by means of a hose that supplies $30$ litres each minute. The pool starts off empty. If $t$ minutes have passed since the filling began, the amount of water in the pool is $V$ litres. There is a small leak in the pool, so water escapes at a rate of $0.01V$ litres per minute. The differential equation relating $V$ and $t$ has the form $\frac{dV}{dt} = a - bV$.

Differential equations

Take $f(x) = \dfrac{5 - x + 6x^2}{(3 - x)(1 + 3x^2)}$.

Integration

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

Algebra

Find the exact value for $\int_{0}^{4x} x\sec^2 x\,dx$.

Integration

For this curve, the parametric equations are $x = 2t - \tan t$, $y = \ln(\sin 2t)$, with $0 < t < \frac{1}{2}\pi$.

Differentiation

On an Argand diagram sketch, shade the set of points that represent complex numbers $z$ satisfying $|z + 2| \leq 2$ and $\operatorname{Im} z \geq 1$.

Complex numbers

Solve the quadratic equation $(1-3i)z^2 - (2+i)z + i = 0$, and give your answers in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Show that $\sqrt{5}\sec x + \tan x = 4$ may be written as $R\cos(x + \alpha) = \sqrt{5}$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.

Trigonometry

For the curve given by $y = \frac{x^3}{e^x - 1}$, there is a stationary point when $x = p$, where $p > 0$.

Numerical solution of equations

Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix}0 \\ 5 \\ 2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}4 \\ -3 \\ -2\end{pmatrix}$. The midpoint of $AC$ is $M$ and $N$ is a point on $BC$, between $B$ and $C$, such that $BN = 2NC$.

Vectors

Coplanar forces with magnitudes $P\text{ N}$, $Q\text{ N}$, $16\text{ N}$ and $22\text{ N}$ act together at a point in the directions indicated in the diagram. These forces are in equilibrium.

Forces and equilibrium

Small smooth spheres $A$ and $B$, with equal radii and masses of $6\text{ kg}$ and $2\text{ kg}$ respectively, are on a smooth horizontal plane. At the start, $A$ is travelling towards $B$ at speed $5\ \text{m\,s}^{-1}$, while $B$ is travelling towards $A$ at speed $3\ \text{m\,s}^{-1}$. After the collision, both $A$ and $B$ travel in the same direction, and the difference between their speeds is $2\ \text{m\,s}^{-1}$.

Momentum

A resisting force of magnitude $1400\text{ N}$ acts on a car whose mass is $1250\text{ kg}$.

Energy, work and power

A block with mass $8\,\text{kg}$ rests on a rough plane that is tilted at $18^\circ$ to the horizontal. It is dragged up the plane by a light string, which is set at an angle of $26^\circ$ above a line of greatest slope. The string has tension $T\,\text{N}$ (see diagram). The coefficient of friction between the block and plane is $0.65$.

Kinematics of motion in a straight line

Particle $P$ travels along the $x$-axis starting from the origin $O$ with initial velocity $-20\,\text{m s}^{-1}$. Its acceleration $a\,\text{m s}^{-2}$ at time $t$ after leaving $O$ is given by $a = 12 - 2t$.

Kinematics of motion in a straight line

In Fig. 6.1, particles $A$ and $B$, with masses $4\,\text{kg}$ and $3\,\text{kg}$ respectively, are fixed to the ends of a light inextensible string that runs over a small smooth pulley. The pulley is attached at the top of a plane inclined at $30^\circ$ to the horizontal. $A$ hangs freely beneath the pulley, while $B$ rests on the inclined plane. The string is taut, and the part of the string from $B$ to the pulley is parallel to a line of greatest slope of the plane.

Forces and equilibrium

A cyclist travels on a straight horizontal road $AB$ of length $50\text{ m}$. The cyclist begins from rest at $A$ and reaches a speed of $6\text{ m s}^{-1}$ at $B$. The cyclist applies a constant driving force with magnitude $100\text{ N}$. A resistance force is present, and the work done against this resistance from $A$ to $B$ is $3560\text{ J}$.

Energy, work and power

Particle $P$, with mass $0.4\text{ kg}$, is in limiting equilibrium on a plane that is inclined at $30^\circ$ to the horizontal.

Newton's laws of motion

A particle with mass $0.3\text{ kg}$ is kept stationary by two light inextensible strings. One string is fixed at an angle of $60^\circ$ to a horizontal ceiling. The other string is fixed at an angle $\alpha^\circ$ to a vertical wall (see diagram). The tension in the string attached to the ceiling is $4\text{ N}$.

Forces and equilibrium

A car with mass $1200\,\text{kg}$ is moving on a straight horizontal road $AB$. A constant resisting force of size $500\,\text{N}$ acts on it. When the car goes through point $A$, its speed is $15\,\text{m s}^{-1}$ and its acceleration is $0.8\,\text{m s}^{-2}$. While it travels from $A$ to $B$, the car keeps working at this power. It takes $53$ seconds for the journey from $A$ to $B$, and the car’s speed at $B$ is $32\,\text{m s}^{-1}$.

Energy, work and power

Block $A$ has mass $80\,\text{kg}$ and is joined by a light, inextensible rope to block $B$, which has mass $40\,\text{kg}$. The rope between the blocks is taut and runs parallel to the line of greatest slope of a plane that is inclined at $20^\circ$ to the horizontal. A force of magnitude $500\,\text{N}$ acting at an angle of $15^\circ$ above that same line of greatest slope is applied to $A$ (see diagram). The blocks move up the plane, there is a resistance force of $50\,\text{N}$ on $B$, and there is no resistance force on $A$.

Newton's laws of motion

Three particles $A$, $B$ and $C$ have masses $0.3\,\text{kg}$, $0.4\,\text{kg}$ and $m\,\text{kg}$ respectively and are at rest on a smooth horizontal plane in a straight line. The separation of $B$ and $C$ is $2.1\,\text{m}$. $A$ is launched directly at $B$ with speed $2\,\text{m s}^{-1}$. Once $A$ collides with $B$, the speed of $A$ becomes $0.6\,\text{m s}^{-1}$, still travelling in the same direction.

Momentum

Particle $P$ moves along a straight line and begins from rest at point $O$. After $t\,\text{s}$ from leaving $O$, its acceleration is $a\,\text{m s}^{-2}$, with $a = 0.3t^{\frac{1}{2}}$ for $0 \le t \le 4$, and $a = -kt^{-3/2}$ for $4 < t \le T$, where $k$ and $T$ are constants.

Kinematics of motion in a straight line

Particle $P$ is projected vertically upwards from the ground with speed $u\,\text{m s}^{-1}$. $P$ reaches its maximum height after $3\,\text{s}$.

Kinematics of motion in a straight line

A box of mass $5\,\text{kg}$ is moved at a constant speed of $1.8\,\text{m s}^{-1}$ for $15\,\text{s}$ up a rough plane inclined at an angle of $20^\circ$ to the horizontal. It travels along the line of greatest slope and acts against a frictional force of $40\,\text{N}$. The force pulling the box is parallel to the line of greatest slope.

Energy, work and power

A ring of mass $4\,\text{kg}$ is placed on a smooth circular rigid wire with centre $C$. The wire is fixed in a vertical plane, and the ring is held in equilibrium by a light string attached to $A$, the topmost point of the circle. The string is inclined at $25^{\circ}$ to the vertical (see diagram).

Forces and equilibrium

A particle $P$ moves along a straight line in the positive direction with constant acceleration. During the $2\text{nd}$ second of its motion, $P$ covers $52\,\text{m}$, and during the $4\text{th}$ second of its motion, it covers $64\,\text{m}$.

Kinematics of motion in a straight line

Particles $X$ and $Y$ travel along a straight line through points $A$ and $B$. Particle $X$ begins from rest at $A$ and heads towards $B$. At the same instant, $Y$ begins from rest at $B$. After $t$ seconds from the start of motion: • the acceleration of $X$ in the direction $AB$ is $(12t + 12)\,\text{m s}^{-2}$, • the acceleration of $Y$ in the direction $AB$ is $(24t - 8)\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A car with mass $1750\,\text{kg}$ is towing a caravan whose mass is $500\,\text{kg}$. A light rigid tow-bar links the car and the caravan. The resistive forces opposing the motion of the car and caravan are $650\,\text{N}$ and $150\,\text{N}$ respectively.

Energy, work and power

Masses of $1.5\ \text{kg}$ and $3\ \text{kg}$ are placed on a plane inclined at angle $\alpha$ to the horizontal, with $\tan \alpha = \frac{3}{4}$. The part of the plane from $A$ to $B$ is smooth, whereas the part from $B$ to $C$ is rough. The $1.5\ \text{kg}$ particle is kept at rest at $A$, and the $3\ \text{kg}$ particle is in limiting equilibrium at $B$. Let $AB = x\ \text{m}$ and $BC = 4\ \text{m}$.

Momentum

The probability distribution of a random variable $X$ is given in the table below.

Discrete random variables

People living in Persham were asked how dependable they thought their internet service was. $12\%$ described it as ‘poor’, $36\%$ described it as ‘satisfactory’ and $52\%$ described it as ‘good’. A random sample of $8$ residents of Persham is chosen.

The normal distribution

The Lions and the Tigers are two basketball clubs. The table shows the heights, in cm, of the 11 players in each of their first team squads.

Representation of data

For a large population, adults’ systolic blood pressure (SBP) follows a normal distribution with mean $125.4$ and standard deviation $18.6$.

The normal distribution

A standard fair $6$-sided die is used in a game. The player rolls once. If the outcome is $2$, $3$, $4$ or $5$, that number becomes the score and there is no second roll. If the outcome is $1$ or $6$, the player rolls again and the score is the total of the two results from the two rolls. Events $A$ and $B$ are defined as follows. $A$: the player's score is $5$, $6$, $7$, $8$ or $9$. $B$: the player has two throws.

Probability

A Social Club has $15$ members, with $8$ men and $7$ women. Its committee is made up of $5$ members.

Permutations and combinations

On any given day, Kino goes to school by bus, by car or on foot with probabilities $0.2$, $0.1$ and $0.7$ respectively. The chance that he is late when he goes by bus is $x$. The chance that he is late when he goes by car is $2x$ and the chance that he is late when he goes on foot is $0.25$. The probability that Kino is late on a randomly selected day is $0.235$.

Probability

The rod lengths manufactured by a company are normally distributed with mean $55.6\text{ mm}$ and standard deviation $1.2\text{ mm}$.

The normal distribution

Three fair $6$-sided dice, with faces labelled $1, 2, 3, 4, 5, 6$, are thrown together repeatedly. For each throw, the score is the total of the numbers showing on the upper faces.

Discrete random variables

The table shows the times, in minutes, that $250$ employees at a particular company took to finish a word processing task: Time taken ($t$ minutes): $0 \leq t < 20$, $20 \leq t < 40$, $40 \leq t < 50$, $50 \leq t < 60$, $60 \leq t < 100$. Frequencies: $32, 46, 96, 52, 24$.

Representation of data

Eric has three coins. One coin is fair, while each of the other two is biased so that the chance of getting a head on any throw is $\frac{1}{4}$, independently of every other throw. Eric throws all three coins at the same time. Events $A$ and $B$ are defined below. $A$: all three coins show matching outcomes $B$: at least one of the biased coins shows a head

Discrete random variables

At a company call centre, $90\%$ of callers are put through straight away to a representative. A random sample of $12$ callers is selected.

The normal distribution