May/June 2021

The curve is defined by $\frac{dy}{dx} = \frac{3}{x^4} + 32x^3$. It is also stated that the curve goes through $\left(\frac{1}{2}, 4\right)$.

Integration

The circle is given by the equation $x^2 + y^2 - 4x + 6y - 77 = 0$.

Coordinate geometry

The curve is described by $y = 2\sqrt{3x + 4} - x$.

Differentiation

The total of the first 20 terms in an arithmetic progression is $405$, while the total of the first 40 terms is $1410$.

Series

Find the first three terms of the expansion of $(3 - 2x)^5$ in ascending powers of $x$.

Series

The diagram illustrates a section of the graph of $y = a\tan(x - b) + c$.

Trigonometry

The fifth, sixth, and seventh terms in a geometric progression are $8k$, $-12$, and $2k$ respectively.

Series

The curve has equation $y = (2k - 3)x^2 - kx - (k - 2)$, with $k$ as a constant. The straight line $y = 3x - 4$ touches the curve.

Differentiation

Prove the identity by using trig identities to show that $\frac{1 - 2\sin^2\theta}{1 - \sin^2\theta} = 1 - \tan^2\theta$.

Trigonometry

The diagram illustrates a symmetrical metal plate. It is formed by cutting out two congruent parts from a circular disc with centre $C$. The outline of the plate is made up of the two arcs $PS$ and $QR$ from the original circle, together with two semicircles having $PQ$ and $RS$ as diameters. The radius of the circle with centre $C$ is $4\text{ cm}$, and $PQ = RS = 4\text{ cm}$ as well.

Circular measure

The functions $f$ and $g$ are given by: $f(x) = (x - 2)^2 - 4$ for $x \geq 2$, $g(x) = ax + 2$ for $x \in \mathbb{R}$, with $a$ as a constant.

Functions

Write $16x^2 - 24x + 10$ in the form $(4x + a)^2 + b$.

Quadratics

Prove that $\dfrac{1 + \sin x}{1 - \sin x} - \dfrac{1 - \sin x}{1 + \sin x} = \dfrac{4 \tan x}{\cos x}$.

Trigonometry

The gradient of a curve is given by $\frac{dy}{dx} = 6(3x - 5)^3 - kx^2$, where $k$ is a constant. The curve passes through a stationary point at $(2, -3.5)$.

Differentiation

The diagram presents a cross-section of seven cylindrical pipes, each with radius $20\text{ cm}$, secured by a thin rope that is pulled taut around them. The centres of the six outer pipes are $A, B, C, D, E$ and $F$. Points $P$ and $Q$ mark the places where the straight parts of the rope touch the pipe centred at $A$.

Circular measure

The graph of $y = f(x)$ becomes $y = 2f(x - 1)$. Describe fully the two separate transformations that have been combined to produce the final image.

Functions

A curve has equation $y = (x - 3)\sqrt{x + 1} + 3$. These points lie on the curve. Any non-exact values are rounded to 4 decimal places. $A\,(2, k)$, $B\,(2.9, 2.8025)$, $C\,(2.99, 2.9800)$, $D\,(2.999, 2.9980)$, $E\,(3, 3)$.

Differentiation

In the expansion of $(4x + \frac{10}{x})^3$, the coefficient of $x$ is $p$. In the expansion of $(2x + \frac{k}{x^2})^5$, the coefficient of $\frac{1}{x}$ is $q$. If $p = 6q$, determine the possible values of $k$.

Series

The function $f$ is given by $f(x) = 2x^2 + 3$ for $x \geq 0$.

Functions

The coordinates of points $A$ and $B$ are $(8, 3)$ and $(p, q)$ respectively. The perpendicular bisector of $AB$ is given by $y = -2x + 4$.

Coordinate geometry

Point $A$ is located at $(1, 5)$, and line $l$ passes through $A$ with gradient $-\frac{2}{3}$. The circle is centred at $(5, 11)$ and has radius $\sqrt{52}$.

Coordinate geometry

In an arithmetic progression, the first, second and third terms are $a$, $\frac{3}{2}a$ and $b$ respectively, where $a$ and $b$ are positive constants. In a geometric progression, the first, second and third terms are $a$, $18$ and $b + 3$ respectively.

Series

The diagram displays a section of the curve with equation $y^2 = x - 2$ together with the lines $x = 5$ and $y = 1$. The shaded area bounded by the curve and the lines is turned through $360^\circ$ around the $x$-axis.

Integration

For the curve $y = f(x)$, the derivative is $f'(x) = 6x^2 - \frac{8}{x^2}$. The curve is known to pass through $(2, 7)$.

Integration

The points $A(-2, 3)$, $B(3, 0)$ and $C(6, 5)$ are positioned on the circumference of a circle whose centre is $D$.

Coordinate geometry

The diagram shows a section of the curve with equation $y = x^{\frac{1}{2}} + k^2 x^{-\frac{1}{2}}$, where $k$ is a positive constant.

Integration

The function $f$ is given by $f(x) = \frac{1}{3}(2x - 1)^{\frac{3}{2}} - 2x$ for $\frac{1}{2} < x < a$. It is stated that $f$ decreases as $x$ increases.

Differentiation

The curve $y = x^2 - 4x + 3$ is touched by the line $y = mx - 6$ at a tangent.

Differentiation

Show that the equation $\dfrac{\tan x + \sin x}{\tan x - \sin x} = k$, where $k$ is a constant, can be rewritten in the form $\dfrac{1 + \cos x}{1 - \cos x} = k$.

Trigonometry

The figure presents triangle $ABC$, where angle $ABC = 90^\circ$ and $AB = 4\text{ cm}$. Sector $ABD$ is part of a circle centred at $A$. Its area is $10\text{ cm}^2$.

Circular measure

Functions $f$ and $g$ are each defined for $x \in \mathbb{R}$, and they are given by $f(x) = x^2 - 2x + 5$, $g(x) = x^2 + 4x + 13$.

Functions

Write down the first four terms, arranged in ascending powers of $x$, of the expansion of $(a - x)^6$.

Series

The functions $f$ and $g$ are given by $f : x \mapsto x^2 - 1$ when $x < 0$, and $g : x \mapsto \frac{1}{2x + 1}$ when $x < -\frac{1}{2}$.

Functions

For a geometric progression, the second term is $24\%$ of the sum to infinity. Determine the possible values of the common ratio.

Series

Solve for $x$ the inequality $|3x - 7| < |4x + 5|$.

Algebra

Expand $\sin(\theta + 30^\circ)$ first, then solve $\sin(\theta + 30^\circ)\csc\theta = 2$ for $0^\circ < \theta < 360^\circ$.

Trigonometry

Demonstrate that $(\sec x + \cos x)^2$ may be written in the form $\sec^2 x + a + b\cos 2x$, with $a$ and $b$ as the constants to determine.

Integration

The curve is given by the parametric equations $x = \ln(2t + 6) - \ln t$, $y = t\ln t$.

Differentiation

The diagram illustrates the curve given by $y = \frac{3x + 2}{\ln x}$. This curve has a minimum point $M$.

Numerical solution of equations

Apply the trapezium rule with three intervals to estimate $\int_{1}^{4} \frac{6}{1 + \sqrt{x}} \, dx$. Give the result correct to $5$ significant figures.

Integration

The polynomial $p(x)$ is defined as $p(x) = ax^3 - 11x^2 - 19x - a$, where $a$ is a constant. It is known that $(x - 3)$ is a factor of $p(x)$.

Logarithmic and exponential functions

Solve $\ln(2 + x) - \ln x = 2\ln 3$.

Logarithmic and exponential functions

The roots of the equation $5|x| = 5 - 2x$ are $x = a$ and $x = b$, with $a < b$.

Algebra

Solve $\sin(2\theta + 30^\circ) = 5\cos(2\theta + 60^\circ)$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

Determine the exact value of $\int_{0}^{2} 6e^{2x+1}\,dx$.

Integration

Determine the quotient when $x^4 - 32x + 55$ is divided by $(x - 2)^2$, and show that the remainder equals $7$.

Logarithmic and exponential functions

The diagram depicts the curve defined by $y = (\ln x)^2 - 2\ln x$. It meets the $x$-axis at $A$ and $B$, and has a minimum at $M$.

Differentiation

The diagram depicts the curve defined by the parametric equations $x = 4t + e^{2t}$, $y = 6t\sin 2t$, for $0 \leq t \leq 1$. The point $P$ on the curve is assigned parameter $p$ and has $y$-coordinate $3$.

Numerical solution of equations

Solve for $x$ in the equation $\ln(2 + x) - \ln x = 2\ln 3$.

Logarithmic and exponential functions

For the equation $5|x| = 5 - 2x$, the two solutions are $x = a$ and $x = b$, with $a < b$.

Algebra

Solve $ \sin(2\theta + 30^{\circ}) = 5\cos(2\theta + 60^{\circ})$ for $0^{\circ} < \theta < 180^{\circ}$.

Trigonometry

Determine the exact value of $\int_{0}^{2} 6e^{2x+1}\,dx$.

Integration

Determine the quotient when $x^4 - 32x + 55$ is divided by $(x - 2)^2$ and show that the remainder is $7$.

Logarithmic and exponential functions

The diagram illustrates the curve with equation $y = (\ln x)^2 - 2\ln x$. The curve intersects the $x$-axis at the points $A$ and $B$, and it has a minimum point $M$.

Differentiation

The diagram represents the curve whose parametric equations are $x = 4t + e^{2t}$, $y = 6t\sin 2t$, where $0 \leq t \leq 1$. Point $P$ lies on the curve and has parameter $p$ and $y$-coordinate $3$.

Numerical solution of equations

Solve $2|3x - 1| < |x + 1|$.

Algebra

The variables $x$ and $t$ are linked by the differential equation $\frac{dx}{dt} = x^2(1 + 2x)$, with the condition that $x = 1$ when $t = 0$.

Differential equations

Determine the real solution of $\frac{2e^x + e^{-x}}{2 + e^x} = 3$, and give your answer to $3$ decimal places. Your solution should clearly indicate that there is only one real root.

Numerical solution of equations

From the equation $\cos(x - 30^{\circ}) = 2\sin(x + 30^{\circ})$, show that $\tan x = \frac{2 - \sqrt{3}}{1 - 2\sqrt{3}}$.

Trigonometry

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

Integration

On an Argand diagram with origin $O$, the roots of this equation are shown by the two distinct points $A$ and $B$.

Complex numbers

A curve is given by the parametric equations $x = \ln(2 + 3t)$, $y = \frac{t}{2 + 3t}$.

Differentiation

The diagram depicts the curve $y = \frac{\tan^{-1} x}{\sqrt{x}}$ and its maximum point $M$, at which $x = a$.

Numerical solution of equations

Taking $O$ as the origin, the position vectors of $A$ and $B$ are $\overrightarrow{OA} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$. The line $l$ is given by $\mathbf{r} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$.

Vectors

The curve is defined by $y = x^{-\frac{2}{3}} \ln x$ for $x > 0$, and it possesses a single stationary point.

Differentiation

Solve for the values of $x$ that satisfy $|2x - 1| < 3|x + 1|$.

Algebra

The diagram depicts trapezium $ABCD$, with $AD = BC = r$ and $AB = 2r$. The acute angles $BAD$ and $ABC$ are each $x$ radians. Circular arcs of radius $r$ centred at $A$ and $B$ intersect at $M$, the midpoint of $AB$.

Numerical solution of equations

Relative to origin $O$, the position vectors of points $A$ and $B$ are $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j}$ and $\overrightarrow{OB} = \mathbf{j} - 2\mathbf{k}$.

Vectors

On an Argand diagram sketch, indicate by shading the set of points for complex numbers $z$ that satisfy the stated inequalities.

Complex numbers

The variables $x$ and $y$ are linked by $x = A(3^{-y})$, with $A$ as a constant.

Logarithmic and exponential functions

Find the exact value of $\int_{0}^{2} \tan^{-1}\left(\tfrac{1}{2}x\right)\,dx$ by using integration by parts.

Integration

The complex number $u$ is defined as $u = 10 - 4\sqrt{6}i$.

Complex numbers

Prove the identity $\cosec 2\theta - \cot 2\theta = \tan \theta$.

Trigonometry

The curve has the property that the gradient at a point with coordinates $(x, y)$ is proportional to $\frac{y}{\sqrt{x+1}}$. It passes through the points with coordinates $(0, 1)$ and $(3, e)$.

Differential equations

The equation for a curve is $y = e^{-5x} \tan^2 x$ where $-\tfrac{1}{2}\pi < x < \tfrac{1}{2}\pi$.

Differentiation

Take $f(x) = \dfrac{14 - 3x + 2x^2}{(2 + x)(3 + x^2)}$.

Algebra

Expand $(1 + 3x)^{\frac{2}{3}}$ into ascending powers of $x$, up to and including the $x^3$ term, with the coefficients simplified.

Algebra

Verify that $-1 + \sqrt{2}i$ is a root of the equation $z^{4} + 3z^{2} + 2z + 12 = 0$.

Complex numbers

Solve the equation $4^x = 3 + 4^{-x}$. Give your answer correct to $3$ decimal places.

Logarithmic and exponential functions

The curve is given in parametric form by $x = t + \ln(t + 2)$ and $y = (t - 1)e^{-2t}$, where $t > -2$.

Differentiation

Define $f(x)$ by $f(x) = \frac{15 - 6x}{(1 + 2x)(4 - x)}$.

Integration

By expanding $\tan(2\theta + 2\theta)$ first, show that the equation $\tan 4\theta = \frac{1}{2}\tan\theta$ can be rewritten as $\tan^4\theta + 2\tan^2\theta - 7 = 0$.

Trigonometry

Sketch an appropriate pair of graphs to show that the equation $\cot \frac{1}{2}x = 1 + e^{-x}$ has exactly one root in the interval $0 < x \leq \pi$.

Numerical solution of equations

For the curve drawn in the diagram, the normal to the curve at the point $P$ with coordinates $(x, y)$ intersects the $x$-axis at $N$. Point $M$ is the perpendicular drop from $P$ to the $x$-axis. The curve is defined so that, for every value of $x$ in the range $0 \leq x < \frac{1}{2}\pi$, the area of triangle $PMN$ is $\tan x$.

Differential equations

The graph depicts the curve $y = \frac{\ln x}{x^4}$ and its highest point $M$.

Integration

The quadrilateral $ABCD$ is a trapezium with $AB \parallel DC$. Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\overrightarrow{OB} = \mathbf{i} + 3\mathbf{j} + \mathbf{k}$ and $\overrightarrow{OC} = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$.

Vectors

The winch works through a force applied by a rope. It is used to haul a load of mass $50\,\text{kg}$ along the line of greatest slope of a plane that is inclined at $60^\circ$ to the horizontal. The winch moves the load a distance of $5\,\text{m}$ up the plane at constant speed. A constant resistance to motion of $100\,\text{N}$ acts.

Energy, work and power

Particles $A$ and $B$ have masses $m\,\text{kg}$ and $0.1\,\text{kg}$ respectively, with $m > 0.1$. They are fastened to the two ends of a light inextensible string, which runs over a fixed smooth pulley so that the particles hang vertically beneath it (see diagram). Initially, each particle is $0.9\,\text{m}$ above level ground, and the system is released from rest; during the time that both particles are moving, the string tension is $1.5\,\text{N}$. Particle $B$ does not reach the pulley.

Energy, work and power

Three particles $P$, $Q$ and $R$, with masses $0.1\,\text{kg}$, $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are initially at rest in a straight line on a smooth horizontal plane. Particle $P$ is sent towards $Q$ with speed $5\,\text{m s}^{-1}$. Once $P$ and $Q$ have collided, $P$ rebounds at speed $1\,\text{m s}^{-1}$. $Q$ then collides with $R$. Immediately after the collision with $Q$, $R$ starts moving with speed $V\,\text{m s}^{-1}$.

Momentum

Two cyclists, Isabella and Maria, are racing. Each one moves along a straight road with constant acceleration, beginning from rest at point $A$. Isabella accelerates for $5\,\text{s}$ at a steady rate of $a\,\text{m s}^{-2}$. After that, she continues at the constant speed she has reached for $10\,\text{s}$, and then she slows to rest at a constant rate over $5\,\text{s}$. Maria accelerates at a constant rate until she reaches $5\,\text{m s}^{-1}$ over a distance of $27.5\,\text{m}$. She then keeps this speed for $10\,\text{s}$ before decelerating to rest at a constant rate over $5\,\text{s}$.

Kinematics of motion in a straight line

A particle travels in a straight line, beginning from rest at $A$ and stopping instantaneously at $B$. At time $t$ after it has left $A$, its acceleration is $a\,\text{m s}^{-2}$, where $a = 6t^2 - 2t$.

Kinematics of motion in a straight line

At point O, three coplanar forces with magnitudes 10 N, 25 N and 20 N act in the directions shown in the diagram.

Forces and equilibrium

A playground slide slopes downward at a fixed angle of $30^\circ$ over 2.5 m. After this, it continues as a horizontal part lying in the same vertical plane as the inclined part. A child of mass 35 kg, modelled as a particle P, begins from rest at the top of the slide and moves directly down the sloping part. She then travels along the horizontal part until she stops (see diagram). As the child moves from the sloping part to the horizontal part, her speed does not change instantaneously. A resistive force acts on the horizontal part of the slide, and the work done against the resistive force on the horizontal part of the slide is 250 J per metre. The sloping part of the slide is smooth.

Energy, work and power

A particle with mass $0.6\,\text{kg}$ is projected at a speed of $4\,\text{m s}^{-1}$ along the line of greatest slope of a smooth plane that is inclined at $10^\circ$ to the horizontal.

Energy, work and power

Three coplanar forces with magnitudes $34\,\text{N}$, $30\,\text{N}$ and $26\,\text{N}$ are applied at a point in the directions indicated in the diagram. Since $\sin \alpha = \frac{5}{13}$ and $\sin \theta = \frac{8}{17}$,

Forces and equilibrium

A ring with mass $0.3\,\text{kg}$ is passed through a horizontal rough rod. The coefficient of friction between the ring and the rod is $0.8$. A force of magnitude $8\,\text{N}$ acts on the ring. It is applied at an angle of $10^\circ$ above the horizontal in the vertical plane containing the rod.

Kinematics of motion in a straight line

A particle of mass $12\,\text{kg}$ rests on a rough plane inclined at $25^\circ$ to the horizontal. A force of magnitude $P\,\text{N}$ is applied at $8^\circ$ above the line of greatest slope of the plane. This force keeps the particle in equilibrium. The coefficient of friction between the particle and the plane is $0.3$.

Energy, work and power

A car with mass $1250\,\text{kg}$ is towing a caravan whose mass is $800\,\text{kg}$ on a straight road. The resistive forces acting on the car and the caravan are $440\,\text{N}$ and $280\,\text{N}$ respectively. The car and caravan are joined by a light rigid tow-bar.

Energy, work and power

Particle $A$ is launched vertically upward from level ground with an initial speed of $30\,\text{m s}^{-1}$. At that same instant, particle $B$ is let go from rest $15\,\text{m}$ directly above $A$. One of the particles has twice the mass of the other. As the motion proceeds, $A$ and $B$ collide and stick together to make particle $C$.

Kinematics of motion in a straight line

Particle $P$ moves along a straight line, begins from rest at point $O$, and is back at rest $16\,\text{s}$ after leaving $O$. If $t\,\text{s}$ has elapsed since it left $O$, the acceleration $a\,\text{m s}^{-2}$ of $P$ is defined by $a = 6 + 4t\quad 0 \leq t < 2,$ $a = 14\quad 2 \leq t < 4,$ $a = 16 - 2t\quad 4 \leq t \leq 16.$ At no instant does the velocity change suddenly.

Kinematics of motion in a straight line

Particles $P$, with mass $0.4\,\text{kg}$, and $Q$, with mass $0.5\,\text{kg}$, are free to move on a smooth horizontal plane. $P$ and $Q$ travel straight towards one another at speeds $2.5\,\text{m s}^{-1}$ and $1.5\,\text{m s}^{-1}$ respectively. After the collision between $P$ and $Q$, the speed of $Q$ is twice that of $P$.

Momentum

A cyclist moves along a straight level road. She is producing power at a steady rate of $150\,\text{W}$. At the instant when her speed is $4\,\text{m s}^{-1}$, her acceleration is $0.25\,\text{m s}^{-2}$. The resistance opposing motion is $20\,\text{N}$.

Energy, work and power

Four forces lie in one plane and act at a single point. Their magnitudes are $20\,\text{N}$, $30\,\text{N}$, $40\,\text{N}$ and $F\,\text{N}$. Their directions are indicated in the diagram, with $\sin \alpha = 0.28$ and $\sin \beta = 0.6$.

Forces and equilibrium

A particle is launched vertically upwards at speed $u\,\text{m s}^{-1}$ from a point on level ground. After $2$ seconds, its height above the ground is $24\,\text{m}$.

Kinematics of motion in a straight line

A car with mass $1400\,\text{kg}$ is pulling a trailer with mass $500\,\text{kg}$ down a straight hill that is inclined at an angle of $5^\circ$ to the horizontal. The car and trailer are joined by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is $20\,\text{m s}^{-1}$ and at the bottom of the hill their speed is $30\,\text{m s}^{-1}$.

Energy, work and power

A particle travels along a straight line and passes through point $A$ at time $t = 0$. After leaving $A$, the particle’s velocity at time $t$ s is $v\,\text{m s}^{-1}$, where $v = 2t^2 - 5t + 3$.

Kinematics of motion in a straight line

Particle $P$, with mass $0.3\,\text{kg}$, is stationary on a rough plane that makes an angle $\theta$ with the horizontal, where $\sin\theta = \frac{7}{25}$. A force of magnitude $4\,\text{N}$, applied horizontally in the vertical plane containing a line of greatest slope of the plane, acts on $P$ (see diagram). The particle is on the point of moving up the plane.

Forces and equilibrium

A bag contains 12 marbles, and no two are the same size. 8 of the marbles are red and 4 of the marbles are blue.

Permutations and combinations

A factory makes a specific kind of metal rod. These rods have lengths that are normally distributed with mean $25.2\,\text{cm}$ and standard deviation $0.4\,\text{cm}$. One random sample of $500$ of these rods is selected.

The normal distribution

How many distinct arrangements can be made from the $8$ letters in the word RELEASED?

Permutations and combinations

For entry to a science college, students must first sit a written test and then take a practical test. Each student may have at most two tries at the written test. A second try is permitted only if the first try is unsuccessful. No student may attempt the practical test more than once. If both written-test attempts are failed, the practical test cannot be taken. The chance that a student passes the written test on the first attempt is $0.8$. If the first written-test attempt is failed, the chance of passing on the second attempt is $0.6$. The chance that a student passes the practical test is always $0.3$.

Probability

The times needed by 200 players to complete a computer puzzle are set out in the table below: Time ($t$ seconds): $0 \leq t < 10$, $10 \leq t < 20$, $20 \leq t < 40$, $40 \leq t < 60$, $60 \leq t < 100$. Number of players: $16$, $54$, $78$, $32$, $20$.

Representation of data

In Questa, $60\%$ of adults go to work by car.

Discrete random variables

Sharma is aware that her cupboard contains 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn. Every tin has the same shape and size, but all the labels have been removed, so Sharma cannot tell what is in any tin. Sharma wants carrots for her meal, so she opens the tins one at a time, chosen at random, until she opens a tin of carrots. The random variable $X$ represents the number of tins she has to open.

Discrete random variables

An ordinary fair die is thrown again and again until a 5 appears. Let the random variable $X$ represent the number of throws needed.

Discrete random variables

The masses of bags of sugar are normally distributed with mean $1.04\text{ kg}$ and standard deviation $\sigma\text{ kg}$. In a random sample of $2000$ bags of sugar, $72$ had a mass greater than $1.10\text{ kg}$.

The normal distribution

On each day Alexa goes to work, the probabilities that she travels by bus, by train or by car are $0.4$, $0.35$ and $0.25$ respectively. If she goes by bus, the chance that she is late is $0.55$. If she goes by train, the chance that she is late is $0.7$. If she goes by car, the chance that she is late is $x$. For a day chosen at random when Alexa goes to work, the probability that she is not late is $0.48$.

Probability

Two fair spinners are used. One has faces labelled $1, 2, 2$ and the other has faces labelled $-2, 0, 1$. Each spinner is spun once, and the number showing when it stops is recorded. The random variable $X$ represents the total of the two numbers.

Discrete random variables

Each day, Richard boards a flight from Astan to Bejin. For any one day, the chance that the flight is early is $0.15$, the chance that it is on time is $0.55$ and the chance that it is late is $0.3$.

Discrete random variables

Determine how many different arrangements can be made from the 8 letters in TOMORROW.

Permutations and combinations