Oct/Nov 2024

For the expansion of $(kx + \frac{2}{x})^4$, with $k$ a positive constant, the term that does not contain $x$ has value $150$.

Series

An arithmetic progression begins with first term $5$ and has common difference $d$, with $d > 0$. The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.

Series

The function $f$ is given by $f(x) = 3 + 6x - 2x^2$ for $x \in \mathbb{R}$.

Functions

The curve $y = x^2 - \frac{a}{x}$ has a stationary point at $(-3, b)$.

Differentiation

The diagram illustrates a sector of a circle, with centre $O$, for which $OB = OC = 15\text{ cm}$. The angle $BOC$ measures $\frac{2}{5}\pi$ radians. On the lines $OB$ and $OC$, points $A$ and $D$ are joined by an arc $AD$ belonging to a circle centred at $O$. The shaded part is enclosed by arcs $AD$ and $BC$ together with the straight segments $AB$ and $DC$. The area of the shaded region is given as $\frac{19}{5}\pi\text{ cm}^2$.

Circular measure

Prove that the curve with equation $x^2 - 3xy - 40 = 0$ and the line with equation $3x + y + k = 0$ intersect for every value of the constant $k$.

Coordinate geometry

The curve satisfies $\frac{dy}{dx} = 4x - 3\sqrt{x} + 1$.

Differentiation

The equations of circles $C_1$ and $C_2$ are $x^2 + y^2 + 6x - 10y + 18 = 0$ and $(x - 9)^2 + (y + 4)^2 - 64 = 0$, respectively.

Coordinate geometry

The diagram displays part of the curve with equation $y = \frac{12}{\sqrt[3]{2x + 1}}$. The point $A$ on the curve has coordinates $(\frac{7}{2}, 6)$.

Integration

It is given that $\beta$ is an angle between $90^\circ$ and $180^\circ$ for which $\sin \beta = a$. Express $\tan^2 \beta - 3 \sin \beta \cos \beta$ in terms of $a$.

Trigonometry

The curve is described by $y = 4 + 5x + 6x^2 - 3x^3$.

Differentiation

The graph shown is the curve given by $y = a\sin(bx) + c$ for $0 \leq x \leq 2\pi$, with $a$, $b$ and $c$ all positive constants.

Trigonometry

For a function $f$ with domain $x > 0$, the derivative is given by $f'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{2}{5}}$. The curve $y = f(x)$ is stated to pass through the point $(1, 0)$.

Differentiation

The first term in this arithmetic progression is $-20$, and the common difference is $5$.

Series

A curve is given by $y = 2x^2 - 3$. The points $A$ and $B$, whose $x$-coordinates are $2$ and $(2 + h)$ respectively, are on the curve.

Differentiation

Determine the term with no $x$ in the expansion of each of the following:

Series

The function $f$ is given by $f(x) = \frac{2x + 1}{2x - 1}$, with the condition $x < \frac{1}{2}$.

Functions

The diagram depicts a metal plate $OABCDE$ made up of sectors from two circles, both with centre $O$. The radii of sectors $AOB$ and $EOF$ are $r$ cm, while sector $COD$ has radius $2r$ cm. The angle $AOB = \text{angle } EOF = \theta$ radians and the angle $COD = 2\theta$ radians. It is given that the plate has perimeter $14$ cm and area $10\text{ cm}^2$.

Circular measure

Express $-2x^2 + 8x + 11$ in the form $-a(x - b)^2 + c$, where $a$, $b$ and $c$ are positive integers, and then find the coordinates of the vertex of the graph with equation $y = -2x^2 + 8x + 11$.

Integration

A circle is described by the equation $x^2 + y^2 + px + 2y + q = 0$, where $p$ and $q$ are constants.

Coordinate geometry

The curve is described by $y = \frac{1}{2}kx^2 - 2kx + 2$, whereas the line is given by $y = kx + p$; here $k$ and $p$ are constants with $0 \le k \le 1$.

Coordinate geometry

In an arithmetic progression, the fourth term is $15$ and the eighth term is $25$.

Series

The coordinates of points $A$ and $B$ are $(4, 3)$ and $(8, -5)$ respectively. A circle of radius $10$ goes through $A$ and $B$.

Coordinate geometry

A curve has equation $y = kx^{\frac{1}{2}} - 4x^2 + 2$, with $k$ denoting a constant.

Differentiation

Determine the exact solution to the equation $\cos \frac{\pi}{6} + \tan 2x + \frac{\sqrt{3}}{2} = 0$ for $-\frac{1}{4}\pi < x < \frac{1}{4}\pi$.

Trigonometry

Determine the coefficients of $x^3$ and $x^4$ in the expansion of $(3-ax)^5$, where $a$ is a constant. Express each answer in terms of $a$.

Series

Find the solutions of $4\sin^4\theta + 12\sin^2\theta - 7 = 0$ for $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

On the diagram, $y=f(x)$ is drawn with solid lines, while $y=g(x)$ is drawn with broken lines.

Functions

A convergent geometric progression has first term $10$. Let the sum of its first $4$ terms be $p$, and let the sum of its first $8$ terms be $q$. It is given that $\frac{q}{p} = \frac{17}{16}$.

Series

The diagram depicts a metal plate $ABCDEF$ formed from five regions. Regions $BCD$ and $DEF$ are semicircles. Region $BAFO$ is a sector of a circle with centre $O$ and radius $20\text{ cm}$. $D$ is located on this circle. Regions $OBD$ and $ODF$ are triangles. The angles $BOD$ and $DOF$ are each $\theta$ radians.

Circular measure

Express $3x^2 - 12x + 14$ in the form $3(x+a)^2 + b$, with $a$ and $b$ as the constants to determine.

Functions

The diagram displays the curves given by $y = x^3 - 3x + 3$ and $y = 2x^3 - 4x^2 + 3$.

Integration

The variables $x$ and $y$ are related by $a^{2y} = e^{3x+k}$, with $a$ and $k$ constant. A graph of $y$ against $x$ is a straight line.

Logarithmic and exponential functions

Solve for the values of $x$ in the inequality $|x - 7| > 4x + 3$.

Algebra

The function $f$ is given by $f(x) = \tan^2\left(\frac{1}{2}x\right)$ for $0 \leq x < \pi$.

Integration

The polynomial $p(x)$ is given by $p(x) = ax^3 - ax^2 - 15x + 18$, with $a$ as a constant. It is stated that $(x + 2)$ is one factor of $p(x)$.

Trigonometry

It is stated that $\int_a^3 \frac{10}{2x+1} \, dx = 7$, with $a$ being a constant larger than 1.

Numerical solution of equations

The curve is given by the parametric equations $x = \frac{e^{2t} - 2}{e^{2t} + 1}$ and $y = e^{3t} + 1$.

Differentiation

Prove that $\cos(\theta + 30^\circ)\cos(\theta + 60^\circ) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta$.

Trigonometry

Use logarithms to show that $5^{8y} = 6^{7x}$ may be written in the form $y = kx$. State the value of the constant $k$ correct to 3 significant figures.

Logarithmic and exponential functions

Define $f(x) = 4\sin^2 3x$.

Algebra

The curve is given by the equation $6e^{-x}y^2 + e^{2x} - 12y + 7 = 0$.

Differentiation

On one set of axes, sketch the graphs of $y = 1 + e^{2x}$ and $y = |x - 4|$.

Numerical solution of equations

Let $p(x)$ denote the polynomial $p(x) = ax^3 + bx^2 - ax + 8$, with $a$ and $b$ as constants. It is stated that $(x + 2)$ divides $p(x)$ exactly, and that the remainder is $24$ when $p(x)$ is divided by $(x - 2)$.

Trigonometry

The diagram displays the curves given by $y = \sqrt[3]{5x^2 + 7}$ and $y = \frac{27}{2x + 5}$ for $x \geq 0$. The curves intersect at $(2, 3)$. Region $A$ is enclosed by the curve $y = \sqrt[3]{5x^2 + 7}$ together with the straight lines $x = 0$, $x = 2$ and $y = 0$. Region $B$ is enclosed by the two curves and the straight line $x = 0$.

Integration

Express $4 \sin \theta \sin (\theta + 60^\circ)$ as $a + R \sin (2\theta - \alpha)$, where $a$ and $R$ are positive integers and $0^\circ < \alpha < 90^\circ$.

Trigonometry

The variables $x$ and $y$ obey the equation $a^y = e^{3x + k}$, with $a$ and $k$ as constants, and the graph of $y$ plotted against $x$ is a straight line.

Logarithmic and exponential functions

Solve for $x$ in the inequality $|x - 7| > 4x + 3$.

Algebra

The function $f$ is given by $f(x)=\tan^2\left(\tfrac{1}{2}x\right)$, with domain $0 \leq x < \pi$.

Integration

The polynomial $p(x)$ is given by $p(x) = ax^3 - ax^2 - 15x + 18$, with $a$ taken as a constant. It is also stated that $(x + 2)$ divides $p(x)$.

Trigonometry

You are told that $\int_a^3 \frac{10}{2x+1}\,dx = 7$, with $a$ being a constant larger than 1.

Numerical solution of equations

The curve is described by the parametric equations $x = \frac{e^{2t} - 2}{e^{2t} + 1}$, $y = e^{3t} + 1$.

Differentiation

Show that $\cos(\theta + 30^\circ)\cos(\theta + 60^\circ) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta$.

Trigonometry

Let p(x) represent the polynomial $4x^3 + ax^2 + 5x + b$, where $a$ and $b$ are constants. It is stated that $(2x + 1)$ is a factor of $p(x)$. In addition, when $p(x)$ is divided by $(x - 4)$, the remainder is 3 times the remainder obtained when $p(x)$ is divided by $(x - 2)$.

Algebra

A sizeable cylindrical tank is being used to hold water. The tank has a circular base of radius $4$ metres. At time $t$ minutes, the water depth in the tank is $h$ metres. A tap is fitted at the bottom of the tank. When the tap is open, water leaves the tank at a rate proportional to the square root of the volume of water in the tank.

Differential equations

Determine the exact value of $\int_{1}^{3} x^2 \ln(3x)\,dx$. Present your answer as $a\ln b + c$, where $a$ and $c$ are rational and $b$ is an integer.

Integration

A curve is described by the equation $\ln(x + y) = 3x^2y$.

Differentiation

Show that the expression $\sec^4 \theta - \tan^4 \theta$ is equal to $1 + 2\tan^2 \theta$.

Trigonometry

By sketching a suitable pair of graphs, show that the equation $2 + e^{-0.2x} = \ln(1 + x)$ has only one root.

Numerical solution of equations

The diagram depicts the curve $y = \sin 2x(1 + \sin 2x)$, with $0 \leq x \leq \tfrac{3}{4}\pi$, together with its minimum point $M$. The shaded area enclosed by the part of the curve above the $x$-axis and the $x$-axis itself is labelled $R$.

Trigonometry

Define $f(x) = \dfrac{5x^2 + 8x + 5}{(1 + 2x)(2 + x^2)}$.

Algebra

If $z = 1 + yi$ and $y$ is a real number, write $\frac{1}{z}$ in the form $a + bi$, with $a$ and $b$ expressed as functions of $y$.

Complex numbers

With respect to the origin $O$, the position vector of point $A$ is $\overrightarrow{OA} = 8i - 5j + 6k$. The line $l$ goes through $A$ and is parallel to the vector $2i + j + 4k$.

Vectors

Expand $\sqrt{9 - 3x}$ as a series in ascending powers of $x$, up to and including the term in $x^2$, and simplify the coefficients.

Algebra

A spherical balloon has volume $V$ and radius $r$. Air is pumped into the balloon at a steady rate of $40\pi t$ from the moment $t = 0$ when $r = 0$. At the same time, air also escapes from the balloon at $0.8\pi r$. The balloon stays spherical throughout.

Differential equations

Define $f(x)$ by $f(x) = \frac{2e^{2x}}{e^{2x} - 3e^{x} + 2}$.

Integration

By plotting a suitable pair of graphs, demonstrate that the equation $\cot 2x = \sec x$ has a single root in the interval $0 < x < \frac{1}{2}\pi$.

Numerical solution of equations

The square roots of $6 - 8i$ can be written in Cartesian form as $x + iy$, with $x$ and $y$ both real and exact.

Complex numbers

Solve the equation $5^x = 5^{x+2} - 10$. State your answer correct to $3$ decimal places.

Logarithmic and exponential functions

For the complex number $u = \frac{(\cos \frac{\pi}{7} + i \sin \frac{\pi}{7})^4}{\cos \frac{\pi}{7} - i \sin \frac{\pi}{7}}$, find the exact value of $\arg u$.

Complex numbers

The variables $x$ and $y$ obey the relation $ay = b^x$, with $a$ and $b$ as constants. In the diagram, the graph of $\ln y$ against $x$ is a straight line that goes through the points $(0.50, 2.24)$ and $(3.40, 8.27)$.

Algebra

Show that the equation $\tan^3 x + 2\tan 2x - \tan x = 0$ can be rewritten as $\tan^4 x - 2\tan^2 x - 3 = 0$ for $\tan x \neq 0$.

Trigonometry

The curve is given in parametric form by $x = \tan^2 2t$ and $y = \cos 2t$, where $0 < t \leq \frac{1}{4}\pi$.

Differentiation

Relative to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\overrightarrow{OA}=\begin{pmatrix}2\\1\\-3\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\4\\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}-3\\-2\\2\end{pmatrix}$.

Vectors

The complex number $z$ obeys $|z| = 2$ and $0 \leq \arg z \leq \frac{1}{4}\pi$.

Complex numbers

A water tank has the form of a cuboid with base area $40000\,\text{cm}^2$. When the time is $t$ minutes, the water depth in the tank is $h$ cm. Water is pumped into the tank at a rate of $50000\,\text{cm}^3$ per minute. Water leaks from the tank through a hole in the bottom at a rate of $600h\,\text{cm}^3$ per minute.

Differential equations

The diagram depicts the curve $y = 2\sin x\sqrt{2 + \cos x}$, for $0 \leq x \leq 2\pi$, together with its minimum point $M$, where $x = a$.

Integration

Define $f(x)$ by $f(x)=2x^3 - 5x^2 + 4$.

Numerical solution of equations

A bacterial population, $P$, after $t$ hours is represented by $P = ae^{kt}$, where $a$ and $k$ are constants. In the diagram, the graph of $\ln P$ against $t$ is a straight line with gradient $\frac{1}{20}$ and it cuts the vertical axis at $(0, 3)$.

Logarithmic and exponential functions

Determine the complex number $z$ that satisfies the equation $$\frac{z - 3i}{z + 3i} = \frac{2 - 9i}{5}.$$ Write your answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Show that $$\cos^4 \theta - \sin^4 \theta - 4 \sin^2 \theta \cos^2 \theta \equiv \cos^2 2\theta + \cos 2\theta - 1.$$ is true.

Trigonometry

The vector equations of the lines $l$ and $m$ are given by $l: \ \mathbf{r} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{k})$ and $m: \ \mathbf{r} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + 5\mathbf{k})$. The two lines $l$ and $m$ meet at the point $P$.

Vectors

For a curve, the parametric equations are $x = 3\sin 2t$ and $y = \tan t + \cot t$, with $0 < t < \tfrac{1}{2}\pi$.

Differentiation

Let $f(x)$ be given by $\frac{7a^2}{(a - 2x)(3a + x)}$, where $a$ is a positive constant.

Algebra

Determine the quotient and remainder when $x^4 + 16$ is divided by $x^2 + 4$.

Integration

A light inextensible string passes over a fixed smooth pulley and links two particles of masses $1.8\,\text{kg}$ and $1.2\,\text{kg}$. The particles are hanging vertically. The system is released from rest.

Newton's laws of motion

A particle with mass $7.5\,\text{kg}$ begins at rest at $A$ and moves down the inclined plane $AB$. Point $B$ lies $12.5$ metres vertically below the level of $A$, as shown in the diagram.

Energy, work and power

At a point, coplanar forces with magnitudes $52\,\text{N}$, $39\,\text{N}$ and $P\,\text{N}$ act in the directions shown in the diagram, and the system remains in equilibrium.

Forces and equilibrium

A bus runs between stops $A$ and $B$. It leaves $A$ from rest and speeds up at a constant rate of $a\,\text{m s}^{-2}$ until its speed is $16\,\text{m s}^{-1}$. It then continues at this fixed speed before slowing at a constant rate of $0.75\,\text{m s}^{-2}$, finally coming to rest at $B$. The whole journey lasts $240\,\text{s}$.

Kinematics of motion in a straight line

Particle $A$ is launched vertically upward from point $O$ at a speed of $80\,\text{m s}^{-1}$. After 1 second, a second particle, $B$, which has the same mass as $A$, is launched vertically upward from $O$ at a speed of $100\,\text{m s}^{-1}$. At a time $T\,\text{s}$ after the first particle is launched, the two particles meet and merge to form particle $C$.

Kinematics of motion in a straight line

A particle with mass $1.2\,\text{kg}$ rests on a rough plane inclined at an angle $\theta$ to the horizontal, where $\sin\theta = \frac{7}{25}$. It is maintained in equilibrium by a horizontal force of magnitude $P\,\text{N}$ acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is $0.15$.

Forces and equilibrium

A car of mass $1200\,\text{kg}$ is moving at speed $v\,\text{m s}^{-1}$, and the resistive force has magnitude $kv\,\text{N}$. The engine’s maximum power is $92.16\,\text{kW}$. The car is moving on a straight, horizontal road.

Newton's laws of motion

A particle $P$ travels along a straight line and passes through point $O$ with velocity $4.2\,\text{m s}^{-1}$. For time $t\,\text{s}$ after $P$ passes $O$, its acceleration $a\,\text{m s}^{-2}$ is $a = 0.6t - 2.7$.

Kinematics of motion in a straight line

At time t seconds after the particle has left a fixed point O, its velocity is v\,\text{m s}^{-1}. The diagram is a velocity-time graph representing the particle's motion from t = 0 up to t = T. It is formed from four straight-line sections. From rest, the particle increases its speed to V\,\text{m s}^{-1} in 4 s, and then slows down at 3\,\text{m s}^{-2} until it is momentarily at rest after 6 s. The particle then moves back towards O, reaches a greatest speed of 3\,\text{m s}^{-1}, and finally comes to rest again when t = T.

Kinematics of motion in a straight line

A block with mass $20\,\text{kg}$ is initially at rest at the top of a plane that slopes at $30^{\circ}$ to the horizontal. It is launched at speed $5\,\text{m s}^{-1}$ along a line of greatest slope down the plane. A resistance force acts on the block. When the block has travelled $2\,\text{m}$ down the plane from the point from which it was projected, the work done against the resistance force is $50\,\text{J}$.

Energy, work and power

A cyclist travels along a straight horizontal road. The combined mass of the cyclist and his bicycle is $90\,\text{kg}$. The cyclist produces a power of $250\,\text{W}$. At the instant when the cyclist’s speed is $5\,\text{m s}^{-1}$, his acceleration is $0.1\,\text{m s}^{-2}$.

Energy, work and power

In the diagram, particles $A$ and $B$ have masses $0.2\,\text{kg}$ and $0.1\,\text{kg}$ respectively. They hang beneath a horizontal ceiling from two strings, $AP$ and $BQ$, which are fixed at points $P$ and $Q$ on the ceiling. The particles are joined by a horizontal string $AB$. Angle $APQ = 45^\circ$ and $BQP = \theta^\circ$. Every string is light and inextensible. The particles are in equilibrium.

Forces and equilibrium

Two particles, $P$ and $Q$, with masses $2m\,\text{kg}$ and $m\,\text{kg}$ respectively, are initially at rest on the same vertical line. Their heights above horizontal ground are $1\,\text{m}$ for $P$ and $2\,\text{m}$ for $Q$. $P$ is thrown vertically upward at speed $2\,\text{m s}^{-1}$, and at that same moment $Q$ is let go from rest.

Momentum

A particle, $P$, moves along a straight path, beginning at point $O$ with velocity $6\,\text{m s}^{-1}$. The acceleration of $P$ at time $t\,\text{s}$ after leaving $O$ is $a\,\text{m s}^{-2}$, where $a = -1.5t^{\frac{1}{2}}$ for $0 \leq t \leq 1$, and $a = 1.5t^{\frac{1}{2}} - 3t^{-\frac{1}{2}}$ for $t > 1$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are attached to the two ends of a light inextensible string. That string runs over a small fixed smooth pulley fixed to the lower end of a rough plane inclined at an angle $\theta$ to the horizontal, where $\sin\theta = 0.6$. Particle $A$ is on the plane, while particle $B$ hangs vertically below the pulley and is $0.25\,\text{m}$ above horizontal ground. The part of the string from $A$ to the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between $A$ and the plane is $1.125$. Particle $A$ is released from rest.

Newton's laws of motion

An athlete of mass $m\,\text{kg}$ is moving on a level road and experiences a constant resistance force of magnitude $24\,\text{N}$. As he increases his speed from $5\,\text{m s}^{-1}$ to $6\,\text{m s}^{-1}$ over a distance of $50\,\text{m}$, the total work done by the athlete is $1541\,\text{J}$.

Energy, work and power

At a point, coplanar forces with magnitudes $16\,\text{N}$, $12\,\text{N}$, $24\,\text{N}$ and $8\,\text{N}$ act in the directions indicated on the diagram.

Forces and equilibrium

A car with mass 1600 kg moves up a slope that makes an angle of $\sin^{-1} 0.08$ with the horizontal. A steady resistive force of 240 N acts on the car.

Energy, work and power

Particles $A$ and $B$, with masses 3 kg and 6 kg respectively, are located on a smooth horizontal plane. At first, $B$ is stationary while $A$ is travelling towards $B$ at speed $8\,\text{m s}^{-1}$. After $A$ and $B$ collide, $A$ has speed $2\,\text{m s}^{-1}$.

Momentum

A particle with mass 12 kg is to be dragged over a rough horizontal surface by a light inextensible string. The string makes an angle of $30^\circ$ to the plane, and its tension is $T$ N (see diagram). The coefficient of friction between the particle and the plane is 0.5.

Newton's laws of motion

A particle travels along a straight line. It begins at rest at time $t = 0$ and has acceleration $0.6\,\text{m s}^{-2}$ for $4\,\text{s}$, so it attains a speed of $V\,\text{m s}^{-1}$. It then continues at $V\,\text{m s}^{-1}$ for $11\,\text{s}$, before slowing uniformly and coming to rest after another $5\,\text{s}$.

Kinematics of motion in a straight line

Particles $A$ and $B$, whose masses are $3\,\text{kg}$ and $5\,\text{kg}$ respectively, are joined by a light inextensible string that runs over a fixed smooth pulley. The string is taut, with each straight section vertical, and the particles are initially held so that $A$ is $1\,\text{m}$ above a horizontal plane while $B$ is $2\,\text{m}$ above the plane (see diagram). They are then released from rest. In the motion that follows, $A$ does not reach the pulley, and after $B$ touches the plane it remains in contact with the plane.

Newton's laws of motion

Nicola rolls a fair ordinary six-sided dice. The random variable $X$ represents how many throws she needs to obtain a 6.

Discrete random variables

The random variable $X$ may assume the values $-2, -1, 0, 2, 3$. It is stated that $P(X = x) = k(x^2 + 2)$, where $k$ is a positive constant.

Discrete random variables

The walking time to school, measured in minutes, was noted for 200 pupils at one school. The results are shown in the table below.

Representation of data

Rahul has two bags, $X$ and $Y$. Bag $X$ holds $4$ red marbles and $2$ blue marbles, while bag $Y$ contains $3$ red marbles and $4$ blue marbles. Rahul also has a biased coin for which the chance of getting a head on a toss is $\frac{1}{4}$. Rahul throws the coin. If a head appears, he selects one marble at random from bag $X$, notes its colour and returns the marble to bag $X$. He then selects a second marble at random from bag $X$. If a tail appears, he selects one marble at random from bag $Y$, notes its colour and discards the marble. He then selects a second marble at random from bag $Y$.

Probability

The weights of the green apples sold by a shop follow a normal distribution with mean $90$ grams and standard deviation $8$ grams.

The normal distribution

At Breven college, the heights of the female students follow a normal distribution: • $90\%$ of the female students are shorter than $182.7$ cm. • $40\%$ of the female students are shorter than $162.5$ cm.

The normal distribution

In how many distinct ways can the $9$ letters of INTELLECT be arranged if the two Ts are adjacent?

Permutations and combinations

In a college, each student selects just one sport from tennis, hockey or netball. The table gives the numbers of Year 1 and Year 2 students at the college who play each sport. One student is selected at random from the 120 students. The events $X$ and $N$ are defined as follows: $X$: the student is in Year 1. $N$: the student plays netball.

Discrete random variables

Find how many different arrangements can be made from the 9 letters in the word ALGEBRAIC.

Permutations and combinations

A fair coin and a fair ordinary six-sided die are tossed together. The random variable $X$ is defined like this: when the coin lands tails, $X$ is double the die score; when the coin lands heads, $X$ equals the die score if it is even, and $X$ is $0$ otherwise.

Discrete random variables

The heights, in metres, of white pine trees follow a normal distribution with mean $19.8$ and standard deviation $2.4$. In one forest, there are $450$ white pine trees.

The normal distribution

Out of a class of $21$ students, $10$ are violinists, $6$ are guitarists and $5$ are pianists. A set of $7$ is to be selected from these $21$ students. The set will contain $4$ violinists, $2$ guitarists and $1$ pianist.

Permutations and combinations

Last Saturday, 15 runners in each team joined a charity run. The table shows the finishing times, in minutes, for the Falcons runners and the Kites runners as they completed the course.

Representation of data