Oct/Nov 2023

Use the binomial theorem to expand $(1 + 3x)^6$ in ascending powers of $x$, stopping at and including the term in $x^2$.

Series

A curve has a stationary point at $(2, -10)$ and its second derivative is $\frac{d^2y}{dx^2} = 6x$.

Differentiation

The circle in the diagram has equation $(x - 4)^2 + (y + 1)^2 = 40$. Two parallel tangents, both having gradient $1$, meet the circle at $A$ and $B$.

Differentiation

The equation of a line is $y = 2cx + 3$ and the equation of a curve is $y = cx^2 + 3x - c$, with $c$ a constant. Showing all necessary working, determine which statement below is correct.

Coordinate geometry

The figure represents a closed cubical container formed from a thin elastic material, filled with water and then frozen. While freezing takes place, the edge length, $x$ cm, of the container rises at the fixed rate of $0.01$ cm per minute. At time $t$ minutes, the volume of the container is $V$ cm$^3$.

Differentiation

Here, the transformation $R$ is a reflection across the $x$-axis, while $T$ is a translation given by $\begin{pmatrix}3\\-1\end{pmatrix}$.

Functions

Show that the equation $4\sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0$ can be rewritten in the form $a\cos^2 x + b\cos x + c = 0$, where $a$, $b$ and $c$ are integers to be found.

Trigonometry

The diagram depicts a motif made up of the major arc $AB$ of a circle with radius $r$ and centre $O$, together with the minor arc $AOB$ of another circle that also has radius $r$ but centre $C$. Point $C$ lies on the circle centred at $O$.

Circular measure

In a geometric progression, the first two terms add to $15$, and the sum to infinity is $\frac{125}{7}$. The common ratio is negative.

Series

The diagram shows the curves with equations $y = 2(2x - 3)^4$ and $y = (2x - 3)^2 + 1$, intersecting at points $A$ and $B$.

Integration

The function $f$ has the rule $f(x) = 4x^2 - 12x + 13$ for $p < x < q$, where $p$ and $q$ are constants. The function $g$ has the rule $g(x) = 3x + 1$ for $x < 8$.

Functions

The coefficient of $x^3$ in the expansion of $(3 + 2ax)^5$ is six times as large as the coefficient of $x^2$ in the expansion of $(2 + ax)^6$.

Series

For the curve $y = f(x)$, the function is defined by $f(x) = (4x - 3)^{\frac{5}{3}} - \frac{20}{3}x$.

Differentiation

The points $A$, $B$ and $C$ have coordinates $(6, 4)$, $(p, 7)$ and $(14, 18)$, respectively, where $p$ is a constant. The line $AB$ is perpendicular to the line $BC$.

Coordinate geometry

Determine the exact solution of the equation $\frac{1}{6}\pi + \tan^{-1}(4x) = -\cos^{-1}\left(\frac{1}{2}\sqrt{3}\right)$.

Trigonometry

The curve is defined by $\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}$. It goes through the point $P(2, 8)$.

Differentiation

The diagram illustrates a coin’s outline. The arcs $AB$, $BC$ and $CA$ are each sections of circles whose centres are $C$, $A$ and $B$ respectively. $ABC$ is an equilateral triangle with side length $2\,\text{cm}$.

Circular measure

The first three terms of a geometric progression are $\sin\theta$, $\cos\theta$ and $2 - \sin\theta$ respectively, where $\theta$ radians is an acute angle.

Series

The curve is given by the equation $y = x^2 - 8x + 5$.

Quadratics

Verify the identity $(2x - 1)(4x^2 + 2x - 1) \equiv 8x^3 - 4x + 1$.

Trigonometry

The functions $f$ and $g$ are given by $f(x) = (x + a)^2 - a$ for $x \leq -a$, and by $g(x) = 2x - 1$ for $x \in \mathbb{R}$, where $a$ is a positive constant.

Functions

The diagram presents the curves with equations $y = 2x^{\frac{1}{2}} + 13x^{-\frac{1}{2}}$ and $y = 3x^{\frac{1}{2}} + 12$. The points where these curves cross are $A$ and $B$.

Coordinate geometry

For a curve, the gradient at a point $(x, y)$ is given by $\frac{dy}{dx} = x - 3x^{-\frac{1}{2}}$. The curve is known to pass through $(4, 1)$.

Integration

In the diagram, points $A$, $B$ and $C$ are on a circle with centre $O$ and radius $r$. The angle $AOB$ measures $2.8$ radians. The shaded region has two arc boundaries. Its upper boundary is an arc from the circle centred at $O$ with radius $r$. Its lower boundary is an arc from a circle centred at $C$ with radius $R$.

Circular measure

The diagram displays part of the curve with equation $y = x + \frac{2}{(2x - 1)^2}$. The lines $x = 1$ and $x = 2$ meet the curve at $P$ and $Q$ respectively, while $R$ is the stationary point on the curve.

Integration

The circle given by $(x - 3)^2 + (y - 5)^2 = 40$ cuts the $y$-axis at the points $A$ and $B$.

Coordinate geometry

Show that the equation $5\cos\theta - \sin\theta\tan\theta + 1 = 0$ can be rewritten in the form $a\cos^2\theta + b\cos\theta + c = 0$, where $a$, $b$ and $c$ are constants to be found.

Trigonometry

Expand $(1 + 2x)^5$ in powers of $x$ from lowest to highest, including terms up to and including $x^2$.

Series

The first three terms of a geometric progression are $2p + 6$, $5p$ and $8p + 2$, respectively.

Series

The straight line is given by $y = 6x - c$, and the curve is given by $y = cx^2 + 2x - 3$, where $c$ is a constant. The line touches the curve at point $P$.

Differentiation

The function $f$ is given by $f(x)=1+\frac{3}{x-2}$, with $x>2$.

Functions

The diagram depicts a section of the graph of $y=\sin(a(x+b))$, where $a$ and $b$ are positive constants.

Functions

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

Differentiation

You are told that $\theta$ is an acute angle measured in degrees, and that $\sin \theta = \frac{2}{3}$.

Trigonometry

The curve is given by the equation $y = 3\tan\frac{1}{2}x\cos 2x$.

Differentiation

Calculate $\int_{4}^{10} \frac{4}{2x-5}\,dx$, and present your answer in the form $\ln a$, where $a$ is an integer.

Integration

On one set of axes, sketch the graphs of $y = |3x - 5|$ and $y = 2x + 7$.

Logarithmic and exponential functions

The polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 + bx - 20$, with $a$ and $b$ as constants. You are told that $(x + 2)$ is a factor of $p(x)$ and that the remainder is $-11$ when $p(x)$ is divided by $(x + 1)$.

Algebra

Show that $\cosec \theta (3 \sin 2\theta + 4 \sin^3 \theta)$ is equal to $4 + 6 \cos \theta - 4 \cos^2 \theta$.

Trigonometry

The curve whose equation is $e^{2x} - 18x + y^3 + y = 11$ has a stationary point at $(p, q)$.

Numerical solution of equations

If the polynomial $ax^3 + 4ax^2 - 7x - 5$ is divided by $(x + 2)$, then the remainder equals $33$.

Algebra

Solve the equation $\sec \theta \cos(\theta - 60^\circ) = 4$ within $-180^\circ < \theta < 180^\circ$.

Trigonometry

The diagram illustrates the curve with equation $y = 6e^{-\frac{1}{2}x}$. The points on the curve with $x$-coordinates $0$ and $2$ are labelled $A$ and $B$ respectively. The shaded region is bounded by the curve, the line through $A$ parallel to the $x$-axis and the line through $B$ parallel to the $y$-axis.

Integration

Sketch, on one set of axes, the graphs of $y = |3 - x|$ and $y = 9 - 2x$.

Logarithmic and exponential functions

Determine the quotient when $6x^3 - 5x^2 - 24x - 4$ is divided by $(2x + 1)$, and show that the remainder equals 6.

Integration

The curve in the diagram has parametric equations $x = 3\ln(2t - 3)$, $y = 4t\ln t$. It meets the $y$-axis at the point $A$. At $B$, the gradient of the curve is 12.

Numerical solution of equations

Prove the identity $\sin 2x(\cot x + 3\tan x) = 4 - 2\cos 2x$.

Integration

Assume that $\theta$ is an acute angle, measured in degrees, with $\sin \theta = \frac{2}{3}$.

Trigonometry

The curve is given by $y = 3 \tan \frac{1}{2} x \cos 2x$.

Differentiation

Find $\int_{4}^{10} \frac{4}{2x - 5} \, dx$, and express your answer in the form $\ln a$, where $a$ is an integer.

Integration

On one set of axes, sketch the graphs of $y = |3x - 5|$ and $y = 2x + 7$.

Logarithmic and exponential functions

The polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 + bx - 20$, where $a$ and $b$ are constants. It is stated that $(x + 2)$ is a factor of $p(x)$ and that dividing $p(x)$ by $(x + 1)$ leaves remainder $-11$.

Algebra

Show that $\cosec\theta(3\sin 2\theta + 4\sin^3 \theta)$ is equal to $4 + 6\cos \theta - 4\cos^2 \theta$.

Trigonometry

For the curve with equation $e^{2x} - 18x + y^3 + y = 11$, the stationary point is $(p, q)$.

Numerical solution of equations

Determine the exact coordinates of the points on the curve $y = \frac{x^2}{1 - 3x}$ where the gradient of the tangent is 8.

Differentiation

Take $f(x) = \dfrac{24x + 13}{(1 - 2x)(2 + x)^2}$.

Algebra

The diagram shows cuboid $OABCDEFG$, with $OA = 3$ units, $OC = 2$ units and $OD = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OD$ and $OC$ respectively. $M$ is the midpoint of $EF$. The position vector of $P$ is $\mathbf{i} + \mathbf{j} + 2\mathbf{k}$.

Vectors

On an Argand diagram, shade the set of points for complex numbers $z$ that satisfy the inequalities $|z - 2i| \le |z + 2 - i|$ and $0 \le \arg(z + 1) \le \frac{1}{4}\pi$.

Complex numbers

The variables $x$ and $y$ satisfy $y = ab^x$, where $a$ and $b$ are constants. The diagram gives the outcome of plotting $\ln y$ against $x$ for two ordered pairs of $x$ and $y$. These points have coordinates $(1, 3.7)$ and $(2.2, 6.46)$.

Logarithmic and exponential functions

The complex number $u$ is defined by $u = \dfrac{3 + 2i}{a - 5i}$, with $a$ taken to be real.

Complex numbers

If $\sin\left(x + \dfrac{\pi}{6}\right) - \sin\left(x - \dfrac{\pi}{6}\right) = \cos\left(x + \dfrac{\pi}{3}\right) - \cos\left(x - \dfrac{\pi}{3}\right)$, determine the exact value of $\tan x$.

Trigonometry

The curve is given in parametric form by $x = \sqrt{t} + 3$ and $y = \ln t$, with $t > 0$.

Differentiation

The variables $x$ and $\theta$ are linked by the differential equation $\frac{x}{\tan \theta} \frac{dx}{d\theta} = x^2 + 3$. You are also told that $x = 1$ when $\theta = 0$.

Differential equations

Using a suitable pair of sketches, show that the equation $\sqrt{x} = e^x - 3$ has just one root.

Numerical solution of equations

The diagram depicts the curve $y = x e^{-\frac{1}{4}x^2}$, with $x \geq 0$, together with its maximum point $M$.

Integration

Draw a sketch of the graph of $y = |4x - 2|$.

Algebra

The line equations $l$ and $m$ are $l: \mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ and $m: \mathbf{r} = \begin{pmatrix} 6 \\ -3 \\ 6 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 4 \\ c \end{pmatrix}$, with $c$ a positive constant. The angle between $l$ and $m$ is $60^\circ$.

Vectors

The variables $x$ and $y$ are linked by the differential equation $x^2 \frac{dy}{dx} + y^2 + y = 0$. Also given is $x = 1$ when $y = 1$.

Differential equations

A curve is described by the parametric equations $x = (\ln t)^2$ and $y = e^{2 - t^2}$, where $t > 0$.

Differentiation

Let $p(x)$ represent the polynomial $2x^3 + ax^2 - 11x + b$. It is stated that $p(x)$ has $(2x - 1)$ as a factor and that dividing $p(x)$ by $(x + 1)$ leaves a remainder of $12$.

Algebra

On an Argand diagram sketch, shade the set of points that correspond to complex numbers $z$ for which $|z - 4 - 3i| \leq 2$ and $\text{Re } z \leq 3$.

Complex numbers

Find the exact value of $\int_{0}^{6} \frac{x(x + 1)}{x^2 + 4} \, dx$.

Integration

The required equation is $\cot x = 2 - \cos x$.

Numerical solution of equations

By writing $3\theta$ as $2\theta + \theta$, establish the identity $\cos 3\theta = 4\cos^3\theta - 3\cos \theta$.

Trigonometry

You are told that $\dfrac{2 + 3ai}{a + 2i} = \lambda(2 - i)$, with $a$ and $\lambda$ both real constants.

Algebra

The curve $y = \sin x \cos 2x$ is drawn for $0 \leq x \leq \pi$, and the maximum point $M$ has $x = a$. The shaded area enclosed by the curve and the $x$-axis is labelled $R$.

Integration

Find the set of $x$ values that satisfy $|2^{x+1} - 2| < 0.5$, and give your result correct to $3$ significant figures.

Logarithmic and exponential functions

The curve shown in the diagram is $y = x \cos 2x$, with $x \geq 0$.

Integration

The line $l$ is described by $\mathbf{r} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 2\mathbf{k})$. The points $A$ and $B$ are given by the position vectors $-2\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $3\mathbf{i} - \mathbf{j} + \mathbf{k}$ respectively.

Vectors

On an Argand diagram, shade the set of points for complex numbers $z$ that satisfy $|z - 1 + 2i| \leq |z|$ and $|z - 2| \leq 1$.

Complex numbers

Let $p(x)$ represent the polynomial $2x^3 + ax^2 + bx + 6$, in which $a$ and $b$ are constants. If $p(x)$ is divided by $(x + 2)$, the remainder is $-38$; if it is divided by $(2x - 1)$, the remainder is $\frac{19}{2}$.

Algebra

Solve the quadratic equation $(3 + i)w^2 - 2w + 3 - i = 0$, and express your answers as $x + iy$, where $x$ and $y$ are real numbers.

Complex numbers

Determine the exact coordinates of the stationary points on the curve $y = \dfrac{e^{3x^2 - 1}}{1 - x^2}$.

Differentiation

Show that the equation $\cot^2 \theta + 2\cos 2\theta = 4$ can be transformed into the form $4\sin^4 \theta + 3\sin^2 \theta - 1 = 0$.

Trigonometry

A curve is defined by $x^3 + y^2 + 3x^2 + 3y = 4$.

Differentiation

The variables $x$ and $y$ satisfy the differential equation $e^{4x}\,\frac{dy}{dx} = \cos^2 3y$. It is given that $y = 0$ when $x = 2$.

Differential equations

Define $f(x) = \dfrac{17x^2 - 7x + 16}{(2 + 3x^2)(2 - x)}$.

Algebra

A particle with mass $1.6\text{ kg}$ is launched at a speed of $20\text{ m s}^{-1}$ up the line of greatest slope on a smooth plane inclined at $\alpha$ to the horizontal, where $\tan \alpha = \frac{3}{4}$.

Energy, work and power

A particle with mass $2.4\text{ kg}$ is in equilibrium, supported by two light inextensible strings, one fixed at point $A$ and the other at point $B$. The strings are inclined at $35^\circ$ and $40^\circ$ to the horizontal (see diagram).

Forces and equilibrium

The diagram gives the velocity-time graph for a bus’s motion. The bus begins from rest and accelerates uniformly for $8$ seconds until it reaches $12.6\text{ m s}^{-1}$. It then continues at this speed for $40$ seconds. After that, it slows down uniformly in two stages. From $48$ to $62$ seconds the bus decelerates at $a\text{ m s}^{-2}$, and from $62$ to $70$ seconds it decelerates at $2a\text{ m s}^{-2}$ until it stops.

Kinematics of motion in a straight line

Particles $P$ and $Q$, whose masses are $6\,\text{kg}$ and $2\,\text{kg}$ respectively, are initially at rest and separated by $12.5\,\text{m}$ on a rough horizontal plane. The coefficient of friction between each particle and the plane is $0.4$. Particle $P$ is then projected towards $Q$ with speed $20\,\text{m s}^{-1}$.

Energy, work and power

The diagram shows a particle $A$, with mass $1.2\,\text{kg}$, resting on a plane inclined at $40^\circ$ to the horizontal, and a particle $B$, with mass $1.6\,\text{kg}$, resting on a plane inclined at $50^\circ$ to the horizontal. The particles are joined by a light inextensible string that passes over a small smooth pulley $P$ fixed at the top of the planes. The sections $AP$ and $BP$ of the string are taut and run parallel to the lines of greatest slope of the two planes. Both planes are rough, and the coefficient of friction between each particle and its plane is the same, $\mu$.

Forces and equilibrium

A car with mass $1300\,\text{kg}$ is travelling along a straight road.

Energy, work and power

A particle travels in a straight line, beginning at point $O$, before coming to instantaneous rest at point $X$. For time $t$ s after leaving $O$, the velocity $v\,\text{m s}^{-1}$ of the particle is defined by $v = 7.2t^2 \quad 0 \leq t \leq 2,$ $v = 30.6 - 0.9t \quad 2 \leq t \leq 8,$ $v = \frac{1600}{t^2} + kt \quad 8 \leq t,$ where $k$ is a constant. It is stated that there is no instantaneous change in velocity at $t = 8$.

Kinematics of motion in a straight line

A block of mass $15\,\text{kg}$ moves down the line of greatest slope on an inclined plane. The top of the plane is $1.6\,\text{m}$ vertically above the level of the bottom. The block’s speed at the top is $2\,\text{m s}^{-1}$, and at the bottom it is $4\,\text{m s}^{-1}$.

Energy, work and power

The diagram illustrates a smooth ring $R$, of mass $m\,\text{kg}$, threaded onto a light inextensible string. A horizontal force with magnitude $2\,\text{N}$ acts on $R$. The two ends of the string are fixed at points $A$ and $B$ on a vertical wall. The section $AR$ of the string is at an angle of $30^\circ$ to the vertical, the section $BR$ is at an angle of $40^\circ$ to the vertical, and the string is taut. The ring is in equilibrium.

Forces and equilibrium

A $10\,\text{kg}$ block is initially stationary on a rough plane tilted at $30^\circ$ to the horizontal. A $120\,\text{N}$ force acts on the block at an angle of $20^\circ$ above the line of greatest slope (see diagram). A force opposes the block’s motion, and $200\,\text{J}$ of work is done against this force as the block moves $5\,\text{m}$ up the plane from rest.

Kinematics of motion in a straight line

Particle $P$ has mass $0.2\,\text{kg}$ and is initially at rest on a rough horizontal plane. A horizontal force of $1.2\,\text{N}$ acts on $P$.

Forces and equilibrium

A particle $A$, whose mass is $0.5\,\text{kg}$, is launched vertically upwards from level ground at a speed of $25\,\text{m s}^{-1}$.

Momentum

A railway engine with mass $120000\text{ kg}$ is hauling a coach with mass $60000\text{ kg}$ along a straight track that rises at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.02$. The engine and coach are linked by a light rigid coupling, parallel to the track. The engine supplies a driving force of $125000\text{ N}$, and the constant resistances to motion are $22000\text{ N}$ on the engine and $13000\text{ N}$ on the coach.

Newton's laws of motion

A particle $X$ moves along a straight line. Let its velocity at time $t$ s after departing from a fixed point $O$ be $v\text{ m s}^{-1}$, where $v = -0.1t^3 + 1.8t^2 - 6t + 5.6$. The acceleration of $X$ is zero when $t = p$ and when $t = q$, with $p < q$.

Kinematics of motion in a straight line

A particle is fired vertically upwards from horizontal ground with speed $u\,\text{m s}^{-1}$. Its height above the ground is $s\,\text{m}$ at 3 seconds and again at 4 seconds after projection.

Kinematics of motion in a straight line

A device used to drive a nail into a block of wood makes a hammerhead fall straight down onto the head of a nail. The hammerhead has mass $1.2\,\text{kg}$ and the nail has mass $0.004\,\text{kg}$ (see diagram). The hammerhead strikes the nail with speed $v\,\text{m s}^{-1}$ and stays in contact with the nail after the collision. Immediately after the impact, the hammerhead and nail together move with speed $40\,\text{m s}^{-1}$.

Energy, work and power

A block of mass $8\,\text{kg}$ moves down a rough plane inclined at $30^\circ$ to the horizontal, and it begins from rest. The coefficient of friction between the block and the plane is $\mu$. The block accelerates uniformly down the plane at $2.4\,\text{m s}^{-2}$.

Newton's laws of motion

The mass of the car is $1600\,\text{kg}$.

Energy, work and power

A particle of weight $80\,\text{N}$ is attached at $B$ to a light string $AB$, which is fixed at $A$. At $B$, a horizontal force of magnitude $P\,\text{N}$ acts, making the string inclined at an angle $\theta^\circ$ to the vertical (see diagram).

Forces and equilibrium

A particle travels along a straight line. When $t\text{ s}$, its acceleration, $a\text{ m s}^{-2}$, is given by $a = 36 - 6t$. The particle’s velocity is $27\text{ m s}^{-1}$ at $t = 2$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $2.4\text{ kg}$ and $3.3\text{ kg}$ respectively, are joined by a light inextensible string that runs over a smooth pulley fixed at the top of a rough plane. The plane is inclined at an angle of $\theta^\circ$ to horizontal ground. Particle $A$ is on the plane and the part of the string from $A$ to the pulley is parallel to a line of greatest slope of the plane. Particle $B$ hangs vertically beneath the pulley and is $1\text{ m}$ above the ground (see diagram). The coefficient of friction between the plane and $A$ is $\mu$.

Forces and equilibrium

The cumulative frequency graph shows how long 120 children took to finish a specific puzzle.

Representation of data

Hazeem keeps throwing two standard fair 6-sided dice together. Each time, the result is the total of the two numbers she gets.

Discrete random variables

A farmer trades eggs. Their weights, measured in grams, are modelled by a normal distribution with mean $80.5$ and standard deviation $6.6$. The eggs are sorted into small, medium or large by weight. An egg is small if it weighs below $76$ grams, and $40\%$ of the eggs fall into the medium category.

The normal distribution

The times, rounded to the nearest minute, for 150 athletes in a charity run are recorded. The findings are summarised in the table.

Representation of data

A red spinner has four sides numbered 1, 2, 3, 4. Each time the spinner is spun, the score is the number on the side where it lands. The random variable $X$ represents this score, and the probability distribution table for $X$ is shown.

Probability

A restaurant has rectangular tables. Each table has space for four people: two seats are placed along each of the table’s longer sides (see diagram). Eight friends have reserved two tables, $X$ and $Y$. Rajid, Sue and Tan are three of these friends.

Permutations and combinations

In a throwing competition, a competitor gets three attempts to throw a ball as far as possible. The random variable $X$ represents how many throws go beyond $30$ metres. The probability distribution table for $X$ is given below.

Discrete random variables

George uses a fair $5$-sided spinner whose faces are labelled $1, 2, 3, 4, 5$. He spins it and records the number shown on the face where it comes to rest.

Discrete random variables

A factory makes one particular kind of electrical component. It is known that $15\%$ of the components made are faulty. A random sample of $200$ components is selected.

The normal distribution

The table below gives the heights, in cm, of the 11 players in each team, the Aces and the Jets.

Representation of data

The heights of club members follow a normal distribution with mean $166\text{ cm}$ and standard deviation $10\text{ cm}$. Find the probability that a member chosen at random from the club has a height below $170\text{ cm}$.

The normal distribution

Freddie owns two bags of marbles. Bag $X$ has 7 red marbles and 3 blue marbles, while bag $Y$ has 4 red marbles and 1 blue marble. Freddie picks one bag at random. A marble is then taken at random from that bag and is not put back. One new red marble is added to each bag. After that, a second marble is taken at random from the same bag as the first marble.

Probability