Oct/Nov 2020

Determine the set of values of $m$ for which the line $y = mx - 3$ and the curve $y = 2x^2 + 5$ do not meet.

Quadratics

The diagram displays a sector $CAB$ from a circle centred at $C$. Inside the sector is a circle with centre $O$ and radius $r$; it touches the sector at $D$, $E$ and $F$, with $COD$ forming a straight line and angle $ACD$ equal to $\theta$ radians.

Circular measure

The functions $f$ and $g$ are defined so that $f(x) = x^2 + 3$ for $x > 0$, while $g(x) = 2x + 1$ for $x > -\frac{1}{2}$.

Functions

The diagram displays a curve given by $y = 4x^2 - 2x$ for $x \geq 0$, together with a straight line given by $y = 3 - x$. The curve meets the $x$-axis at $A\,(4, 0)$ and intersects the line at $B$ and $C$.

Differentiation

The derivative of a curve is given by $\frac{dy}{dx} = \frac{1}{(x - 3)^2} + x$. The curve is known to pass through the point $(2, 7)$.

Integration

Air is being pumped into a spherical balloon, and its volume is rising at a constant rate of $50\,\text{cm}^3\,\text{s}^{-1}$.

Differentiation

In the diagram, the lower curve is given by $y = \cos \theta$. The upper curve is produced by a combination of transformations applied to $y = \cos \theta$.

Trigonometry

When $(2x^2 + \frac{a}{x})^6$ is expanded, the coefficients attached to $x^6$ and $x^3$ match.

Series

A curve has equation $y = 2 + \sqrt{25 - x^2}$.

Differentiation

Show that $\dfrac{\sin \theta}{1 - \sin \theta} - \dfrac{\sin \theta}{1 + \sin \theta} = 2 \tan^2 \theta$.

Trigonometry

A geometric progression starts with first term $a$, has common ratio $r$, and its sum to infinity is $S$. A different geometric progression also begins with first term $a$, but its common ratio is $R$ and its sum to infinity is $2S$.

Series

The diagram depicts a circle with centre $A$ that passes through $B$. A second circle is centred at $B$ and passes through $A$. The tangent at $B$ to the first circle cuts the second circle at $C$ and $D$. The coordinates of $A$ are $(-1, 4)$ and the coordinates of $B$ are $(3, 2)$.

Coordinate geometry

The coefficient attached to $x^3$ in the expansion of $(1 + kx)(1 - 2x)^5$ is $20$.

Series

The diagram presents a section of the curve $y = \frac{2}{(3 - 2x)^2} - x$ together with its minimum point $M$, which is situated on the $x$-axis.

Integration

The curve is given by $y = 3\cos 2x + 2$ for $0 \leq x \leq \pi$.

Trigonometry

The first three terms of a geometric progression are $2p + 6$, $-2p$ and $p + 2$, in that order, where $p$ is positive.

Series

The curve is given by $y = 2x^2 + m(2x + 1)$, where $m$ is constant, and the line is given by $y = 6x + 4$.

Coordinate geometry

The sum, $S_n$, of the first $n$ terms in an arithmetic progression is $S_n = n^2 + 4n$. The $k$th term of the progression exceeds $200$.

Series

The functions $f$ and $g$ are given by $f(x) = 4x - 2$, where $x \in \mathbb{R}$, and $g(x) = \frac{4}{x + 1}$, where $x \in \mathbb{R}$ and $x \ne -1$.

Functions

Prove that $\left(\frac{1}{\cos x} - \tan x\right)\left(\frac{1}{\sin x} + 1\right) = \frac{1}{\tan x}$.

Trigonometry

The point $(4,7)$ is on the curve $y = f(x)$, and it is known that $f'(x) = 6x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}$.

Differentiation

From the diagram, $ABC$ is an isosceles triangle with $AB = BC = r$ cm, and angle $BAC = \theta$ radians. Point $D$ is located on $AC$, and $ABD$ is a sector of a circle with centre $A$.

Circular measure

The centre of a circle is at $B\,(5, 1)$, and $A\,(-1, -2)$ lies on the circle.

Coordinate geometry

Write $x^2 + 6x + 5$ in the form $(x + a)^2 + b$, where $a$ and $b$ are constants.

Quadratics

The curve is given by $y = \frac{1}{k}x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}$ for $x > 0$, with $k$ a positive constant.

Integration

A circle centred at $C$ has equation $(x - 8)^2 + (y - 4)^2 = 100$.

Coordinate geometry

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

Integration

Solve the equation $3\tan^2\theta + 1 = \frac{2}{\tan^2\theta}$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

A curve is given by $y = 3x^2 - 4x + 4$, while a straight line is given by $y = mx + m - 1$, where $m$ is a constant.

Quadratics

In the expansion of $(a + bx)^7$, with $a$ and $b$ being non-zero constants, the coefficients of $x$, $x^2$ and $x^4$ are, respectively, the first, second and third terms of a geometric progression.

Series

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

Functions

The first two terms in an arithmetic progression are $\frac{1}{\cos^2 \theta}$ and $-\frac{\tan^2 \theta}{\cos^2 \theta}$, in that order, where $0 < \theta < \frac{1}{2}\pi$.

Series

A curve is described by $y = 2x + 1 + \frac{1}{2x + 1}$, with $x > -\frac{1}{2}$.

Differentiation

The diagram shows arc $AB$ as part of a circle centred at $O$ with radius $8\text{ cm}$. Arc $BC$ belongs to a circle centred at A with radius $12\text{ cm}$, and $AOC$ lies on a straight line.

Circular measure

Assume that $\ln(2x + 1) - \ln(x - 3) = 2$.

Logarithmic and exponential functions

The polynomial $p(x)$ is defined as $p(x) = x^3 + ax^2 + bx + 16$, where $a$ and $b$ are constants. It is stated that $(x + 2)$ is a factor of $p(x)$ and that the remainder is $72$ when $p(x)$ is divided by $(x - 2)$.

Algebra

The diagram depicts the curve $y = 2 + e^{-2x}$. It cuts the $y$-axis at $A$, and $B$ lies on the curve with $x$-coordinate $1$. The shaded part is enclosed by the curve and the line segment $AB$.

Integration

Solve for $x$ in $|2x - 5| = |x + 6|$.

Logarithmic and exponential functions

The values produced by the iterative formula $x_{n+1} = \frac{6 + 8x_n}{8 + x_n^2}$ with starting value $x_1 = 2$ tend to $\alpha$.

Numerical solution of equations

You are told that $3\sin 2\theta = \cos \theta$, with $\theta$ an angle satisfying $0^\circ < \theta < 90^\circ$.

Trigonometry

The parametric equations $x = 3t - 2\sin t$, $y = 5t + 4\cos t$ define a curve, with $0 \leq t \leq 2\pi$. At the two points $P$ and $Q$ on this curve, the gradient is $\frac{5}{2}$.

Trigonometry

The curve is defined by the equation $y = f(x)$, where $f(x) = \frac{4x^3 + 8x - 4}{2x - 1}$.

Integration

Find the value of $\theta$ that satisfies $7\cot\theta = 3\cosec\theta$ for $0^\circ < \theta < 90^\circ$.

Trigonometry

Starting from $\frac{2^{3x+2} + 8}{2^{3x} - 7} = 5$, determine $2^{3x}$ and then, by taking logarithms, determine $x$ correct to $4$ significant figures.

Logarithmic and exponential functions

Sketch the graphs of $y=\left|\frac{1}{2}x-a\right|$ and $y=\frac{3}{2}x-\frac{1}{2}a$ on one shared diagram, where $a$ is a positive constant.

Algebra

The diagram presents the curve with equation $y=\frac{x-2}{x^2+8}$. The shaded area is enclosed by the curve together with the lines $x=14$ and $y=0$.

Integration

The equation for a curve is $2e^{2x}y - y^3 + 4 = 0.$

Differentiation

Evaluate $\displaystyle \int \left( \frac{8}{4x + 1} + \frac{8}{\cos^2(4x + 1)} \right)\,dx.$

Integration

The graph is defined by the equation $y = f(x)$, where $f(x) = x^4 - 5x^3 + 6x^2 + 5x - 15$. The diagram shows that the curve intersects the $x$-axis at $A$ and $B$, whose coordinates are $(a, 0)$ and $(b, 0)$ respectively.

Numerical solution of equations

The equation $\ln(2x + 1) - \ln(x - 3) = 2$ is given.

Logarithmic and exponential functions

The polynomial $p(x)$ is given by $p(x) = x^3 + ax^2 + bx + 16$, with $a$ and $b$ constant. You are told that $(x + 2)$ is a factor of $p(x)$ and that dividing $p(x)$ by $(x - 2)$ leaves a remainder of $72$.

Algebra

The diagram displays the curve $y = 2 + e^{-2x}$. This curve meets the $y$-axis at $A$, and the point $B$ on the curve has $x$-coordinate $1$. The shaded region is enclosed by the curve and the line segment $AB$.

Integration

Determine the values of $x$ for which $|2x - 5| = |x + 6|$.

Algebra

The values produced by the iterative formula $x_{n+1} = \frac{6 + 8x_n}{8 + x_n^2}$, starting from the initial value $x_1 = 2$, tend towards $\alpha$.

Numerical solution of equations

It is stated that $3\sin 2\theta = \cos \theta$ where $\theta$ is an angle such that $0^{\circ} < \theta < 90^{\circ}$.

Trigonometry

A curve is given by the parametric equations $x = 3t - 2\sin t$, $y = 5t + 4\cos t$, with $0 \leq t \leq 2\pi$. At the two points $P$ and $Q$ on the curve, the gradient equals $\frac{5}{2}$.

Trigonometry

The curve is defined by $y = f(x)$, with $f(x) = \dfrac{4x^3 + 8x - 4}{2x - 1}$.

Differentiation

Solve this inequality: $2 - 5x > 2|x - 3|$.

Algebra

The diagram depicts the curve $y = (2 - x)e^{-\frac{1}{2}x}$ together with its minimum point $M$.

Integration

The two lines are given by $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$, where $a$ is a constant.

Vectors

On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy $|z| \ge 2$ and $|z - 1 + i| \le 1$.

Complex numbers

A curve is given by the parametric equations $x = 3 - \cos 2\theta$, $y = 2\theta + \sin 2\theta$, where $0 < \theta < \frac{1}{2}\pi$.

Differentiation

Solve $\log_{10}(2x + 1) = 2\log_{10}(x + 1) - 1$. Give the answers correct to 3 decimal places.

Logarithmic and exponential functions

Sketch a suitable pair of graphs to show that the equation $\cosec x = 1 + e^{-\frac{1}{2}x}$ has exactly two roots for $0 < x < \pi$.

Numerical solution of equations

Rewrite $\sqrt{6}\cos\theta + 3\sin\theta$ in the form $R\cos(\theta - \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and state $\alpha$ correct to 2 decimal places.

Trigonometry

Verify that $-1 + \sqrt{5}i$ satisfies the equation $2x^3 + x^2 + 6x - 18 = 0$.

Complex numbers

The coordinates $(x, y)$ of a typical point on the curve satisfy the differential equation $x \frac{dy}{dx} = (1 - 2x^2) y$, for $x > 0$. It is given that $y = 1$ when $x = 1$.

Differential equations

Define $f(x)$ by $f(x) = \dfrac{8 + 5x + 12x^2}{(1 - x)(2 + 3x)^2}$.

Algebra

Solve the equation $\ln(1 + e^{-3x}) = 2$. State your answer correct to $3$ decimal places.

Logarithmic and exponential functions

The diagram illustrates the curve $y = \sqrt{x}\cos x$, for $0 \le x \le \tfrac{3}{2}\pi$, together with its lowest point $M$, where $x = a$. The shaded area between the curve and the $x$-axis is labelled $R$.

Numerical solution of equations

Expand $\sqrt[3]{1 + 6x}$ as a series in ascending powers of $x$, including terms up to and including $x^3$, and simplify the coefficients.

Algebra

The variables $x$ and $y$ satisfy the relation $2^y = 3^{1-2x}$.

Logarithmic and exponential functions

Show that the equation $\tan(\theta + 60^\circ) = 2\cot\theta$ may be rearranged into the form $\tan^2\theta + 3\sqrt{3}\tan\theta - 2 = 0$.

Trigonometry

The sketch shows the curve defined by the parametric equations $x = \tan\theta$, $y = \cos^2\theta$, for $-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi$.

Differentiation

The complex number $u$ is specified as $u = \dfrac{7 + i}{1 - i}$.

Complex numbers

The variables $x$ and $t$ are linked by the differential equation $e^{3t}\,\frac{dx}{dt} = \cos^2 2x$, for $t \geq 0$. It is given that $x = 0$ when $t = 0$.

Differential equations

Taking origin $O$ as the reference point, the position vectors of the points $A$, $B$, $C$ and $D$ are $\vec{OA} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}$, $\vec{OC} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ and $\vec{OD} = \begin{pmatrix} 3 \\ 2 \\ 3 \end{pmatrix}$.

Vectors

Define $f(x)$ as $\dfrac{7x + 18}{(3x + 2)(x^2 + 4)}$.

Integration

Solve this inequality: $2 - 5x > 2|x - 3|$.

Algebra

The diagram displays the curve $y = (2 - x)e^{-\frac{1}{2}x}$ together with its minimum point $M$.

Integration

The two lines are given by the equations $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - \kappa + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$, where $a$ is a constant.

Vectors

For a sketch of an Argand diagram,

Complex numbers

For the curve, the parametric equations are $x = 3 - \cos 2\theta$ and $y = 2\theta + \sin 2\theta$, with $0 < \theta < \frac{1}{2}\pi$.

Differentiation

Find the solutions of $\log_{10}(2x + 1) = 2\log_{10}(x + 1) - 1$. Give your answers correct to 3 decimal places.

Logarithmic and exponential functions

By sketching an appropriate pair of graphs, show that the equation $\cosec x = 1 + e^{-\frac{1}{2}x}$ has exactly two roots in the interval $0 < x < \pi$.

Numerical solution of equations

Express $\sqrt{6}\cos\theta + 3\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and quote $\alpha$ correct to 2 decimal places.

Trigonometry

Verify that $-1 + \sqrt{5}i$ satisfies the equation $2x^3 + x^2 + 6x - 18 = 0$.

Complex numbers

For a general point on the curve, the coordinates $(x, y)$ satisfy the differential equation $x\frac{dy}{dx} = (1 - 2x^2)y$, for $x > 0$. It is also given that $y = 1$ when $x = 1$.

Differential equations

Define $f(x) = \dfrac{8 + 5x + 12x^2}{(1 - x)(2 + 3x)^2}$.

Algebra

A particle $B$ with mass $5\,\text{kg}$ is initially stationary on a smooth horizontal table. A particle $A$ with mass $2.5\,\text{kg}$ travels along the table at $6\,\text{m s}^{-1}$ and strikes $B$ head-on. During the collision, the two particles join together.

Momentum

A car with mass $1400\,\text{kg}$ is travelling on a straight horizontal road and experiences a resistance of magnitude $350\,\text{N}$.

Energy, work and power

The coplanar forces, with magnitudes $8\,\text{N}$, $12\,\text{N}$, $10\,\text{N}$ and $P\,\text{N}$, act on a point in the directions shown in the diagram, and the system is in equilibrium.

Forces and equilibrium

Particle $P$ travels along a straight line. It leaves point $O$ from rest, and after $t$ s from leaving $O$ its acceleration is $a\text{ m s}^{-2}$, with $a = 6t - 18$.

Kinematics of motion in a straight line

A light inextensible string passes over a fixed smooth pulley and joins particles of masses $0.8\text{ kg}$ and $0.2\text{ kg}$. The system starts from rest, with each particle $0.5\text{ m}$ above a horizontal floor (see diagram). During the motion that follows, the $0.2\text{ kg}$ particle does not reach the pulley.

Newton's laws of motion

A car with mass $1500\,\text{kg}$ is towing a trailer of mass $750\,\text{kg}$ up a straight hill that is $800\,\text{m}$ long and inclined at an angle of $\sin^{-1} 0.08$ to the horizontal. The resistive forces acting on the car and trailer are $400\,\text{N}$ and $200\,\text{N}$ respectively. A light rigid tow-bar connects the car to the trailer. At the foot of the hill, the car and trailer move at $30\,\text{m s}^{-1}$, and at the top they move at $20\,\text{m s}^{-1}$.

Energy, work and power

Points $A$, $B$ and $C$ are on the line of greatest slope of a plane inclined at $30^\circ$ to the horizontal, with $AB = 1\,\text{m}$ and $BC = 1\,\text{m}$, as shown in the diagram. A particle of mass $0.2\,\text{kg}$ is let go from rest at $A$ and slides down the plane. The section from $A$ to $B$ is smooth, while the section from $B$ to $C$ is rough, with coefficient of friction $\mu$ between the plane and the particle.

Energy, work and power

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are initially at rest on a smooth horizontal plane. $P$ is launched towards $Q$ at speed $2\,\text{m s}^{-1}$.

Momentum

A car with mass $1800\,\text{kg}$ is moving along a straight horizontal road. The car’s engine delivers constant power. A steady resistive force of $650\,\text{N}$ acts against the motion.

Energy, work and power

A block of mass $m\,\text{kg}$ is in equilibrium beneath a horizontal ceiling, supported by two strings as the diagram shows. One string makes an angle of $45^\circ$ to the horizontal and has tension $T\,\text{N}$. The other string makes an angle of $60^\circ$ to the horizontal and has tension $20\,\text{N}$.

Forces and equilibrium

The diagram illustrates a velocity-time graph used to model a car’s motion. It is made up of four straight-line sections. Starting from rest, the car accelerates at a steady rate of $2\,\text{m s}^{-2}$ to reach a speed of $20\,\text{m s}^{-1}$ in $T\,\text{s}$. It then decelerates at a constant rate for $5\,\text{s}$ before moving at the constant speed $V\,\text{m s}^{-1}$ for $27.5\,\text{s}$. After this, the car decelerates to rest at a steady rate over $5\,\text{s}$.

Kinematics of motion in a straight line

A particle is launched vertically upwards at a speed of $40\,\text{m s}^{-1}$ next to a building of height $h$ m.

Kinematics of motion in a straight line

A $5\,\text{kg}$ block is on a plane that is inclined at $30^\circ$ to the horizontal, and the coefficient of friction between the block and the plane is $\mu$.

Forces and equilibrium

Particle $P$ travels along a straight line, beginning at point $O$ with velocity $1.72\text{ m s}^{-1}$. Its acceleration $a\text{ m s}^{-2}$, $t\text{ s}$ after leaving $O$, is defined by $a = 0.1t^{\frac{3}{2}}$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.3\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string. This string passes over a fixed smooth pulley, attached between a horizontal plane and the top of an inclined plane. At the start, the particles are at rest, with $A$ on the horizontal plane and $B$ on the inclined plane, which is inclined at $30^\circ$ to the horizontal. The string is taut, and $B$ moves along the line of greatest slope of the inclined plane. A force of magnitude $3.5\,\text{N}$ acts on $B$ down the plane (see diagram).

Energy, work and power

From a point on the ground, particle $P$ is projected vertically upwards at speed $v\,\mathrm{m\,s^{-1}}$. It reaches its greatest height in $3\,\mathrm{s}$.

Kinematics of motion in a straight line

A box with mass $5\,\mathrm{kg}$ is dragged a distance of $15\,\mathrm{m}$ at steady speed along a rough plane that is tilted at an angle of $20^\circ$ to the horizontal. It travels along the line of greatest slope, and a frictional force of $40\,\mathrm{N}$ acts against the motion. The pulling force is parallel to the line of greatest slope.

Energy, work and power

A string is tied to a block of mass $4\,\mathrm{kg}$ that is in limiting equilibrium on a rough horizontal table. The string is inclined at $24^\circ$ above the horizontal, and its tension is $30\,\mathrm{N}$.

Forces and equilibrium

Two small smooth spheres $A$ and $B$, with equal radii and masses $4\,\text{kg}$ and $m\,\text{kg}$ respectively, are on a smooth horizontal plane. At the start, sphere $B$ is stationary and $A$ is moving towards $B$ at $6\,\text{m s}^{-1}$. Following the collision $A$ has speed $1.5\,\text{m s}^{-1}$ and $B$ has speed $3\,\text{m s}^{-1}$.

Momentum

A particle $P$ travels along a straight line. It begins at a point $O$ on the line and, after $t$ s from leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 4t^2 - 20t + 21$.

Kinematics of motion in a straight line

A car with mass $1600\ \text{kg}$ is towing a caravan with mass $800\ \text{kg}$. A light rigid tow-bar joins the car to the caravan. The resistances to the motion of the car and caravan are $400\ \text{N}$ and $250\ \text{N}$ respectively.

Energy, work and power

In the diagram, particles $A$ and $B$, having masses $2\ \text{kg}$ and $3\ \text{kg}$ respectively, are joined by the ends of a light inextensible string. This string passes over a small fixed smooth pulley attached to the top of two inclined planes. Particle $A$ is placed on plane $P$, which makes an angle of $10^\circ$ with the horizontal. Particle $B$ is placed on plane $Q$, which makes an angle of $20^\circ$ with the horizontal. The string is taut, and each section of it is parallel to the line of greatest slope of the plane on which it lies.

Forces and equilibrium

Two fair dice, one red and one blue, are rolled. Event $A$ is "the number on the red die is a multiple of $3$". Event $B$ is "the total of the two numbers is at least $9$."

Probability

The chance that a student at a large music college takes part in the band is $0.6$. If a student is in the band, the chance that she also performs in the choir is $0.3$. If a student is not in the band, the chance that she sings in the choir is $x$. The probability that a student selected at random from the college does not sing in the choir is $0.58$.

Probability

Kayla is taking part in a throwing event. A throw counts as a success when the distance thrown is more than $30$ metres. The probability that Kayla gets a success on any throw is $0.25$.

Discrete random variables

The random variable $X$ can take the values $1$, $2$, $3$ and $4$, with each value occurring with probability $\frac{1}{4}$. Two independent values of $X$ are selected at random. If the two selected values of $X$ are identical, then the random variable $Y$ is that value. If they are not identical, then $Y$ is the larger value of $X$ minus the smaller value of $X$.

Discrete random variables

Davin’s daily time spent on his games machine is normally distributed, with mean $3.5$ and standard deviation $0.9$.

The normal distribution

The times, $t$ minutes, needed by $150$ students to finish a particular challenge are set out in the cumulative frequency table below.

Representation of data

Determine how many different arrangements can be made from the $10$ letters in SHOPKEEPER if all $3$ Es are kept together.

Permutations and combinations

A fair die with six faces labelled $1, 2, 3, 4, 5, 6$ is rolled again and again until a $4$ appears.

Discrete random variables

A bag has $5$ red balls and $3$ blue balls inside it. Sadie selects $3$ balls at random from the bag, without replacement. The random variable $X$ denotes the number of red balls that she selects.

Discrete random variables