May/June 2018

Find the first three terms, written in ascending powers of $x$, in the expansion of $(1 - 2x)^5$.

Series

The curve given by $y = x^3 - 2x^2 + 5x$ goes through the origin.

Differentiation

A point travels along the curve $y = 2x + \frac{5}{x}$ so that the $x$-coordinate rises at a steady rate of $0.02$ units per second.

Differentiation

A curve is defined by $\frac{dy}{dx} = \frac{12}{(2x + 1)^2}$. The point $(1, 1)$ is on the curve.

Coordinate geometry

Show that $(\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta) \equiv \sin^3 \theta + \cos^3 \theta$.

Trigonometry

The diagram depicts a kite $OABC$, with $AC$ as the axis of symmetry. The coordinates of $A$ and $C$ are $(0,4)$ and $(8,0)$, respectively, and $O$ is the origin.

Coordinate geometry

The diagram depicts a circle with centre $O$ and radius $r$ cm. Points $A$ and $B$ are on the circle, and $AT$ is tangent to the circle. Angle $AOB = \theta$ radians, and $OBT$ is a straight line.

Circular measure

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

Coordinate geometry

Find the possible values of the first term of a geometric progression with second term $12$ and sum to infinity $54$.

Series

For $x \in \mathbb{R}$, the functions $f$ and $g$ are given by $f : x \mapsto \frac{1}{2}x - 2$, $g : x \mapsto 4 + x - \frac{1}{2}x^2$.

Functions

In the expansion of $(2 + \frac{x}{2})^6 + (a + x)^5$, the coefficient of $x^2$ comes to $330$.

Series

Solve the equation $2\cos x + 3\sin x = 0$, for $0^\circ \leq x \leq 360^\circ$.

Trigonometry

The diagram presents a segment of the curve $y = \frac{x}{2} + \frac{6}{x}$. The line $y = 4$ cuts the curve at the points $P$ and $Q$.

Integration

The curve is defined by $y = x^2 - 6x + k$, with $k$ a constant.

Quadratics

A firm that made salt from sea water switched to a different process. Each week, the quantity of salt produced rose by $2\%$ of the amount produced in the previous week. It is stated that during the first week after the change the firm produced $8000\text{ kg}$ of salt.

Series

The function $f$ is defined by $f(x) = a + b\cos x$ for $0 \leq x \leq 2\pi$. It is given that $f\left(\tfrac{1}{3}\pi\right) = 5$ and $f(\pi) = 11$.

Functions

The diagram represents a three-dimensional shape. The base $OAB$ is a horizontal triangle with angle $AOB$ equal to $90^\circ$. The side $OBCD$ is a rectangle, and the side $OAD$ lies in a vertical plane. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OB$ respectively, and the unit vector $\mathbf{k}$ is vertical. The position vectors of $A$, $B$ and $D$ are given by $\overrightarrow{OA} = 8\mathbf{i}$, $\overrightarrow{OB} = 5\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}$.

Coordinate geometry

In the diagram, $A$ and $B$ lie on a circle of centre $O$ and radius $r$. The tangents at $A$ and $B$ intersect at $T$. The shaded region is enclosed by the minor arc $AB$ together with the lines $AT$ and $BT$. Angle $AOB$ measures $2\theta$ radians.

Circular measure

The function $f$ is defined as $f : x \mapsto 7 - 2x^2 - 12x$ for $x \in \mathbb{R}$.

Quadratics

The points A and B are located at (h, h) and (4h + 6, 5h) respectively. The perpendicular bisector of AB is given by 3x + 2y = k.

Coordinate geometry

The curve satisfies $\frac{dy}{dx} = \sqrt{4x + 1}$, and the point $(2, 5)$ lies on it.

Differentiation

Rewrite $3x^2 - 12x + 7$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

Quadratics

The function $f$ is one-one and is given by $f(x) = (x - 2)^2 + 2$ for $x \geq c$, where $c$ is a constant.

Functions

The diagram displays a section of the curve $y = (x + 1)^2 + (x + 1)^{-1}$ together with the line $x = 1$. Point $A$ is the curve's minimum point.

Integration

Determine the coefficient of $\frac{1}{x}$ in the expansion of $\left(x - \frac{2}{x}\right)^5$.

Series

The common ratio in the geometric progression is $0.99$. Write the sum of the first $100$ terms as a percentage of the sum to infinity, and give the result correct to $2$ significant figures.

Series

The graph of the equation $y = f(x)$ goes through the point $A\,(3,1)$ and intersects the $y$-axis at $B$.

Integration

The diagram depicts triangle $OAB$ with angle $OAB = 90^\circ$ and $OA = 5\ \text{cm}$. Arc $AC$ is part of a circle centred at $O$. Its length is $6\ \text{cm}$, and it cuts $OB$ at $C$.

Integration

Points $A$ and $B$ have coordinates $(-3k - 1,\ k + 3)$ and $(k + 3,\ 3k + 5)$ respectively, where $k$ is a constant $(k \neq -1)$.

Coordinate geometry

Express $\dfrac{\tan^2 \theta - 1}{\tan^2 \theta + 1}$ as $a\sin^2 \theta + b$, with $a$ and $b$ as the constants to determine.

Trigonometry

At the point $A$, the tangent to the curve $y = x^3 - 9x^2 + 24x - 12$ is parallel to the line $y = 2 - 3x$. Find the equation of the tangent at $A$.

Differentiation

The diagram depicts a pyramid $OABCD$ with a horizontal rectangular base $OABC$. The edges $OA$ and $AB$ are $8$ units and $6$ units long respectively. Point $E$ lies on $OB$ in such a way that $OE = 2$ units. The vertex $D$ of the pyramid is $7$ units vertically above $E$. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OE$ respectively.

Coordinate geometry

Solve the equation $3e^{2x} - 82e^{x} + 27 = 0$, and give your answers in the form $k\ln 3$.

Logarithmic and exponential functions

The variables $x$ and $y$ are linked by $y = A \times B^{\ln x}$, with $A$ and $B$ as constants. The plot of $\ln y$ against $\ln x$ is a straight line that goes through the points $(2.2,\,4.908)$ and $(5.9,\,11.008)$, as illustrated in the diagram.

Algebra

Without using a calculator, determine the exact value of $\int_{0}^{2} 4\,e^{-x}(e^{3x} + 1)\,dx$.

Integration

The diagram presents the curve whose equation is $y = \frac{5\ln x}{2x + 1}$. It meets the $x$-axis at $P$ and has a maximum at $M$.

Numerical solution of equations

The curve is expressed parametrically by $x = 2\cos 2\theta + 3\sin \theta$, $y = 3\cos \theta$, with $0 < \theta < \frac{1}{2}\pi$.

Differentiation

The cubic polynomial $f(x)$ is written as $f(x) = x^3 + ax^2 + 14x + a + 1$, with $a$ as a constant. It is stated that $(x + 2)$ is a factor of $f(x)$.

Algebra

Express $5\cos \theta - 2\sin \theta$ as $R\cos(\theta + \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the value of $\alpha$ correct to 4 decimal places.

Trigonometry

Solve $|3x - 2| < |x + 5|$.

Algebra

The curve is defined by $y = 3\ln(2x + 9) - 2\ln x$.

Differentiation

Determine the quotient when $x^4 - 2x^3 + 8x^2 - 12x + 13$ is divided by $x^2 + 6$ and confirm that the remainder is $1$.

Algebra

Find the solution of the equation $2\ln(2x) - \ln(x + 3) = 4\ln 2$.

Logarithmic and exponential functions

The curve is defined by $y^3\sin 2x + 4y = 8$.

Differentiation

You are given that $\int_{0}^{a} (1 + e^{\frac{1}{2}x})^{2} \, dx = 10$, where $a$ is a positive constant.

Numerical solution of equations

Show that $2\cosec^2 2x(1-\cos 2x)\equiv \sec^2 x$.

Trigonometry

Solve for $x$ in the inequality $|3x - 2| < |x + 5|$.

Algebra

The equation of the curve is $y = 3\ln(2x + 9) - 2\ln x$.

Differentiation

Determine the quotient when $x^4 - 2x^3 + 8x^2 - 12x + 13$ is divided by $x^2 + 6$, and show that the remainder equals $1$.

Algebra

Find the value of $x$ in the equation $2\ln(2x) - \ln(x + 3) = 4\ln 2$.

Logarithmic and exponential functions

The curve is defined by $y^3\sin 2x + 4y = 8$.

Differentiation

The information provided is that $\int_0^a (1 + e^{\frac{1}{2}x})^2\,dx = 10$, with $a$ a positive constant.

Numerical solution of equations

Show that $2\cosec^2 2x(1 - \cos 2x)$ is equal to $\sec^2 x$.

Trigonometry

With all working shown, solve the equation $\ln(x^4 - 4) = 4\ln x - \ln 4$, and give your answer correct to $2$ decimal places.

Logarithmic and exponential functions

Point $P$ is given by the position vector $3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$. Line $l$ is defined by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + u(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

Vectors

If $\sin(x - 60^\circ) = 3\cos(x - 45^\circ)$, determine the exact value of $\tan x$.

Trigonometry

The curve is given by $y = \dfrac{e^{3x}}{\tan\left(\tfrac{1}{2}x\right)}$.

Differentiation

The polynomial $x^4 + 2x^3 + ax + b$, with $a$ and $b$ treated as constants, is exactly divisible by $x^2 - x + 1$.

Algebra

Take $I = \displaystyle\int_{\frac{1}{4}}^{\frac{3}{4}} \sqrt{\dfrac{x}{1 - x}}\, dx$.

Integration

In a particular chemical reaction, the quantity $x$ grams of a substance is falling. The differential equation connecting $x$ and $t$, where $t$ is the time in seconds since the reaction began, is $\frac{dx}{dt} = -k x \sqrt{t}$, where $k$ is a positive constant. It is stated that $x = 100$ at the beginning of the reaction.

Differential equations

Show all working and, without using a calculator, solve the equation $z^2 + (2\sqrt{6})z + 8 = 0$, writing your answers in the form $x + iy$, where $x$ and $y$ are exact real values.

Complex numbers

Let $a$ be a positive constant for which $\int_0^a x e^{-\frac{1}{2}x} \, dx = 2$.

Numerical solution of equations

Define $f(x)$ by $f(x) = \dfrac{12x^2 + 4x - 1}{(x - 1)(3x + 2)}$.

Algebra

Show all necessary working and solve the equation $3|2^x - 1| = 2^x$, giving answers accurate to $3$ significant figures.

Logarithmic and exponential functions

The lines $l$ and $m$ are given by $\mathbf{r} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} + s(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + t(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$, respectively.

Vectors

With all required working shown, solve the equation $\cot \theta + \cot(\theta + 45^\circ) = 2$, for $0^\circ < \theta < 180^\circ$.

Trigonometry

The diagram shows the tangent to the curve at $P$, where $P$ has coordinates $(x, y)$, and this tangent cuts the $x$-axis at $T$. $N$ is the point where the perpendicular from $P$ meets the $x$-axis. For every $x$, the curve has a positive gradient, and $TN = 2$.

Differential equations

Prove that $\frac{2 \sin x - \sin 2x}{1 - \cos 2x} = \frac{\sin x}{1 + \cos x}$.

Integration

The curve is defined by the equation $x^2(x + 3y) - y^3 = 3$.

Differentiation

The diagram depicts triangle $ABC$, where $AB = AC = a$ and angle $BAC = \theta$ radians. Two semicircles are drawn externally on $AB$ and $AC$ as diameters. An arc of a circle centred at $A$ connects $B$ to $C$. The shaded segment has the same area as the two semicircles combined.

Numerical solution of equations

For this question, a calculator must not be used. The complex numbers $-3\sqrt{3} + i$ and $\sqrt{3} + 2i$ are called $u$ and $v$ respectively.

Complex numbers

The diagram depicts the curve $y = (x + 1)e^{-\frac{1}{3}x}$ together with its maximum point $M$.

Integration

Define $f(x) = \dfrac{x - 4x^2}{(3 - x)(2 + x^2)}$.

Algebra

Expand $\frac{4}{\sqrt{4 - 3x}}$ as a series in ascending powers of $x$, including terms up to and including $x^2$, and simplify the coefficients.

Algebra

The position vectors of points $A$ and $B$ are $2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ and $4\mathbf{i} + \mathbf{j} + \mathbf{k}$, respectively. The line $l$ is given by $\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + u(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$.

Vectors

By displaying all required working, solve $5^{2x} = 5^x + 5$. Give the solution correct to 3 decimal places.

Logarithmic and exponential functions

With full working shown, determine the value of $\int_{0}^{\frac{1}{6}\pi} x\cos 3x \, dx$, expressing your answer in terms of $\pi$.

Integration

The curve given by $y = \frac{\ln x}{3 + x}$ has a stationary point when $x = p$.

Numerical solution of equations

By expanding $(\cos^2 x + \sin^2 x)^3$ first, or by another valid approach, prove that $\cos^6 x + \sin^6 x = 1 - \frac{3}{4}\sin^2 2x$.

Trigonometry

Write $\frac{1}{4 - y^2}$ as a partial fraction expression.

Differential equations

For this question, calculator use is not allowed.

Trigonometry

The curve is given by $2x^3 - y^3 - 3xy^2 = 2a^3$, where $a$ is a non-zero constant.

Differentiation

Determine the complex number $z$ that satisfies the equation $3z - iz^* = 1 + 5i$, where $z^*$ is the complex conjugate of $z$.

Complex numbers

A particle $P$ is launched vertically upwards at speed $24\,\text{m s}^{-1}$ from a position $5\,\text{m}$ above ground level.

Kinematics of motion in a straight line

The diagram depicts three coplanar forces acting at point $O$. Their magnitudes are $6\,\text{N}$, $8\,\text{N}$ and $10\,\text{N}$. The angle formed by the $6\,\text{N}$ force and the $8\,\text{N}$ force is $90^\circ$. The forces are in equilibrium.

Forces and equilibrium

A particle $P$ of mass $8\,\text{kg}$ lies on a smooth plane inclined at $30^\circ$ to the horizontal. A force of magnitude $100\,\text{N}$, acting in the vertical plane that contains the line of greatest slope and making an angle of $\theta^\circ$ with that line, acts on $P$ (see diagram).

Forces and equilibrium

A particle $P$ travels along a straight line from $O$. After $t$ s from leaving $O$, its displacement from $O$ is $s$ m, where $s = t^3 - 4t^2 + 4t$, and its velocity is $v\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A sprinter takes part in a race of $200\,\text{m}$. The total time taken to complete the race is $20\,\text{s}$. He begins from rest and speeds up uniformly for $6\,\text{s}$, reaching a speed of $12\,\text{m s}^{-1}$. He then keeps this speed for the following $10\,\text{s}$, before slowing down uniformly and crossing the finishing line with speed $V\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

The mass of the car is $1250\,\text{kg}$.

Energy, work and power

The diagram depicts a triangular block whose sloping faces make angles of $45^{\circ}$ and $30^{\circ}$ with the horizontal. Particle $A$, with mass $0.8\text{ kg}$, is on the face inclined at $45^{\circ}$, while particle $B$, with mass $1.2\text{ kg}$, is on the face inclined at $30^{\circ}$. The particles are joined by a light inextensible string that passes over a small smooth pulley $P$ fixed at the top of the faces. The segments $AP$ and $BP$ of the string run parallel to the lines of greatest slope on the respective faces. The particles are let go from rest with both parts of the string taut. During the ensuing motion, neither particle reaches the pulley and neither one reaches the bottom of a face.

Forces and equilibrium

A man of mass $80\,\text{kg}$ runs along a horizontal road while a constant resistance force of magnitude $P\,\text{N}$ acts opposite his motion. The total work done by the man as he increases his speed from $4\,\text{m s}^{-1}$ to $5.5\,\text{m s}^{-1}$ over a distance of $60\,\text{m}$ is $1200\,\text{J}$.

Forces and equilibrium

A train with mass $240\,000\,\text{kg}$ is moving up a slope that makes an angle of $4^\circ$ to the horizontal. A constant resistive force of magnitude $18\,000\,\text{N}$ acts on the train. At the moment when its speed is $15\,\text{m s}^{-1}$, the train’s deceleration is $0.2\,\text{m s}^{-2}$.

Energy, work and power

The diagram shows three coplanar forces with magnitudes $3\,\text{N}$, $2\,\text{N}$ and $P\,\text{N}$.

Forces and equilibrium

A particle $P$ travels along the straight line $ABCD$ with uniform acceleration. The lengths $AB$ and $BC$ are $100\,\text{m}$ and $148\,\text{m}$, respectively. It takes the particle $4\,\text{s}$ to move from $A$ to $B$ and another $4\,\text{s}$ to move from $B$ to $C$.

Kinematics of motion in a straight line

A particle with mass $20\,\text{kg}$ rests on a rough plane inclined at an angle of $60^\circ$ to the horizontal. Equilibrium is produced by a force of magnitude $P\,\text{N}$ acting on the particle, parallel to a line of greatest slope on the plane. The maximum possible value of $P$ is twice the minimum possible value of $P$.

Forces and equilibrium

A particle $P$ travels along a straight line through a point $O$. At time $t\,\text{s}$, the acceleration, $a\,\text{m s}^{-2}$, of $P$ is $a = 6 - 0.24t$. The particle is instantaneously at rest at $t = 20$.

Kinematics of motion in a straight line

The diagram shows a particle $A$ of mass $1.6\,\text{kg}$ on a horizontal plane, and a particle $B$ of mass $2.4\,\text{kg}$ on a plane inclined at an angle of $30^\circ$ to the horizontal. The particles are joined by a light inextensible string that goes over a small smooth pulley $P$ fixed at the top of the inclined plane. The distance $AP$ is $2.5\,\text{m}$ and the distance of $B$ from the bottom of the inclined plane is $1\,\text{m}$. A barrier at the bottom of the inclined plane stops any further movement of $B$. The section $BP$ of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest, with both sections of the string taut.

Newton's laws of motion

The diagram illustrates the velocity-time graph for a train travelling from rest at one station to rest at the next. It is made up of three straight line segments. The distance separating the two stations is $9040\text{ m}$.

Kinematics of motion in a straight line

A small ball is thrown straight downwards from point $A$, which is $7.2\text{ m}$ above level ground, with initial speed $5\text{ m s}^{-1}$. It strikes the ground at speed $V\text{ m s}^{-1}$ and then bounces straight up with speed $\frac{1}{2}V\text{ m s}^{-1}$. After the bounce, the ball rises to its greatest height at $B$.

Kinematics of motion in a straight line

At a point, coplanar forces with magnitudes $8\,\text{N}$, $12\,\text{N}$ and $18\,\text{N}$ act in the directions indicated in the diagram.

Forces and equilibrium

Particles $A$ and $B$, whose masses are $0.8\,\text{kg}$ and $1.6\,\text{kg}$, are joined by a light inextensible string. $A$ is on a smooth plane inclined at angle $\theta$ to the horizontal, with $\sin \theta = \frac{3}{5}$. A small smooth pulley $P$ is fixed at the top of the plane, and $B$ hangs freely (see diagram). The part $AP$ of the string is parallel to the line of greatest slope of the plane. The particles are let go from rest while both parts of the string remain taut.

Energy, work and power

A particle with mass $3\,\text{kg}$ lies on a rough plane that is tilted at an angle of $20^\circ$ to the horizontal.

Forces and equilibrium

A car with mass $1400\,\text{kg}$ moving at speed $v\,\text{m}\,\text{s}^{-1}$ is subject to a resistive force of magnitude $40v\,\text{N}$. On a straight level road, the car’s highest possible constant speed is $56\,\text{m}\,\text{s}^{-1}$.

Energy, work and power

A particle $P$ travels along a straight line and begins at point $O$. Its velocity $v\text{ m s}^{-1}$ at time $t\text{ s}$ is defined by $v = 12t - 4t^2$ for $0 \le t \le 2$, and $v = 16 - 4t$ for $2 \le t \le 4$.

Kinematics of motion in a straight line

A small ball $B$ is launched from point $O$ on level ground. Its initial velocity has horizontal and vertically upward components of $18\,\text{m s}^{-1}$ and $25\,\text{m s}^{-1}$ respectively.

Representation of data

A non-uniform rod $AB$ has length $0.5\,\text{m}$ and weight $8\,\text{N}$. It is freely hinged at the fixed point $A$. The rod is inclined at $30^\circ$ to the horizontal, with $B$ positioned above the level of $A$. Equilibrium is maintained by a force of magnitude $12\,\text{N}$ acting in the vertical plane containing the rod, making an angle of $30^\circ$ to $AB$, and applied at $B$ (see diagram).

Representation of data

Starting from point O, a particle P of mass 0.4\,\text{kg} is projected horizontally over a smooth horizontal plane. After t\,\text{s}, the velocity of P is v\,\text{m s}^{-1}. A force of magnitude 0.8t\,\text{N} acts away from O, and another force of magnitude 2e^{-t}\,\text{N} acts opposite to the motion of P.

Representation of data

A small object is launched from point $O$ with speed $V\,\text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal. After $t$ s from projection, its horizontal displacement from $O$ is $x$ m and its vertically upward displacement is $y$ m.

Representation of data

Particle $P$, with mass $0.7\,\text{kg}$, is connected by a light elastic string to a fixed point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. The string has natural length $0.5\,\text{m}$ and modulus of elasticity $20\,\text{N}$. The particle $P$ is projected up the line of greatest slope through $O$ from a point $A$ below the level of $O$. The initial kinetic energy of $P$ is $1.8\,\text{J}$ and the initial elastic potential energy in the string is also $1.8\,\text{J$.

Probability

Particle $P$ has mass $0.2\,\text{kg}$ and is fastened to one end of a light inextensible string of length $0.6\,\text{m}$. At the other end, the string is joined to a particle $Q$ of mass $0.3\,\text{kg}$. It then passes through a small hole $H$ in a smooth horizontal surface. A light elastic string with natural length $0.3\,\text{m}$ and modulus of elasticity $15\,\text{N}$ connects $Q$ to a fixed point $A$, which lies $0.4\,\text{m}$ vertically below $H$. Particle $P$ travels on the surface in a horizontal circle centred at $H$ (see diagram).

Probability

A solid cone of uniform density has height $1.2\,\text{m}$ and base radius $0.5\,\text{m}$. A uniform object is formed by boring a cylindrical hole of radius $0.2\,\text{m}$ through the cone along its axis of symmetry (see diagram).

Representation of data

A small ball $B$ is launched from point $O$ on level ground in the direction of point $A$, which is $12\,\text{m}$ above the ground. After $0.9\,\text{s}$, $B$ has moved $20\,\text{m}$ horizontally and lies directly beneath $A$ (see diagram).

Probability

A light elastic string has one end fixed at point $O$. Its other end is connected to a particle $P$ with mass $0.4\,\text{kg}$. The string’s natural length is $0.6\,\text{m}$ and its modulus of elasticity is $24\,\text{N}$. The particle is let go from rest at $O$.

Representation of data

$ABC$ is an object formed from a uniform wire with two straight sections, $AB$ and $BC$, where $AB = a$, $BC = x$ and angle $ABC = 90^\circ$. If the object is hung freely from $A$ and is in equilibrium, the angle between $AB$ and the horizontal is $\theta$ (see diagram).

Representation of data

A particle $P$ is fired from a point $O$ on horizontal ground with initial speed $20\,\text{m s}^{-1}$ at an angle of projection $30^\circ$. After $t$ s, the horizontal displacement of $P$ from $O$ is $x$ m and the vertical displacement upwards from $O$ is $y$ m.

Representation of data

A uniform object is formed by connecting a solid cone of height $0.8\,\text{m}$ and base radius $0.6\,\text{m}$ to a cylinder. The cylinder has length $0.4\,\text{m}$ and radius $0.5\,\text{m}$. Through the cylinder, a cylindrical hole of length $0.4\,\text{m}$ and radius $x\,\text{m}$ is drilled along the axis of symmetry. A flat face of the cylinder is fixed to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is arranged with points on the base of the cone and the base of the cylinder touching a horizontal surface (see diagram). The object is just about to topple.

Representation of data

Particle $P$ has mass $0.2\,\text{kg}$ and is fastened to one end of a light inextensible string of length $0.6\,\text{m}$. The free end of that string is fixed at point $A$. $P$ is also attached to one end of a second light inextensible string of length $0.6\,\text{m}$, and the other end is fixed at point $B$ directly below. The particle travels in a horizontal circle of radius $0.3\,\text{m}$, centred at the midpoint of $AB$, with both strings remaining straight (see diagram).

Discrete random variables

A particle $P$ with mass $0.2\,\text{kg}$ is let go from rest at a point $O$ above horizontal ground. At time $t\,\text{s}$ after release, the downward velocity of $P$ is $v\,\text{m s}^{-1}$. A force of magnitude $0.6t\,\text{N}$ acts vertically downwards on $P$. A vertically upwards force of magnitude $ke^{-t}\,\text{N}$, where $k$ is a constant, also acts on $P$.

Probability

A small ball $B$ is launched from point $O$ on horizontal ground. Its initial velocity has horizontal and vertically upward components of $18\,\text{m s}^{-1}$ and $25\,\text{m s}^{-1}$ respectively.

Probability

A non-uniform rod $AB$ with length $0.5\,\text{m}$ and weight $8\,\text{N}$ is freely hinged at a fixed point at $A$. It is inclined at an angle of $30^{\circ}$ to the horizontal, with $B$ above the level of $A$. The rod is maintained in equilibrium by a force of magnitude $12\,\text{N}$, acting in the vertical plane containing the rod, at an angle of $30^{\circ}$ to $AB$ and applied at $B$ (see diagram).

Representation of data