May/June 2019

In the expansion of $(2x + \frac{k}{x})^6$, where $k$ is a constant, the term independent of $x$ is $540$.

Series

The curve satisfying $\frac{d^2 y}{dx^2} = 2x - 5$ has a stationary point at $(3, 6)$.

Differentiation

The diagram displays a section of the curve $y = \frac{3}{\sqrt{1 + 4x}}$ and a point $P\,(2, 1)$ that lies on the curve. The normal to the curve at $P$ cuts the $x$-axis at $Q$.

Integration

At the point $P$ on the curve, the line $4y = x + c$, with $c$ constant, touches the curve $y^2 = x + 3$ as a tangent.

Coordinate geometry

A sector in a circle with radius $r$ cm has area $A$ cm$^2$. Write down the perimeter of the sector in terms of $r$ and $A$.

Circular measure

The diagram depicts a trapezium $ABCD$ with $A$, $B$ and $C$ at $(4, 0)$, $(0, 2)$ and $(h, 3h)$ respectively. $BC$ is parallel to $AD$, $ngle ABC = 90^\circ$ and $CD$ is parallel to the $x$-axis.

Coordinate geometry

The function $f$ is given by $f(x) = -2x^2 + 12x - 3$ for every $x \in \mathbb{R}$.

Quadratics

Prove that $\left(\frac{1}{\cos x} - \tan x\right)^2 = \frac{1 - \sin x}{1 + \sin x}$.

Trigonometry

The sketch represents a three-dimensional solid whose base $OABC$ and top face $DEFG$ are matching horizontal squares. The parallelograms $OAED$ and $CBFG$ each lie in vertical planes. Point $M$ is the midpoint of $AF$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OC$ respectively, while $\mathbf{k}$ points vertically upwards. The position vectors of $A$ and $D$ are given by $\overrightarrow{OA} = 8\mathbf{i}$ and $\overrightarrow{OD} = 3\mathbf{i} + 10\mathbf{k}$.

Coordinate geometry

In a geometric progression, the 3rd and 4th terms are $48$ and $32$ respectively. Find the sum to infinity of the progression.

Series

For $0 \leq x \leq 2\pi$, the function $f$ is given by $f(x) = 2 - 3 \cos x$.

Functions

Find the coefficient of $x$ when $\left(\frac{2}{x} - 3x\right)^5$ is expanded.

Series

In an arithmetic progression, the total of the first ten terms is the same as the total of the next five terms. The first term is $a$.

Series

The diagram shows a segment of the curve $y = \sqrt{4x + 1} + \frac{9}{\sqrt{4x + 1}}$ together with the minimum point $M$.

Integration

The coordinates of the points $A$ and $B$ are $(1, 3)$ and $(9, -1)$ respectively. The perpendicular bisector of $AB$ cuts the $y$-axis at point $C$. Determine the coordinates of $C$.

Coordinate geometry

The curve is defined by $\frac{dy}{dx} = x^3 - \frac{4}{x^2}$. The point $P(2, 9)$ is on the curve.

Differentiation

The angle $x$ satisfies $\sin x = a + b$ and $\cos x = a - b$, with $a$ and $b$ being constants.

Trigonometry

In the diagram, a semicircle has diameter $AB$, centre $O$ and radius $r$. Point $C$ is on the circumference, and $AOC = \theta$ radians. The perimeter of sector $BOC$ is twice the perimeter of sector $AOC$.

Trigonometry

The curve is given by $y = 3\cos 2x$, and the line is given by $2y + \frac{3x}{\pi} = 5$.

Trigonometry

The functions $f$ and $g$ are given by $f: x \mapsto 3x - 2, \; x \in \mathbb{R}$ and $g: x \mapsto \frac{2x + 3}{x - 1}, \; x \in \mathbb{R}, \; x \ne 1$.

Functions

Relative to the origin $O$, the position vectors of points $A$ and $B$ are $\vec{OA} = \begin{pmatrix} 6 \\ -2 \\ -6 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 3 \\ k \\ -3 \end{pmatrix}$, with $k$ a constant.

Coordinate geometry

The curve $C_1$ is given by the equation $y = x^2 - 4x + 7$. The curve $C_2$ is given by the equation $y^2 = 4x + k$, with $k$ constant. The tangent to $C_1$ at the point where $x = 3$ is also tangent to $C_2$ at the point $P$.

Differentiation

The function $f$ is given by $f(x) = x^2 - 4x + 8$ for $x \in \mathbb{R}$.

Quadratics

The diagram presents a segment of the curve with equation $y = (3x + 4)^{\tfrac{1}{2}}$ together with the tangent to the curve at point $A$. The $x$-coordinate of $A$ is $4$.

Differentiation

The first three terms in the binomial expansion of $(2x - \frac{1}{2x})^5$ are $32x^5 - 40x^3 + 20x$. Find the other three terms of the expansion.

Series

The figure depicts triangle $ABC$, which is right-angled at $A$. Angle $ABC = \frac{\pi}{5}$ radians, and $AC = 8\,\text{cm}$. Points $D$ and $E$ are located on $BC$ and $BA$ respectively. Sector $ADE$ is part of a circle with centre $A$, and $BDC$ is tangent to arc $DE$ at $D$.

Integration

The function $f$ is given by $f(x) = \frac{48}{x - 1}$ for $3 \leq x \leq 7$. The function $g$ is given by $g(x) = 2x - 4$ for $a \leq x \leq b$, where $a$ and $b$ are constants.

Functions

Two heavyweight boxers decide that they would achieve better results if they entered a lighter weight class. For each boxer, this means a total loss of 13 kg. By the end of week 1, each has lost 1 kg, and in every later week their weight loss is a little smaller than in the week before. Boxer $A$ loses 0.98 kg in week 2. It is stated that his weekly weight loss is in arithmetic progression.

Series

The figure depicts a solid $ABCDEF$ whose horizontal base $ABC$ is a triangle with a right angle at $A$. $AB$ and $AC$ are $8$ units and $6$ units long respectively, and $M$ is the midpoint of $AB$. Point $D$ is $7$ units directly above $A$. Triangle $DEF$ is in a horizontal plane, with $DE$, $DF$ and $FE$ parallel to $AB$, $AC$ and $CB$ respectively, and $N$ is the midpoint of $FE$. The lengths of $DE$ and $DF$ are $4$ units and $2$ units respectively. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $\overrightarrow{AB}$, $\overrightarrow{AC}$ and $\overrightarrow{AD}$ respectively.

Coordinate geometry

Points $A$ and $B$ have coordinates $(1,3)$ and $(9,-1)$ respectively, and $D$ is the mid-point of $AB$. Point $C$ is located at $(x,y)$, where $x$ and $y$ are variables.

Coordinate geometry

A curve is defined by $\frac{dy}{dx} = 3x^2 + ax + b$. Its stationary points are located at $(-1, 2)$ and $(3, k)$.

Differentiation

The function $f : x \mapsto p\sin^2 2x + q$ is defined on $0 \leq x \leq \pi$, with $p$ and $q$ being positive constants. The diagram displays the graph of $y = f(x)$.

Functions

Show that the relation $\ln(x^3 - 4x) - \ln(x^2 - 2x) = \ln(x + 2)$ holds.

Logarithmic and exponential functions

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

Logarithmic and exponential functions

Determine the equation of the normal to the curve $x^2 \ln y + 2x + 5y = 11$ at the point $(3, 1)$.

Differentiation

Find the exact value of $\int \tan^2 3x \, dx$.

Integration

The polynomial $p(x)$ has the form $p(x) = 5x^3 + ax^2 + bx - 16$, with $a$ and $b$ as constants. It is known that $(x - 2)$ divides $p(x)$ exactly and that the remainder is $27$ when $p(x)$ is divided by $(x + 1)$.

Algebra

The diagram depicts the curve whose equation is $y = \frac{8 + x^3}{2 - 5x}$. The maximum point is labelled $M$.

Numerical solution of equations

Show that $\cosec^2 \theta \equiv 2\cosec 2\theta \cot \theta$.

Trigonometry

The polynomial $p(x)$ is given by $p(x) = 4x^3 + (k + 1)x^2 - mx + 3k$, where $k$ and $m$ are constants.

Algebra

Find the values of $x$ that satisfy the equation $|4 + 2x| = |3 - 5x|$.

Logarithmic and exponential functions

Determine the exact coordinates of the stationary point on the curve given by $y = \frac{3x}{\ln x}$.

Differentiation

Find the exact value of $\int_{0}^{\frac{1}{2}\pi} (4 \sin 2x + 2 \cos^2 x)\, dx$. Include all the working needed.

Integration

Find the quotient and remainder when $2x^3 + x^2 - 8x$ is divided by $(2x + 1)$.

Integration

The diagram depicts the curve given by the parametric equations $x = 3t - 6e^{-2t}$, $y = 4t^2 e^{-t}$, for $0 \le t \le 2$. At the point $P$ on this curve, the $y$-coordinate is $1$.

Numerical solution of equations

Express $4\sin \theta + 4\cos \theta$ as $R\sin(\theta + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Trigonometry

The polynomial $p(x)$ is given by $p(x) = 4x^3 + (k + 1)x^2 - mx + 3k$, with $k$ and $m$ as constants.

Algebra

Find the solutions of $|4 + 2x| = |3 - 5x|$.

Logarithmic and exponential functions

Determine the exact coordinates of the stationary point on the curve given by $y = \frac{3x}{\ln x}$.

Differentiation

Find the exact value of $\int_{0}^{\frac{1}{2}\pi} \left(4\sin 2x + 2\cos^2 x\right) \, dx$. Show all working needed.

Integration

Find the quotient and remainder after dividing $2x^3 + x^2 - 8x$ by $(2x + 1)$.

Integration

The diagram represents the curve with parametric equations $x = 3t - 6e^{-2t}$, $y = 4t^2 e^{-t}$, for $0 \leq t \leq 2$. Point $P$ lies on the curve, and its $y$-coordinate is $1$.

Numerical solution of equations

Express $4\sin\theta + 4\cos\theta$ as $R\sin(\theta + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Trigonometry

Apply the trapezium rule over $3$ intervals to estimate the value of $\int_{0}^{3} \lvert 2^{x} - 4 \rvert\, dx$.

Numerical solution of equations

No calculator may be used anywhere in this question. The complex number $(\sqrt{3}) + i$ is represented by $u$.

Complex numbers

With all essential working shown, solve the equation

Logarithmic and exponential functions

Calculate the gradient of the curve $x^{3} + 3xy^{2} - y^{3} = 1$ at the point $(1, 3)$.

Differentiation

By rewriting the equation $\cot \theta - \cot(\theta + 45^\circ) = 3$ as a quadratic in $\tan \theta$ first, solve it for $0^\circ < \theta < 180^\circ$.

Trigonometry

Find the derivative of $\frac{1}{\sin^2 \theta}$ with respect to $\theta$.

Differential equations

By expanding $\sin(2x + x)$ first, show that $\sin 3x = 3\sin x - 4\sin^3 x$.

Integration

The sketch displays the curves $y = 4\cos\frac{1}{2}x$ and $y = \frac{1}{4 - x}$, for $0 \le x < 4$. At the point where $x = a$, the tangents drawn to the two curves are at right angles.

Numerical solution of equations

Take $f(x) = \frac{16 - 17x}{(2 + x)(3 - x)^2}$.

Algebra

The diagram displays rectangular axes $Ox$, $Oy$ and $Oz$, together with four points $A$, $B$, $C$ and $D$, whose position vectors are $\overrightarrow{OA} = 3\mathbf{i}$, $\overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j}$, $\overrightarrow{OC} = \mathbf{i} + 3\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$.

Vectors

Determine the coefficient of $x^3$ in the expansion of $(3 - x)(1 + 3x)^{\frac{1}{3}}$ when written in ascending powers of $x$.

Algebra

The graph depicts the curve $y = \sin 3x \cos x$ for $0 \leq x \leq \frac{1}{2}\pi$, together with its minimum point $M$. The shaded region $R$ lies between the curve and the $x$-axis.

Trigonometry

With all necessary working shown, solve the equation $9^x = 3^x + 12$. Give your answer correct to $2$ decimal places.

Logarithmic and exponential functions

Showing all necessary working, solve the equation $\cot 2\theta = 2\tan \theta$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

Determine the exact coordinates of the point on the curve $y = \dfrac{x}{1 + \ln x}$ where the tangent has gradient $\dfrac{1}{4}$.

Differentiation

For this question, calculators are not allowed. You are told that the complex number $-1 + \sqrt{3}i$ is a root of the equation $kx^3 + 5x^2 + 10x + 4 = 0$, where $k$ is a real constant.

Algebra

The figure shows $A$ at the midpoint of the semicircle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts the semicircle at $B$ and $C$. The angle $OAB$ is $x$ radians. The shaded region enclosed by $AB$, $AC$ and the arc centred at $A$ has area equal to one half of the semicircle.

Numerical solution of equations

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

Differential equations

Define $f(x)$ by $f(x) = \dfrac{10x + 9}{(2x + 1)(2x + 3)^2}$.

Integration

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

Vectors

Apply logarithms to solve $5^{3-2x} = 4(7^x)$, and give your answer correct to $3$ decimal places.

Logarithmic and exponential functions

The equation of the line $l$ is $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + u(2\mathbf{i} - \mathbf{j} - 2\mathbf{k})$.

Vectors

Show that the integral $\int_{0}^{\frac{\pi}{4}} x^2 \cos 2x \, dx$ equals $\frac{1}{32}(\pi^2 - 8)$.

Integration

Set $f(\theta) = \dfrac{1 - \cos 2\theta + \sin 2\theta}{1 + \cos 2\theta + \sin 2\theta}$.

Integration

The curve is defined by $y = \dfrac{1 + e^{-x}}{1 - e^{-x}}$, with $x > 0$.

Differentiation

The differential equation satisfied by the variables $x$ and $y$ is $(x + 1)\frac{dy}{dx} = y^2 + 5$, and the condition $y = 2$ applies when $x = 0$.

Differential equations

The diagram displays the curve $y = x^4 - 2x^3 - 7x - 6$. The curve meets the $x$-axis at the points $(a, 0)$ and $(b, 0)$, where $a < b$. It is stated that $b$ is an integer.

Numerical solution of equations

Inside the interval $0 \leq x \leq \pi$, the curve $y = \sin\left(x + \frac{1}{3}\pi\right)\cos x$ has two stationary points.

Differentiation

For this question, calculator use is not allowed. The complex number $u$ is defined by $u = \frac{4i}{1 - (\sqrt{3})i}$.

Complex numbers

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

Algebra

The diagram shows a system of coplanar forces acting through a single point. Their magnitudes are $78\,\text{N}$, $50\,\text{N}$ and $112\,\text{N}$, and the angles $\alpha$ and $\theta$ are measured to the horizontal as shown.

Forces and equilibrium

Particle $P$ is fired vertically upwards from a point $3\,\text{m}$ above level ground, with initial speed $25\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

The lorry’s mass is $12000\,\text{kg}$.

Energy, work and power

A particle with mass $1.3\,\text{kg}$ is at rest on a rough plane inclined at angle $\theta$ to the horizontal, where $\tan\theta = \frac{12}{5}$. The coefficient of friction between the particle and the plane is $\mu$.

Forces and equilibrium

A particle $P$ travels along a straight line starting from a fixed point $O$. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = t^2 - 8t + 12$ for $0 \leq t \leq 8$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are joined by a light inextensible string. Particle $A$ is supported on a smooth plane inclined at an angle of $\theta^\circ$ to the horizontal. The string goes over a small smooth pulley $P$ fixed at the top of the plane, and $B$ hangs freely $0.5\,\text{m}$ above horizontal ground (see diagram). The particles are released from rest with both parts of the string taut.

Newton's laws of motion

Forces that are coplanar, with magnitudes $40\,\mathrm{N}$, $32\,\mathrm{N}$, $P\,\mathrm{N}$ and $17\,\mathrm{N}$, act at a point in the directions indicated in the diagram. The system is in a state of equilibrium.

Forces and equilibrium

A car travels along a straight path with starting speed $u\,\mathrm{m\,s^{-1}}$ and uniform acceleration $a\,\mathrm{m\,s^{-2}}$. It needs $5\,\mathrm{s}$ to cover the initial $80\,\mathrm{m}$ and $8\,\mathrm{s}$ to cover the initial $160\,\mathrm{m}$.

Kinematics of motion in a straight line

A particle with mass $13\,\text{kg}$ lies on a rough plane inclined at an angle of $\theta$ to the horizontal, where $\tan \theta = \frac{5}{12}$. The coefficient of friction between the particle and the plane is $0.3$. A force of magnitude $T\,\text{N}$, acting parallel to the line of greatest slope, pulls the particle $2.5\,\text{m}$ up the plane at a constant speed.

Energy, work and power

A force resisting motion with constant magnitude $350\,\text{N}$ acts on a car of mass $1250\,\text{kg}$. The car’s engine provides a constant driving force of $1200\,\text{N}$. The car moves along a road inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.05$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are linked by a light inextensible string that goes over a fixed smooth pulley. Each particle is $0.5\,\text{m}$ above the ground. They hang vertically, as shown in the diagram. The particles are released from rest. In the motion that follows, $B$ does not reach the pulley and $A$ comes to rest once it reaches the ground.

Newton's laws of motion

A car’s mass is $1000\,\text{kg}$. If it moves at a constant speed of $v\,\text{m s}^{-1}$, with $v>2$, the resistive force on the car is $(Av + B)\,\text{N}$, where $A$ and $B$ are constants. On a horizontal road, the car can maintain a steady speed of $18\,\text{m s}^{-1}$ when the engine output is $36\,\text{kW}$. It can also go up a hill making an angle of $\theta$ to the horizontal, with $\sin\theta = 0.05$, at a constant speed of $12\,\text{m s}^{-1}$ when the engine output is $21\,\text{kW}$.

Forces and equilibrium

Particles $P$ and $Q$ depart from the fixed point $A$ at the same instant and move along the same straight line. After $t$ seconds, the velocity of $P$ is $6t(t - 3)\,\text{m s}^{-1}$, while the velocity of $Q$ is $(10 - 2t)\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A bus travels in a straight line from one bus stop to another. It begins from rest and accelerates at $2.1\,\text{m s}^{-2}$ for $5\,\text{s}$. It then continues at a constant speed for $24\,\text{s}$ before slowing down with a constant deceleration and coming to rest after a further $6\,\text{s}$.

Kinematics of motion in a straight line

Three coplanar forces with magnitudes $12\,\text{N}$, $24\,\text{N}$ and $30\,\text{N}$ are applied at one point in the directions indicated in the diagram.

Forces and equilibrium

A car with mass $1400\,\text{kg}$ is moving uphill on a slope that makes an angle of $4^{\circ}$ to the horizontal. A constant resistive force of magnitude $1550\,\text{N}$ acts on the car.

Energy, work and power

Particles $A$ and $B$, having masses $1.3\,\text{kg}$ and $0.7\,\text{kg}$ respectively, are linked by a light inextensible string passing over a smooth fixed pulley. Particle $A$ is positioned $1.75\,\text{m}$ above the floor, while particle $B$ is $1\,\text{m}$ above the floor (see diagram). The system is let go from rest with the string taut, and the particles travel vertically. When the particles are level with each other the string breaks.

Newton's laws of motion

A particle with mass $18\,\text{kg}$ is placed on a plane inclined at an angle of $30^\circ$ to the horizontal. It is projected along the line of greatest slope of the plane with speed $20\,\text{m s}^{-1}$.

Energy, work and power

A particle $P$ travels in a straight line. Its acceleration $a\,\text{m s}^{-2}$ at time $t\,\text{s}$ is $a = 6t - 12$. The displacement of $P$ from a fixed point $O$ on the line is $s\,\text{m}$. It is known that $s = 5$ when $t = 1$ and $s = 1$ when $t = 3$.

Kinematics of motion in a straight line

A particle $P$ with mass $0.3\ \text{kg}$ is connected to the fixed point $A$ by a light inextensible string of length $0.8\ \text{m}$. The point $O$ lies $0.15\ \text{m}$ vertically beneath $A$. Particle $P$ travels at constant speed $v\ \text{m s}^{-1}$ round a horizontal circle centred at $O$ (see diagram).

Probability

A particle is projected at speed $V\ \text{m s}^{-1}$ at an angle of $\theta^\circ$ above the horizontal. After $4\ \text{s}$, the particle has speed $16\ \text{m s}^{-1}$ and is moving at $30^\circ$ above the horizontal. Find $V$ and $\theta$.

Representation of data

The diagram depicts the cross-section, taken through the centre of mass, of a uniform solid object. The object consists of a cylinder with radius $0.2\,\text{m}$ and length $0.7\,\text{m}$, from which a hemisphere of radius $0.2\,\text{m}$ has been cut away at one end. Point $A$ marks the centre of the plane face at the opposite end of the object. [$\text{The volume of a hemisphere is } \frac{2}{3}\pi r^3$ ]

Representation of data

A small ball is launched at speed $25\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal from point $O$ on level ground. After $t\,\text{s}$, the ball’s horizontal displacement from $O$ is $x\,\text{m}$ and its vertical upward displacement is $y\,\text{m}$.

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $0.5\,\text{m}$ and modulus of elasticity is $6\,\text{N}$. The opposite end of the string is fixed at point $O$. The particle $P$ is let go from rest at the point $(0.5 + x)\,\text{m}$ vertically below $O$. The particle $P$ then comes to instantaneous rest at $O$.

Probability

$ABC$ is a triangular uniform lamina with $AB = 0.3\,\text{m}$, $BC = 0.6\,\text{m}$ and a right angle at $B$ (see diagram).

Representation of data

A particle $P$ with mass $0.5\,\text{kg}$ is connected to a fixed point $O$ by a light elastic string whose natural length is $1\,\text{m}$ and whose modulus of elasticity is $16\,\text{N}$. Particle $P$ is projected vertically upwards from $O$ at speed $6\,\text{m s}^{-1}$. When $P$ is at displacement $x\,\text{m}$ above $O$, a resisting force of magnitude $0.1x^2\,\text{N}$ acts on it. After projection, the upward speed of $P$ is $v\,\text{m s}^{-1}$.

Probability

A small ball is launched from a point $O$ on level ground at an angle of $30^\circ$ above the horizontal. After $t$ s from projection, the ball’s vertical displacement upward from $O$ is $(14t - kt^2)$ m, where $k$ is a constant.

Representation of data

A uniform lamina $ABCEFG$ is made from the square $ABDG$ by cutting away the smaller square $CDFE$ from one corner. $AB = 0.7\,\text{m}$ and $DF = 0.3\,\text{m}$ (see diagram).

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is connected to a fixed point $A$ by a light inextensible string of length $0.5\,\text{m}$. The point $A$ is $0.3\,\text{m}$ above a smooth horizontal surface. The particle $P$ moves round a horizontal circle on the surface at constant angular speed $5\,\text{rad s}^{-1}$.

Probability

A particle $P$ with mass $0.5\,\text{kg}$ is attached to one end of a light elastic string whose natural length is $0.8\,\text{m}$ and modulus of elasticity is $16\,\text{N}$. The opposite end of the string is fixed at point $O$. Particle $P$ is released from rest at the point $0.8\,\text{m}$ vertically below $O$. When the extension of the string is $x\,\text{m}$, the downward velocity of $P$ is $v\,\text{m s}^{-1}$ and a force of $25x^2\,\text{N}$ opposes the motion of $P$.

Probability

A light elastic string has natural length $a\text{ m}$ and modulus of elasticity $\lambda\text{ N}$. Its tension is $4\text{ N}$ when the length is $1.6\text{ m}$, and it is $6\text{ N}$ when the length is $2\text{ m}$. One end of the string is fixed at a point $O$ on a smooth horizontal surface, while the other end is connected to a particle $P$ of mass $0.2\text{ kg}$. The particle $P$ moves at constant speed on the surface in a circle with centre $O$ and radius $1.9\text{ m}$.

Discrete random variables

A particle is launched at speed $15\text{ m s}^{-1}$ at an angle of $\theta^\circ$ above the horizontal. Four seconds after launch, the particle's speed is $30\text{ m s}^{-1}$.

Probability

Fig. 1 depicts an object formed from a uniform wire of length $0.8\,\text{m}$. It has a straight section $AB$ and a semicircular section $BC$, with $A$, $B$ and $C$ lying on one straight line. The semicircle has radius $r\,\text{m}$, and the centre of mass of the object is $0.1\,\text{m}$ from line $ABC$.

Representation of data

A particle $P$ with mass $0.3\,\text{kg}$ is joined to a fixed point $A$ by a light inextensible string of length $0.8\,\text{m}$. The fixed point $O$ lies $0.15\,\text{m}$ directly beneath $A$. Particle $P$ travels at constant speed $v\,\text{m s}^{-1}$ round a horizontal circle centred at $O$ (see diagram).

Probability

A particle is launched at speed $V\,\text{m s}^{-1}$ at an angle of $\theta^\circ$ above the horizontal. After $4\,\text{s}$, its speed is $16\,\text{m s}^{-1}$ and its direction of motion is $30^\circ$ above the horizontal. Determine $V$ and $\theta$.

Probability

The diagram presents a cross-section taken through the centre of mass of a uniform solid body. This body is a cylinder with radius $0.2\,\text{m}$ and length $0.7\,\text{m}$, and a hemisphere of radius $0.2\,\text{m}$ has been cut away from one end. Point $A$ marks the centre of the plane face at the opposite end of the body.

Representation of data

A small ball is launched with speed $25\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal from a point $O$ on level ground. After $t\,\text{s}$ from launch, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical displacement upwards from $O$ is $y\,\text{m}$.

Representation of data