Oct/Nov 2018

With all necessary working shown, solve the equation $4x - 11x^2 + 6 = 0$.

Quadratics

The curve is described by $y = \frac{1}{2}(4x - 3)^{-1}$. Point $A$ on the curve has coordinates $(1, \frac{1}{2})$.

Differentiation

The one-one function $f$ is given by $f(x) = (x - 3)^2 - 1$ for $x < a$, where $a$ is a constant. State the largest possible value of $a$.

Functions

The line is given by the equation $y = x + 1$ and the curve is given by the equation $y = x^2 + bx + 5$.

Coordinate geometry

The coordinates of points $A$ and $B$ are $(3a, -a)$ and $(-a, 2a)$ respectively, and $a$ is a positive constant.

Coordinate geometry

The series has first term $6$ and second term $2$.

Series

Show that the equation $$\frac{\cos \theta - 4}{\sin \theta} - \frac{4 \sin \theta}{5 \cos \theta - 2} = 0$$ may be rewritten as $$9 \cos^2 \theta - 22 \cos \theta + 4 = 0.$$

Trigonometry

A curve has a stationary point at $(3, 9\tfrac{1}{2})$ and its equation is such that $\frac{dy}{dx} = ax^2 + a^2x$, where $a$ is a non-zero constant.

Differentiation

The diagram depicts a section of the curve with equation $y = k(x^3 - 7x^2 + 12x)$, where $k$ is a constant. The curve cuts the line $y = x$ at the origin $O$ and again at the point $A(2, 2)$.

Integration

The diagram depicts a solid figure $OABCDEF$ with a horizontal rectangular base $OABC$, where $OA = 6$ units and $AB = 3$ units. The vertical edges $OF$, $AD$ and $BE$ measure $6$ units, $4$ units and $4$ units respectively. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OF$ respectively.

Coordinate geometry

The diagram depicts triangle $OAB$, where angle $ABO$ is a right angle, angle $AOB = \frac{\pi}{5}$ radians, and $AB = 5\text{ cm}$. Arc $BC$ is part of a circle centred at $A$ and intersects $OA$ at $C$. Arc $CD$ is part of a circle centred at $O$ and intersects $OB$ at $D$.

Integration

Determine the coefficient of $\frac{1}{x^2}$ in the expansion of $(3x + \frac{2}{3x^2})^7$.

Series

The curve is given by $y = 2x + \frac{12}{x}$, while the line has equation $y + x = k$, where $k$ is a constant.

Quadratics

The diagram depicts a section of the curve $y = 3\sqrt{(4x + 1)} - 2x$. The curve meets the $y$-axis at $A$, and the curve's stationary point is $M$.

Integration

With all working shown, determine $\int_{1}^{4} \left( \sqrt{x} + \frac{2}{\sqrt{x}} \right) \, dx.$

Integration

The diagram depicts part of the curve $y = x(9 - x^2)$ together with the line $y = 5x$, which meet at the origin $O$ and at the point $R$. Point $P$ is on the line $y = 5x$ between $O$ and $R$, and its $x$-coordinate is $t$. Point $Q$ is on the curve, and $PQ$ is parallel to the $y$-axis.

Differentiation

The functions $f$ and $g$ are given by $f: x \mapsto 2 - 3\cos x$ for $0 \leq x \leq 2\pi$, and $g: x \mapsto \frac{1}{2}x$ for $0 \leq x \leq 2\pi$.

Functions

In an arithmetic progression, the first three terms are $4$, $x$ and $y$ respectively. In a geometric progression, the first three terms are $x$, $y$ and $18$ respectively. You are told that both $x$ and $y$ are positive.

Series

The diagram depicts triangle $ABC$, where $BC = 20\text{ cm}$ and angle $ABC = 90^\circ$. The perpendicular drawn from $B$ to $AC$ meets $AC$ at $D$, with $AD = 9\text{ cm}$. The angle $BCA$ is $\theta$.

Trigonometry

The diagram depicts a solid cylinder on a horizontal circular base with centre $O$ and radius $4$ units. Points $A$, $B$ and $C$ are on the circumference of the base, with $AB$ as a diameter and angle $BOC = 90^\circ$. Points $P$, $Q$ and $R$ are on the top face of the cylinder directly above $A$, $B$ and $C$ respectively. The cylinder has height $12$ units. $M$ is the midpoint of $CR$ and $N$ is on $BQ$ with $BN = 4$ units. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OB$ and $OC$ respectively, and unit vector $\mathbf{k}$ points vertically upwards.

Coordinate geometry

The diagram depicts an isosceles triangle $ACB$ with $AB = BC = 8\text{ cm}$ and $AC = 12\text{ cm}$. Arc $XC$ is taken from a circle centred at $A$ with radius $12\text{ cm}$, while arc $YC$ comes from a circle centred at $B$ with radius $8\text{ cm}$. Points $A$, $B$, $X$ and $Y$ all lie on one straight line.

Circular measure

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

Quadratics

Determine the coefficient of $\frac{1}{x^3}$ in the expansion of $(x - \frac{2}{x})^7$.

Series

The diagram gives part of the curve $y = 2(3x - 1)^{-\frac{1}{3}}$ together with the lines $x = \frac{2}{3}$ and $x = 3$. The curve meets the line $x = \frac{2}{3}$ at the point $A$.

Integration

Write $2x^2 - 12x + 11$ in the form $a(x + b)^2 + c$, with $a$, $b$ and $c$ as constants.

Functions

The function $f$ has definition $f(x) = x^3 + 2x^2 - 4x + 7$ for $x \geq -2$.

Differentiation

The diagram depicts arc $BC$ of a circle centred at $A$ with radius $5\,\text{cm}$. The arc length $BC$ is $4\,\text{cm}$. Point $D$ is positioned so that $BD$ is perpendicular to $BA$ and $DC$ is parallel to $BA$.

Circular measure

The coordinates of two points $A$ and $B$ are $(-1, 1)$ and $(3, 4)$, respectively. The line $BC$ is perpendicular to $AB$ and meets the $x$-axis at $C$.

Coordinate geometry

In an arithmetic progression, the first term is $a$ and the common difference is $3$. The $n$th term equals $94$, and the sum of the first $n$ terms is $1420$.

Series

The diagram depicts the solid figure $OABCDEFG$ with a horizontal rectangular base $OABC$, where $OA = 8$ units and $AB = 6$ units. The rectangle $DEFG$ is in a horizontal plane, with $D$ situated $7$ units vertically above $O$ and $DE$ parallel to $OA$. The sides $DE$ and $DG$ are $4$ units and $2$ units long respectively. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively.

Coordinate geometry

Show that $\frac{\tan \theta + 1}{1 + \cos \theta} + \frac{\tan \theta - 1}{1 - \cos \theta} = \frac{2(\tan \theta - \cos \theta)}{\sin^2 \theta}$ is true.

Trigonometry

A curve goes through $(0, 11)$ and is defined by an equation for which $\frac{dy}{dx} = ax^2 + bx - 4$, where $a$ and $b$ are constants.

Differentiation

The curve is given by $y = 2x^2 - 3x + 1$ and the line is given by $y = kx + k^2$, where $k$ is a constant.

Quadratics

Solve for $x$ in the equation $\lvert 9x - 2 \rvert = \lvert 3x + 2 \rvert$.

Logarithmic and exponential functions

Show that $\int_{1}^{7} \frac{6}{2x + 1}\,dx = 125$.

Integration

Solve the equation $\sec^2\theta = 3\csc\theta$ within $0^\circ < \theta < 180^\circ$.

Trigonometry

The diagram depicts the curve with equation $y = x^4 + 2x^3 + 2x^2 - 12x - 32$. It crosses the $x$-axis at the points with coordinates $(\alpha, 0)$ and $(\beta, 0)$.

Numerical solution of equations

The curve is given by the parametric equations $x = t + \ln(t + 1)$, $y = 3te^{2t}$.

Differentiation

The sketch displays the curve with equation $y = \sqrt{1 + 3\cos^2\left(\tfrac{1}{2}x\right)}$ for $0 \leq x \leq \pi$. The region $R$ is enclosed by the curve, the axes and the line $x = \pi$.

Integration

The diagram represents the curve given by $y = \sin 2x + 3\cos 2x$ for $0 \leq x \leq \pi$. At points $P$ and $Q$ on the curve, the gradient is $3$.

Differentiation

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

Algebra

If $9^x + 3^x = 240$, determine the value of $3^x$ and then, by using logarithms, find $x$ correct to $4$ significant figures.

Logarithmic and exponential functions

The curve is represented in the diagram by $y = 5\sin 2x - 3\tan 2x$ for all $x$ such that $0 \le x < \frac{1}{4}\pi$.

Differentiation

Determine the gradient of the curve $4x + 3ye^{2x} + y^2 = 10$ at the point $(0, 2)$.

Differentiation

For the curve defined by $y = 5e^{2x} - 8x^2 - 20$, there is only one point where it intersects the $x$-axis. The coordinates of this point are $(p, 0)$.

Numerical solution of equations

Show that $\int_{1}^{6} \frac{12}{3x + 2} \, dx = \ln 256$.

Integration

Use the factor theorem to demonstrate that $(2x + 3)$ is a factor of $8x^3 + 4x^2 - 10x + 3$.

Trigonometry

Solve for $x$ in $|9x - 2| = |3x + 2|$.

Logarithmic and exponential functions

Show that evaluating $\int_{1}^{7} \frac{6}{2x + 1}\,dx$ gives $125$.

Integration

Find the solutions of $\sec^2 \theta = 3\csc \theta$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

The diagram presents the curve whose equation is $y = x^4 + 2x^3 + 2x^2 - 12x - 32$. The curve intersects the $x$-axis at the points with coordinates $(\alpha, 0)$ and $(\beta, 0)$.

Numerical solution of equations

A curve is defined by the parametric equations $x = t + \ln(t + 1)$ and $y = 3te^{2t}$.

Differentiation

The sketch shows the curve given by $y = \sqrt{(1 + 3\cos^2(\tfrac{1}{2}x))}$ for $0 \le x \le \pi$. The region $R$ lies between the curve, the coordinate axes and the line $x = \pi$.

Integration

The sketch presents the curve with equation $y = \sin 2x + 3\cos 2x$ for $0 \le x \le \pi$. At the points $P$ and $Q$ on the curve, the gradient is $3$.

Differentiation

Find the values of $x$ that satisfy the inequality $2|2x - a| < |x + 3a|$, where $a$ is a positive constant.

Algebra

The planes $m$ and $n$ are given by the equations $3x + y - 2z = 10$ and $x - 2y + 2z = 5$, respectively. The line $l$ is given by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

Vectors

With all necessary working shown, solve the equation $\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4$, and give your answer correct to $2$ decimal places.

Logarithmic and exponential functions

By plotting an appropriate pair of graphs, demonstrate that the equation $x^3 = 3 - x$ has exactly one real root.

Numerical solution of equations

The curve is given in parametric form by $x = 2\sin\theta + \sin 2\theta$, $y = 2\cos\theta + \cos 2\theta$, where $0 < \theta < \pi$.

Differentiation

The differential equation satisfied by a general point $(x, y)$ on the curve is $x\frac{dy}{dx} = (2 - x^2)y$. The curve also passes through $(1, 1)$.

Differential equations

Show that the equation $(\sqrt{2})\cosec x + \cot x = \sqrt{3}$ may be written in the form $R\sin(x - \alpha) = \sqrt{2}$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Trigonometry

The diagram represents the curve $y=5\sin^2 x\cos^3 x$ on $0\leq x\leq \tfrac{1}{2}\pi$, together with its maximum point $M$. The shaded region $R$ lies between the curve and the $x$-axis.

Integration

Show all working, and write the complex number $\frac{2 + 3i}{1 - 2i}$ in the form $r e^{i\theta}$, with $r > 0$ and $-\pi < \theta \leq \pi$. State $r$ and $\theta$ correct to 3 significant figures.

Complex numbers

Define $f(x)=\frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}$.

Integration

Solve $3|2x - 1| > |x + 4|$.

Algebra

The line $l$ is represented by $\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$. The plane $p$ is represented by $(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0$. The line $l$ meets the plane $p$ at the point $A$.

Vectors

With all necessary working shown, solve the equation $\sin(\theta - 30^\circ) + \cos \theta = 2 \sin \theta$, for $0^\circ < \theta < 180^\circ$.

Trigonometry

Evaluate $\int \frac{\ln x}{x^3}\, dx$.

Integration

Using all necessary working, Solve the equation

Logarithmic and exponential functions

A curve has equation $y = x \ln(8 - x)$. Its gradient is 1 at exactly one point, where $x = a$.

Numerical solution of equations

For this curve, the gradient at a general point with coordinates $(x, y)$ is proportional to $\frac{y^2}{x}$. It also passes through the points with coordinates $(1, 1)$ and $(e, 2)$.

Differential equations

The curve is given by $y = \frac{3\cos x}{2 + \sin x}$, with $-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi$.

Integration

Define $f(x) = \dfrac{7x^{2} - 15x + 8}{(1 - 2x)(2 - x)^{2}}$.

Algebra

Without a calculator, rewrite the complex number $\frac{2 + 6i}{1 - 2i}$ in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Find all values of $x$ that satisfy the inequality $2|2x - a| < |x + 3a|$, where $a$ is a positive constant.

Algebra

The planes $m$ and $n$ are given by the equations $3x + y - 2z = 10$ and $x - 2y + 2z = 5$ respectively. The line $l$ is represented by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

Vectors

With all required working shown, solve the equation $\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4$, and give your answer correct to 2 decimal places.

Logarithmic and exponential functions

By drawing a suitable pair of graphs, show that the equation $x^3 = 3 - x$ has one and only one real root.

Numerical solution of equations

A curve is given parametrically by $x = 2\sin\theta + \sin 2\theta$ and $y = 2\cos\theta + \cos 2\theta$, with $0 < \theta < \pi$.

Differentiation

For a general point on the curve, the coordinates $(x, y)$ satisfy the differential equation $x\frac{dy}{dx} = (2 - x^2)y$. The curve also goes through $(1, 1)$.

Integration

Show that the equation $\sqrt{2}\,\cosec x + \cot x = \sqrt{3}$ can be written in the form $R\sin(x - \alpha) = \sqrt{2}$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Trigonometry

The graph depicts the curve $y = 5\sin^2 x\cos^3 x$ on $0 \leq x \leq \tfrac{1}{2}\pi$, together with its highest point $M$. The shaded area $R$ lies between the curve and the $x$-axis.

Integration

Show all working needed, and write the complex number $\frac{2 + 3i}{1 - 2i}$ in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. State $r$ and $\theta$ correct to $3$ significant figures.

Complex numbers

Take $f(x)$ to be $\frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}$.

Integration

A particle of mass $0.2\,\text{kg}$ travels along a straight path and is acted on by a constant resistive force of $1.5\,\text{N}$. At the moment its speed is $2.5\,\text{m s}^{-1}$, a constant force of magnitude $F\,\text{N}$ is applied in the direction of motion. If its speed $5$ seconds later is $4.5\,\text{m s}^{-1}$, determine the value of $F$.

Forces and equilibrium

A high-speed train, with a mass of $490\,000\,\text{kg}$, travels along a straight horizontal track at a steady speed of $85\,\text{m s}^{-1}$. Its engines provide $4080\,\text{kW}$ of power.

Energy, work and power

A van with mass $2500\,\text{kg}$ moves down a hill of length $0.4\,\text{km}$ that is inclined at $4^\circ$ to the horizontal. A constant resistive force of $600\,\text{N}$ acts, and the van’s speed rises from $20\,\text{m s}^{-1}$ to $30\,\text{m s}^{-1}$ during the descent.

Energy, work and power

Particles $A$ and $B$, with masses $m\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string. The string goes over a fixed smooth pulley, and the particles hang freely beneath it. The system is let go from rest, with both particles $0.8\,\text{m}$ above horizontal ground. Particle $A$ reaches the ground at a speed of $0.6\,\text{m s}^{-1}$.

Newton's laws of motion

Three coplanar forces, with magnitudes $15\,\text{N}$, $25\,\text{N}$ and $30\,\text{N}$, act at point $B$ on the line $ABC$ in the directions indicated in the diagram.

Forces and equilibrium

A particle is launched from point P with initial speed $u\,\text{m s}^{-1}$ up the line of greatest slope $PQR$ on a rough inclined plane. The distances $PQ$ and $QR$ are each $0.8\,\text{m}$. The particle takes $0.6\,\text{s}$ to move from $P$ to $Q$ and $1\,\text{s}$ to move from $Q$ to $R$.

Newton's laws of motion

A particle travels along a straight line and begins from rest at point $O$. At time $t$ after leaving $O$, its acceleration is $a\,\text{m s}^{-2}$, where $a = 5.4 - 1.62t$.

Kinematics of motion in a straight line

A smooth ring $R$ of mass $m\ \text{kg}$ is attached to a light inextensible string $ARB$. The string’s ends are fixed at points $A$ and $B$, with $A$ directly above $B$. The string is taut, and angle $ARB = 90^{\circ}$. The angle between the segment $AR$ of the string and the vertical is $45^{\circ}$. The ring is kept in equilibrium in this arrangement by a force of magnitude $2.5\,\text{N}$, acting on the ring in the direction $BR$ (see diagram).

Forces and equilibrium

A block of mass $5\,\text{kg}$ is pulled up a rough plane by a rope, and the plane is inclined at $6^{\circ}$ to the horizontal. The rope runs parallel to the line of greatest slope of the plane, and the block travels at constant speed. The coefficient of friction between the block and the plane is $0.3$.

Forces and equilibrium

A particle travels in a straight line with velocity $v\,\text{m s}^{-1}$ at time $t$ seconds. The diagram presents a velocity-time graph that represents the particle’s motion from $t = 0$ up to $t = T$. It is made up of four straight line segments. The particle attains its greatest velocity $V\,\text{m s}^{-1}$ at $t = 10$.

Kinematics of motion in a straight line

Two particles $P$ and $Q$, whose masses are $0.4\ \text{kg}$ and $0.7\ \text{kg}$ respectively, are joined to the two ends of a light inextensible string. This string runs over a fixed smooth pulley attached to the edge of a rough plane. The coefficient of friction between $P$ and the plane is $0.5$. The plane is inclined at an angle $\alpha$ to the horizontal, with $\tan \alpha = \frac{3}{4}$. Particle $P$ is on the plane and particle $Q$ is suspended vertically. The section of string from $P$ to the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude $X\ \text{N}$, acting directly down the plane, is applied to $P$.

Forces and equilibrium

A particle travels along a straight line from point $O$ with initial velocity $1\ \text{m s}^{-1}$. Its acceleration $a\ \text{m s}^{-2}$ at time $t$ s after leaving $O$ is given by $a = 1.2t^2 - 0.6t$.

Kinematics of motion in a straight line

A car with mass $1200\ \text{kg}$ travels on a straight horizontal road at a steady speed of $15\ \text{m s}^{-1}$. A constant resistive force of $350\ \text{N}$ acts on it.

Energy, work and power

A particle with mass $0.3\,\text{kg}$ is let go from rest above a tank of water. It moves straight down and takes $0.8\,\text{s}$ to arrive at the water surface. Its speed does not change suddenly as it enters the water. The water depth in the tank is $1.25\,\text{m}$. A force from the water acts on the particle and opposes its motion. The work done against this resisting force from the moment the particle enters the water until it reaches the tank bottom is $1.2\,\text{J}$. On reaching the bottom, it rebounds vertically upwards with initial speed $7\,\text{m s}^{-1}$. While moving up through the water, it is acted on by a constant resistance force of $1.8\,\text{N}$. The particle is instantaneously at rest $t$ seconds after rebounding from the bottom of the tank.

Energy, work and power

A small smooth ring $R$, with mass $0.2\,\text{kg}$, is placed on a light inextensible string $ARB$. The two ends of the string are fixed at points $A$ and $B$ on a roof sloping at $45^\circ$ to the horizontal. A horizontal force of magnitude $P\,\text{N}$, acting in the plane $ARB$, is exerted on the ring. The section $BR$ of the string is perpendicular to the roof, and the section $AR$ makes an angle of $70^\circ$ with the horizontal (see diagram). The system is in equilibrium.

Forces and equilibrium

A block is driven across a horizontal floor by a force of magnitude $50\,\text{N}$ acting at an angle of $20^\circ$ to the horizontal (see diagram). The coefficient of friction between the block and the floor is $0.3$. Since the block moves with constant speed,

Forces and equilibrium

A particle with mass $1.2\,\text{kg}$ travels along the straight line $AB$. It is launched from $A$ towards $B$ at a speed of $7.5\,\text{m s}^{-1}$ and a resistance force acts on it. The work done against this resistance force as it moves from $A$ to $B$ is $25\,\text{J}$.

Energy, work and power

At time $t = 0$, with $t$ measured in seconds, a runner leaves point $P$. The runner is initially at rest, then accelerates at $1.2\,\text{m s}^{-2}$ for $5\,\text{s}$. During the following $12\,\text{s}$ the runner continues at a constant speed, and then decelerates uniformly over $3\,\text{s}$ until coming to rest at $Q$. A cyclist leaves $P$ at time $t = 10$ and accelerates uniformly for $10\,\text{s}$, after which the cyclist immediately decelerates uniformly to rest at $Q$ at time $t = 30$.

Kinematics of motion in a straight line

Two particles $P$ and $Q$, with masses $0.3\text{ kg}$ and $0.5\text{ kg}$ respectively, are fastened to the two ends of a light inextensible string. The string goes over a fixed smooth pulley, and the particles hang freely beneath it (see diagram). $Q$ is kept stationary with the string taut at a height of $h\text{ m}$ above a horizontal floor (see diagram). $Q$ is then let go, and both particles begin to move. The pulley is high enough for $P$ never to reach it at any stage. The time for $Q$ to reach the floor is $0.6\text{ s}$.

Newton's laws of motion

A van with mass $3200\text{ kg}$ is moving on a level road. The van’s engine delivers constant power of $36\text{ kW}$, and the van experiences a steady resistive force.

Energy, work and power

A particle travels along a straight line. It begins from rest at a point $O$ on the line. After $t\,\text{s}$ from leaving $O$, the particle’s acceleration $a\,\text{m s}^{-2}$ is defined by $a = 25 - t^2$ for $0 \leq t \leq 9$.

Kinematics of motion in a straight line

A small ball $B$ is launched from a point on level ground at a speed of $30\,\text{m s}^{-1}$ and at an angle of $60^\circ$ above the horizontal.

Probability

A single uniform body is formed by joining the flat base of a solid hemisphere to the flat base of a solid cone, so that the body has an axis of symmetry. The cone has base radius $0.3\,\text{m}$, and the hemisphere has radius $0.2\,\text{m}$. The body is resting on a horizontal plane, with point $A$ on the curved surface of the hemisphere and point $B$ on the rim of the cone in contact with the plane (see diagram).

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is tied to a fixed point $O$ by a light elastic string whose natural length is $0.5\,\text{m}$ and whose modulus of elasticity is $20\,\text{N}$. $P$ is released from rest at $O$.

Probability

A particle $P$ of mass $0.5\,\text{kg}$ is fired along a smooth horizontal surface in the direction of a fixed point $A$. At the outset, $P$ is at $O$ on the surface, and after projection its displacement from $O$ is $x\,\text{m}$ and its velocity is $v\,\text{m s}^{-1}$. Particle $P$ is attached to $A$ by a light elastic string with natural length $0.8\,\text{m}$ and modulus of elasticity $16\,\text{N}$. The distance $OA$ is $1.6\,\text{m}$ (see diagram). The motion of $P$ is opposed by a force of magnitude $24x^2\,\text{N}$.

Representation of data

A particle $P$ with mass $0.1\,\text{kg}$ is fastened to one end of a light inextensible string of length $0.5\,\text{m}$. The other end of the string is fixed at point $A$. Particle $P$ travels round a circle whose centre $O$ lies on a smooth horizontal surface $0.3\,\text{m}$ below $A$. The string tension has magnitude $T\,\text{N}$ and the magnitude of the force exerted on $P$ by the surface is $R\,\text{N}$.

Probability

Figure 1 presents the cross-section $ABCDE$ through the centre of mass $G$ of a uniform prism. The cross-section is a rectangle $ABCF$ with triangle $DEF$ removed; $AB = 0.6\,\text{m}$, $BC = 0.7\,\text{m}$ and $DF = EF = 0.3\,\text{m}$.

Representation of data

A small object is launched from point $O$ at the foot of a plane inclined at $45^\circ$ to the horizontal, with speed $24\,\text{m s}^{-1}$. Its projection angle is $15^\circ$ above the line of greatest slope of the plane (see diagram). After $t\,\text{s}$, the object's horizontal displacement from $O$ is $x\,\text{m}$ and its vertically upward displacement from $O$ is $y\,\text{m}$.

Representation of data

A small ball $B$ is fired from a point on horizontal ground with speed $38\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal.

Representation of data

A single solid body is formed by joining a cone to a cylinder so that the base circumference of the cone matches the flat circular face of the cylinder. Both the cone and the cylinder have radius $0.3\,\text{m}$ and height $0.4\,\text{m}$.

Representation of data

Particle $P$, with mass $0.4\,\text{kg}$, is launched horizontally on a smooth horizontal plane from point $O$. After projection, its velocity is $v\,\text{m s}^{-1}$ and its displacement from $O$ is $x\,\text{m}$. A force of magnitude $8x\,\text{N}$ acting away from $O$ acts on $P$, while another force of magnitude $(2e^{-x}+4)\,\text{N}$ acts opposite to the motion of $P$. One end of a light elastic string with natural length $0.5\,\text{m}$ is fixed to $O$, and the other end is attached to $P$.

Representation of data

A small object is fired horizontally at speed $V\,\text{m s}^{-1}$ from point $O$ above level ground. After $t\,\text{s}$, its horizontal displacement from $O$ is $x\,\text{m}$ and its displacement vertically upwards from $O$ is $y\,\text{m}$.

Representation of data

A particle $P$ with mass $0.7\,\text{kg}$ is connected to a fixed point $O$ by a light elastic string whose natural length is $0.6\,\text{m}$ and whose modulus of elasticity is $15\,\text{N}$. Particle $P$ is projected vertically downward from $A$, where $A$ is $0.8\,\text{m}$ vertically beneath $O$. The initial speed of $P$ is $2\,\text{m s}^{-1}$.

Probability

The diagram represents a uniform lamina $ABCDEFGH$. It is built from a quarter-circle $OAB$ with radius $r\,\text{m}$, a rectangle $DEFG$, and two isosceles right-angled triangles $COD$ and $GOH$. In the rectangle, $DG = EF = r\,\text{m}$ and $DE = FG = x\,\text{m}$.

Representation of data

A rough horizontal rod $AB$ with length $0.45\,\text{m}$ turns at a steady angular speed of $6\,\text{rad s}^{-1}$ about a vertical axis through $A$. A small ring $R$ of mass $0.2\,\text{kg}$ is able to slide along the rod. A particle $P$ of mass $0.1\,\text{kg}$ is fixed at the midpoint of a light inextensible string of length $0.6\,\text{m}$. One end of the string is fastened to $R$ and the other end is fastened to $B$, with angle $RPB = 60^\circ$ (see diagram). As the system rotates, $R$ and $P$ each travel in horizontal circles. $R$ is in limiting equilibrium.

Probability

A small ball $B$ is launched from a point on horizontal ground with speed $30\ \text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal.

Probability