May/June 2011

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

Series

The line $x - y + 4 = 0$ meets the curve $y = 2x^2 - 4x + 1$ at the points $P$ and $Q$. The coordinates of $P$ are given as $(3, 7)$.

Quadratics

Functions $f$ and $g$ are defined on $x \in \mathbb{R}$ by $f : x \mapsto 2x + 1$, $g : x \mapsto x^2 - 2$.

Functions

A spherical balloon's volume is rising at a steady rate of $50\text{ cm}^3$ per second.

Differentiation

Sketch the graph of $y = (x - 2)^2$.

Integration

The diagram depicts a prism $ABCDPQRS$ with a horizontal square base $APSD$ of side length $6\text{ cm}$. The cross-section $ABCD$ is a trapezium, and the vertical edges $AB$ and $DC$ measure $5\text{ cm}$ and $2\text{ cm}$ respectively. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $AD$, $AP$ and $AB$ respectively.

Coordinate geometry

Show that $2\tan^2\theta\sin^2\theta = 1$ may be rewritten in the form $2\sin^4\theta + \sin^2\theta - 1 = 0$.

Trigonometry

The variables $x$, $y$ and $z$ are restricted to positive values, and they satisfy $z = 3x + 2y$ together with $xy = 600$.

Differentiation

The curve satisfies $\frac{dy}{dx} = \frac{3}{(1 + 2x)^2}$, and the point $(1, \frac{1}{2})$ lies on it.

Differentiation

A television quiz show is broadcast every day. On day 1 the prize money is $\$1000$. If nobody wins it, the prize fund is raised for day 2. This pattern of increase continues each day until someone wins. The television company considered the following two different models for raising the prize money. Model 1: Increase the prize money by $\$1000$ each day. Model 2: Increase the prize money by $10\%$ each day. On each day that the prize money is not won, the television company makes a donation to charity. The donation is $5\%$ of the prize on that day. After $40$ days the prize money has still not been won.

Integration

In the diagram, $OAB$ is an isosceles triangle with $OA = OB$ and angle $AOB = 2\theta$ radians. Arc $PST$ has centre $O$ and radius $r$, and line $ASB$ is tangent to arc $PST$ at $S$.

Circular measure

Calculate $\int \left(x^3 + \frac{1}{x^3}\right)\,dx$.

Integration

A circle is split into $6$ sectors so that the sector angles form an arithmetic progression. The angle of the largest sector is $4$ times the angle of the smallest sector. If the circle has radius $5\,\text{cm}$, determine the perimeter of the smallest sector.

Series

The diagram shows a section of the curve $y = 4\sqrt{x} - x$. The curve has its maximum at $M$ and cuts the $x$-axis at $O$ and $A$.

Integration

Determine the $x^2$ and $x^3$ terms in the expansion of $(1 - \tfrac{3}{2}x)^6$.

Series

For the equation $x^2 + px + q = 0$, where $p$ and $q$ are constants, the roots are $-3$ and $5$.

Quadratics

The curve is described by the equation $y = \frac{4}{3x - 4}$, and the point $P\,(2,2)$ lies on it.

Differentiation

Prove the identity $\frac{\cos\theta}{\tan\theta(1 - \sin\theta)} = 1 + \frac{1}{\sin\theta}$.

Trigonometry

The function $f$ is given by $f : x \mapsto \frac{x + 3}{2x - 1}$, $x \in \mathbb{R}$, $x \neq \tfrac{1}{2}$.

Functions

Line $L_1$ goes through points $A\,(2,5)$ and $B\,(10,9)$. Line $L_2$ runs parallel to $L_1$ and passes through the origin. Point $C$ is on $L_2$ in such a way that $AC$ is perpendicular to $L_2$.

Coordinate geometry

With respect to the origin $O$, the position vectors for the points $A$, $B$ and $C$ are $\overrightarrow{OA} = \begin{pmatrix}2\\3\\5\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}4\\2\\3\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}10\\0\\6\end{pmatrix}$.

Coordinate geometry

The function $f$ is specified by $f(x) = 3 - 4\cos^k x$, for $0 \leq x \leq \pi$, and $k$ is a constant.

Functions

In $(a + x)^5 + (1 - 2x)^6$, the coefficient of $x^3$ equals $90$, with $a$ positive. Find the value of $a$.

Series

The functions $f$ and $g$ are specified by $f : x \mapsto 3x - 4$, $x \in \mathbb{R}$, and $g : x \mapsto 2(x - 1)^3 + 8$, $x > 1$.

Functions

Determine the values of $m$ for which the line $y = mx + 4$ meets the curve $y = 3x^2 - 4x + 7$ at two separate points.

Quadratics

The line $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are positive constants, cuts the $x$-axis at $P$ and the $y$-axis at $Q$. Given that $PQ = \sqrt{45}$ and that the gradient of the line $PQ$ is $-\frac{1}{2}$, determine the values of $a$ and $b$.

Coordinate geometry

Differentiate $\frac{2x^3 + 5}{x}$ with respect to $x$.

Integration

The diagram shows $OABCDEFG$ as a rectangular block with $OA = OD = 6$ cm and $AB = 12$ cm. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ run parallel to $OA$, $OC$ and $OD$ respectively. Point $P$ is the midpoint of $DG$, $Q$ is at the centre of the square face $CBFG$, and $R$ is located on $AB$ so that $AR = 4$ cm.

Coordinate geometry

A geometric progression has third term $20$ and sum to infinity equal to three times the first term. Find the first term.

Series

In the diagram, $AB$ forms an arc of a circle with centre $O$ and radius $6$ cm, and $AOB = \frac{\pi}{3}$ radians. The line $AX$ touches the circle at $A$, and $OBX$ lies on a straight line.

Circular measure

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

Trigonometry

A curve satisfies $\frac{dy}{dx} = \frac{2}{\sqrt{x}} - 1$ and $P(9, 5)$ lies on the curve.

Differentiation

Solve the equation $|3x + 4| = |2x + 5|$.

Algebra

The curve is described parametrically by $x = 3t + \sin 2t$, $y = 4 + 2\cos 2t$.

Differentiation

The variables $x$ and $y$ obey the equation $y = Kx^m$, with $K$ and $m$ as constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line that passes through the points $(0, 2.0)$ and $(6, 10.2)$, as illustrated in the diagram.

Numerical solution of equations

The polynomial $f(x)$ is given by $f(x) = 3x^3 + ax^2 + ax + a$, with $a$ as a constant.

Algebra

Find the value of $\frac{dy}{dx}$ at $x = 4$ for $y = x\ln(x - 3)$.

Differentiation

Find the value of $\int 4e^x(3 + e^{2x})\,dx$.

Integration

By using a suitable pair of graphs, show that the equation $e^{2x} = 14 - x^2$ has exactly two real roots.

Numerical solution of equations

Express $4\sin\theta - 6\cos\theta$ as $R\sin(\theta - \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and the value of $\alpha$ accurate to 2 decimal places.

Trigonometry

Employ logarithms to find the solution of $3^x = 2^{x+2}$, giving the answer correct to 3 significant figures.

Logarithmic and exponential functions

The diagram displays the curve $y = \sqrt{1 + x^3}$. Region A is enclosed by the curve together with the lines $x = 0$, $x = 2$ and $y = 0$. Region B is enclosed by the curve together with the lines $x = 0$ and $y = 3$.

Integration

The sequence $x_1, x_2, x_3, \ldots$ defined by $x_1 = 1$, $x_{n+1} = \frac{1}{2}\sqrt[3]{x_n^2 + 6}$ tends to the value $\alpha$.

Numerical solution of equations

Calculate the value of $\int_0^{\frac{2\pi}{3}} \sin\left(\frac{1}{2}x\right)\,dx$.

Integration

The curve is defined by $x^2 + 2y^2 + 5x + 6y = 10$. Determine the equation of the tangent to the curve at the point $(2, -1)$. Present your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

Differentiation

The curve $y = 4x^2 \ln x$ contains a single stationary point.

Differentiation

The cubic polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 + bx + 10$, with $a$ and $b$ as constants. It is also stated that $(x + 2)$ is a factor of $p(x)$ and that, on division of $p(x)$ by $(x + 1)$, the remainder equals 24.

Algebra

Show that $\sin^2 2\theta (\cosec^2 \theta - \sec^2 \theta) = 4 \cos 2\theta$.

Trigonometry

Apply logarithms to solve the equation $3^x = 2^{x+2}$, and give the answer correct to 3 significant figures.

Logarithmic and exponential functions

The diagram illustrates the curve $y = \sqrt{1 + x^3}$. Region $A$ is enclosed by the curve together with the lines $x = 0$, $x = 2$ and $y = 0$. Region $B$ is enclosed by the curve together with the lines $x = 0$ and $y = 3$.

Integration

The sequence $x_1, x_2, x_3, \ldots$ specified by $x_1 = 1$, $x_{n+1} = \frac{1}{2}\sqrt[3]{x_n^2 + 6}$ has limit $\alpha$.

Numerical solution of equations

Find the numerical value of $\int_0^{\frac{2\pi}{3}} \sin\left(\frac{1}{2}x\right)\,dx$.

Integration

The curve is given by $x^2 + 2y^2 + 5x + 6y = 10$. Determine the equation of the tangent to the curve at the point $(2, -1)$. Present your answer in the form $ax + by + c = 0$, with $a$, $b$ and $c$ all integers.

Differentiation

The graph of $y = 4x^2 \ln x$ contains a single stationary point.

Differentiation

The cubic polynomial $p(x)$ is specified by $p(x) = 6x^3 + ax^2 + bx + 10$, where $a$ and $b$ are constants. It is given that $(x + 2)$ is a factor of $p(x)$ and that, when $p(x)$ is divided by $(x + 1)$, the remainder is $24$.

Algebra

Prove that $\sin^2 2\theta (\cosec^2 \theta - \sec^2 \theta)$ simplifies to $4 \cos 2\theta$.

Trigonometry

Expand $\sqrt[3]{(1 - 6x)}$ as a series in ascending powers of $x$ up to and including the term in $x^3$, with the coefficients simplified.

Algebra

The population of a particular bird species in a wooded region is monitored over a number of years. At time $t$ years, the bird count is $N$, with $N$ taken to be a continuous variable. The change in the bird population is represented by $\dfrac{dN}{dt} = \dfrac{N(1800 - N)}{3600}$. It is given that $N = 300$ when $t = 0$.

Differential equations

Find $\frac{dy}{dx}$ for each of these cases:

Differentiation

The coordinates of points $A$ and $B$ are $(-1, 2, 5)$ and $(2, -2, 11)$, respectively. The plane $p$ passes through $B$ and is perpendicular to $AB$.

Vectors

The polynomial $f(x)$ is given by $f(x) = 12x^3 + 25x^2 - 4x - 12$.

Logarithmic and exponential functions

The curve defined by $6e^{2x} + ke^x + e^{2y} = c$, where $k$ and $c$ are constants, passes through the point $P$ with coordinates $(\ln 3, \ln 2)$.

Differentiation

The diagram depicts a circle with centre $O$ and radius $10$ cm. The chord $AB$ splits the circle into two parts whose areas are in the ratio $1 : 4$, and the task is to determine the length of $AB$. The angle $AOB$ is $\theta$ radians.

Numerical solution of equations

The integral $I$ is given by $I = \int_0^2 4t^3 \ln(t^2 + 1)\,dt$.

Integration

The complex number $u$ is specified as $u = \dfrac{6 - 3i}{1 + 2i}$.

Complex numbers

Prove that $\cos 4\theta + 4\cos 2\theta = 8\cos^4 \theta - 3$.

Trigonometry

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

Algebra

The diagram illustrates the curve $y = x^2 e^{-x}$.

Integration

Show that the equation $\log_2(x + 5) = 5 - \log_2 x$ may be expressed as a quadratic equation in $x$.

Logarithmic and exponential functions

Solve the equation $\cos \theta + 4\cos 2\theta = 3$, and give all of its solutions for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

The figure presents a semicircle $ACB$ with centre $O$ and radius $r$. The tangent drawn at $C$ intersects $AB$ extended at $T$. The angle $BOC$ is $x$ radians. The shaded area is the same as the area of the semicircle.

Numerical solution of equations

The parametric equations for a curve are $x = \ln(\tan t)$ and $y = \sin^2 t$, where $0 < t < \frac{1}{2}\pi$.

Differentiation

For one curve, the gradient at a point $(x, y)$ is proportional to $xy$. At the point $(1, 2)$ the gradient is 4.

Differential equations

The complex number $u$ is given by $u = \frac{5}{a + 2i}$, where $a$ is a real constant. Write $u$ in the form $x + iy$, with $x$ and $y$ both real.

Complex numbers

Express $\frac{5x - x^2}{(1 + x)(2 + x^2)}$ as partial fractions.

Algebra

The two planes are described by the equations $x + 2y - 2z = 7$ and $2x + y + 3z = 5$.

Vectors

Apply logarithms to solve the equation $5^{2x-1} = 2(3^x)$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

Taking O as the origin, the vector equations of lines $l$ and $m$ are $\mathbf{r} = 2\mathbf{i} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k})$ and $\mathbf{r} = 2\mathbf{j} + 6\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$ respectively.

Vectors

The curve $y = \frac{\ln x}{x^3}$ has a single stationary point. Determine the $x$-coordinate of this point.

Differentiation

Show that $\int_0^1 (1 - x)e^{-\frac{1}{2}x} \, dx$ is equal to $4e^{-\frac{1}{2}} - 2$.

Integration

Show that the equation $\tan(60^\circ + \theta) + \tan(60^\circ - \theta) = k$ may be rearranged into the form $(2\sqrt{3})(1 + \tan^2 \theta) = k(1 - 3\tan^2 \theta)$.

Trigonometry

The polynomial $ax^3 + bx^2 + 5x - 2$, with $a$ and $b$ as constants, is written as $p(x)$. It is known that $(2x - 1)$ is a factor of $p(x)$, and that dividing $p(x)$ by $(x - 2)$ leaves a remainder of $12$.

Algebra

By sketching a suitable pair of graphs, show that the equation $\cot x = 1 + x^2$, with $x$ measured in radians, has a single root for $0 < x < \frac{1}{2}\pi$.

Numerical solution of equations

Find the roots of $z^2 + (2\sqrt{3})z + 4 = 0$, and give the answers in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

The graph shows the curve $y = 5\sin^3 x \cos^2 x$ for $0 \leq x \leq \frac{1}{2}\pi$, together with its highest point $M$.

Integration

In this chemical reaction, compound $X$ is produced from compounds $Y$ and $Z$. After $t$ seconds from the start of the reaction, the masses in grams of $X$, $Y$ and $Z$ are $x$, $10 - x$ and $20 - x$ respectively. At every instant, the rate at which $X$ is formed is proportional to the product of the masses of $Y$ and $Z$ present at that instant. When $t = 0$, $x = 0$ and $\frac{dx}{dt} = 2$.

Differential equations

A car with mass $700\,\text{kg}$ moves along a straight horizontal road. The resistance to motion is constant at $600\,\text{N}$.

Energy, work and power

A crane lifts a load with mass $1250\,\text{kg}$ from rest on level ground to rest at a height of $1.54\,\text{m}$ above the ground. The work done in overcoming resistance to motion is $5750\,\text{J}$.

Energy, work and power

A small smooth ring $R$ with weight $8.5\,\text{N}$ is carried by a light inextensible string passing through it. The two string ends are fixed at points $A$ and $B$, with $A$ vertically above $B$. A horizontal force of magnitude $15.5\,\text{N}$ acts on $R$, and the ring is in equilibrium with angle $ARB = 90^\circ$. The segment $AR$ of the string is inclined at angle $\theta$ to the horizontal, while $BR$ is inclined at angle $\theta$ to the vertical (see diagram). The tension in the string is $T\,\text{N}$.

Forces and equilibrium

A block with mass $11\,\text{kg}$ is initially stationary on a rough plane that is inclined at $30^\circ$ to the horizontal. A force is applied to the block up the plane, along a line of greatest slope. When the force has magnitude $2X\,\text{N}$, the block is just about to move down the plane, and when the force has magnitude $9X\,\text{N}$, the block is just about to move up the plane.

Forces and equilibrium

A train leaves station $A$ at rest and moves along a straight track to station $B$, where it finally stops. For the first $600\,\text{s}$ it has constant acceleration $0.025\,\text{m s}^{-2}$, then it continues at constant speed for the next $2600\,\text{s}$, and at the end it slows down with constant deceleration $0.0375\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A particle moves along a straight line from point $P$ to point $Q$. $t$ seconds after leaving $P$, its velocity is $v\,\text{m s}^{-1}$, where $v = 4t - \frac{1}{16}t^3$. The distance $PQ$ is $64\,\text{m}$.

Kinematics of motion in a straight line

Masses $A$ and $B$, with masses $1.2\,\text{kg}$ and $2.0\,\text{kg}$ respectively, are fixed to the two ends of a light inextensible string that passes over a fixed smooth pulley. $A$ is kept at rest while $B$ hangs freely, and both vertical sections of the string are straight. When $A$ is released, it moves upwards. In the later motion, it does not reach the pulley.

Energy, work and power

A load is dragged over horizontal ground for a distance of $76\,\text{m}$ by means of a rope. The rope makes an angle of $5^\circ$ above the horizontal, and the rope tension is $65\,\text{N}$.

Energy, work and power

An object of mass $8\,\text{kg}$ moves down the line of greatest slope on an inclined plane. At the top of the plane, its initial speed is $3\,\text{m s}^{-1}$, and at the bottom its speed is $8\,\text{m s}^{-1}$. The work done against the resistance to motion of the object is $120\,\text{J}$.

Energy, work and power

The velocity-time graph shown represents a parachutist’s vertical descent. The motion has four phases: free fall with the parachute shut; slowing down at a steady rate with the parachute open; descending at constant speed with the parachute open; and stopping immediately on reaching the ground.

Forces and equilibrium

The three coplanar forces in the diagram act at a point $P$ and are balanced.

Forces and equilibrium

Particles $P$ and $Q$ are launched vertically upwards from level ground at the same moment. Their initial speeds are $12\,\text{m s}^{-1}$ and $7\,\text{m s}^{-1}$ respectively, and $t$ seconds after launch their heights above the ground are $h_P$ and $h_Q$ respectively. On the way back down to the ground, each particle comes to rest.

Kinematics of motion in a straight line

A small smooth ring $R$, with mass $0.6\,\text{kg}$, is placed on a light inextensible string of length $100\,\text{cm}$. One end of the string is fixed at point $A$. A small bead $B$ of mass $0.4\,\text{kg}$ is attached to the other end of the string, and it is threaded on a fixed rough horizontal rod that passes through $A$. The system is in equilibrium, with $B$ located $80\,\text{cm}$ from $A$.

Forces and equilibrium

A walker moves along a straight road through points $A$ and $B$, with speeds $0.9\,\text{m s}^{-1}$ and $1.3\,\text{m s}^{-1}$ at $A$ and $B$ respectively. The acceleration from $A$ to $B$ is constant, $0.004\,\text{m s}^{-2}$. A cyclist departs from $A$ at the same moment as the walker. She begins from rest and rides along the straight road, reaching $B$ at the same instant as the walker. After time $t$ seconds from leaving $A$, the cyclist’s speed is $kt^2\,\text{m s}^{-1}$, where $k$ is a constant.

Kinematics of motion in a straight line

A block is dragged across a level floor for $50\,\text{m}$ by a rope making an angle of $\alpha^\circ$ with the floor. The rope tension is $180\,\text{N}$, and the work done by the tension is $8200\,\text{J}$.

Energy, work and power

A car with mass $1250\,\text{kg}$ moves on a straight horizontal road while its engine delivers power at a steady rate of $P\,\text{W}$. The resistive force opposing the car’s motion remains constant and has magnitude $R\,\text{N}$. At a speed of $19\,\text{m s}^{-1}$, its acceleration is $0.6\,\text{m s}^{-2}$, and at a speed of $30\,\text{m s}^{-1}$, its acceleration is $0.16\,\text{m s}^{-2}$.

Forces and equilibrium

Particle $P$ is launched from the upper end of a smooth ramp with speed $u\,\text{m s}^{-1}$ and moves along the steepest descent. The ramp is $6.4\,\text{m}$ long and makes an angle of $30^\circ$ to the horizontal. At the same moment that $P$ is launched, a second particle $Q$ is let go from rest at a point $3.2\,\text{m}$ vertically above the foot of the ramp (see diagram). If $P$ and $Q$ arrive at the bottom of the ramp at the same time,

Energy, work and power

The velocity-time graphs in the diagram describe the motion of two particles $P$ and $Q$, both moving in the same direction along a straight line. $P$ and $Q$ begin from the same point $X$ on the line, although $Q$ begins $T\,\text{s}$ after $P$. For the first $20\,\text{s}$ of motion, each particle travels at speed $2.5\,\text{m s}^{-1}$. Once each has been moving for $20\,\text{s}$, its speed increases instantaneously to $4\,\text{m s}^{-1}$ and then stays at that speed. It is given that $P$ has covered $70\,\text{m}$ at the moment $Q$ begins.

Kinematics of motion in a straight line

A small block of mass $1.25\,\text{kg}$ rests on a horizontal surface. Three horizontal forces, with the magnitudes and directions shown in the diagram, act on the block. The angle $\theta$ is such that $\cos\theta = 0.28$ and $\sin\theta = 0.96$. A horizontal frictional force also acts on the block, and the block is in equilibrium.

Forces and equilibrium

A lorry with mass $15\,000\,\text{kg}$ travels up a hill that is $500\,\text{m}$ long at constant speed. The slope is at an angle of $2.5^\circ$ to the horizontal. The resistance to the lorry’s motion stays constant at $800\,\text{N}$. On the way back, the lorry arrives at the top of the hill with speed $20\,\text{m s}^{-1}$ and moves downhill with a constant driving force of $2000\,\text{N}$. The resistance to the lorry’s motion is again constant at $800\,\text{N}$.

Energy, work and power

A particle moves along a straight line from $A$ to $B$ in $20\,\text{s}$. At $t$ seconds after departing from $A$, its acceleration is $a\,\text{m s}^{-2}$, where $a = \frac{3}{160}t^2 - \frac{1}{800}t^3$. It is stated that the particle is at rest when it reaches $B$.

Kinematics of motion in a straight line

A particle is launched from level ground with speed $15\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal. Calculate how long the particle takes to strike the ground.

Representation of data

$AOB$ is a uniform lamina shaped as a quadrant of a circle, with centre $O$ and radius $0.6\,\text{m}$ (see diagram).

Representation of data

A light elastic string with natural length $1.2\,\text{m}$ and modulus of elasticity $24\,\text{N}$ is fixed between points $A$ and $B$ on a smooth horizontal plane, with $AB = 1.2\,\text{m}$. A particle $P$ is attached at the midpoint of the string. $P$ is projected at $0.5\,\text{m s}^{-1}$ across the surface in a direction at right angles to $AB$ (see diagram). $P$ is momentarily at rest when it is $0.25\,\text{m}$ from $AB$.

Representation of data

A particle $P$ leaves rest at point $O$ and moves in a straight line. The acceleration of $P$ is $(15 - 6x)\,\text{m s}^{-2}$, where $x\,\text{m}$ denotes the displacement of $P$ from $O$.

Representation of data

A uniform triangular lamina has weight $19\,\text{N}$, with $AB = 0.22\,\text{m}$ and $AC = BC = 0.61\,\text{m}$. Its plane is vertical. Point $A$ is in contact with a rough horizontal plane, and $AB$ is vertical. The lamina is kept in equilibrium by a light elastic string of natural length $0.7\,\text{m}$, which passes over a small smooth peg $P$ and is fixed to $B$ and $C$. The part of the string joined to $B$ is horizontal, whereas the part joined to $C$ is vertical (see diagram).

Probability

A particle $P$ is launched from point $O$ on level ground. After $0.4\,\text{s}$ from the instant of projection, $P$ is $5\,\text{m}$ above the ground and $12\,\text{m}$ horizontally from $O$.

Representation of data

A slender groove is made along a diameter on the top face of a horizontal disc with centre $O$. Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are placed in the groove, and the coefficient of friction between each particle and the groove is $\mu$. The particles are joined to opposite ends of a light inextensible string of length $1\,\text{m}$. The disc turns with angular velocity $\omega\,\text{rad s}^{-1}$ about a vertical axis through $O$, and the particles travel in horizontal circles (see diagram).

Representation of data

A uniform rod $AB$ has weight $16\,\text{N}$ and is hinged freely at $A$ to a fixed point. A force with magnitude $4\,\text{N}$, acting at right angles to the rod, is applied at $B$ (see diagram). If the rod is in equilibrium,

Representation of data

A uniform lamina $ABCD$ is made up of a semicircle $BCD$ with centre $O$ and diameter $0.4\,\text{m}$, together with an isosceles triangle $ABD$ whose base is $BD = 0.4\,\text{m}$ and whose perpendicular height is $h\,\text{m}$. The lamina’s centre of mass is at $O$.

Representation of data

A particle $P$ with mass $0.5\,\text{kg}$ is connected to the vertex $V$ of a fixed solid cone by a light inextensible string. $P$ is on the smooth curved surface of the cone and travels in a horizontal circle of radius $0.1\,\text{m}$, with its centre on the cone’s axis. The cone’s semi-vertical angle is $60^\circ$ (see diagram).

Probability

One end of a light elastic string whose natural length is $0.5\,\text{m}$ and whose modulus of elasticity is $12\,\text{N}$ is fastened to a fixed point $O$. The other end is joined to a particle $P$ of mass $0.24\,\text{kg}$. $P$ is projected vertically upwards with speed $3\,\text{m s}^{-1}$ from a point $0.8\,\text{m}$ vertically beneath $O$.

Probability

Particle $P$, with mass $0.4\,\text{kg}$, travels in a straight line on a horizontal surface, and its velocity at time $t\,\text{s}$ is $v\,\text{m s}^{-1}$. A horizontal force of magnitude $k\sqrt{v}\,\text{N}$ acts against the motion of $P$. When $t = 0$, $v = 9$, and when $t = 2$, $v = 4$.

Representation of data

Particle $P$ is projected from a point $O$, which is $80\,\text{m}$ vertically above horizontal ground, with speed $26\,\text{m s}^{-1}$ at an angle of $30^\circ$ below the horizontal.

Representation of data