Oct/Nov 2011

Determine the term in the expansion of $(2x + \frac{1}{x^2})^6$ that does not contain $x$.

Series

The diagram depicts the curve $y = \sqrt{1 + 2x}$, which crosses the $x$-axis at $A$ and the $y$-axis at $B$. The $y$-coordinate of point $C$ on the curve is $3$.

Integration

The functions $f$ and $g$ are given by $f: x \mapsto 2x^2 - 8x + 10$ for $0 \leq x \leq 2$, and $g: x \mapsto x$ for $0 \leq x \leq 10$.

Functions

The curve is defined by $y = 3x^3 - 6x^2 + 4x + 2$. Show that its gradient is never negative.

Differentiation

On one set of axes, sketch the graphs of $y = \cos 2\theta$ and $y = \frac{1}{2}$ for $0 \leq \theta \leq 2\pi$.

Trigonometry

A function $f$ is defined for $x \in \mathbb{R}$ and satisfies $f'(x) = 2x - 6$. Its range is $f(x) \geq -4$.

Differentiation

The diagram shows a metal plate $OABC$, made up of a sector $OAB$ of a circle with centre $O$ and radius $r$, along with a triangle $OCB$ that is right-angled at $C$. Angle $AOB = \theta$ radians, and $OC$ is perpendicular to $OA$.

Circular measure

An arithmetic progression has sixth term $23$ and total of the first ten terms $200$. Determine the seventh term.

Series

The diagram gives the garden's dimensions in metres, and the garden is L-shaped. Its perimeter is $48\text{ m}$.

Differentiation

With origin $O$, point $A$ has position vector $4\mathbf{i} + 7\mathbf{j} - p\mathbf{k}$, while point $B$ has position vector $8\mathbf{i} - \mathbf{j} - p\mathbf{k}$, where $p$ is a constant.

Coordinate geometry

The line is given by $y = kx + 6$ and the curve is given by $y = x^2 + 3x + 2k$, with $k$ a constant.

Differentiation

Find the first 3 terms when $(2 - y)^5$ is expanded in increasing powers of $y$.

Series

There are $25$ terms in an arithmetic progression, and the first term is $-15$. The total of every term in the progression is $525$. Calculate the common difference of the progression.

Series

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f: x \mapsto 3x + a$ and $g: x \mapsto b - 2x$, where $a$ and $b$ are constants. It is given that $f(2) = 10$ and $g^{-1}(2) = 3$.

Functions

Taking origin $O$, the position vectors of points $A$ and $B$ are $overrightarrow{OA} = 5\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ and $\overrightarrow{OB} = 2\mathbf{i} + 7\mathbf{j} + p\mathbf{k}$, where $p$ is a constant.

Coordinate geometry

A curve has equation $y^2 + 2x = 13$, and a line has equation $2y + x = k$, with $k$ a constant.

Coordinate geometry

On the same set of axes, sketch the graphs of $y = \sin x$ and $y = \cos 2x$ for $0^{\circ} \leq x \leq 180^{\circ}$.

Trigonometry

The diagram depicts circle $C_1$ touching circle $C_2$ at point $X$. Circle $C_1$ is centred at $A$ and has radius $6\text{ cm}$, whereas circle $C_2$ is centred at $B$ and has radius $10\text{ cm}$. Points $D$ and $E$ are on $C_1$ and $C_2$ respectively, and $DE$ is parallel to $AB$. Angle $DAX = \frac{\pi}{3}$ radians and angle $EBX = \theta$ radians.

Trigonometry

The curve satisfies $\frac{dy}{dx} = 5 - \frac{8}{x^2}$. At the point $P$ on the curve, the normal is given by the line $3y + x = 17$. The $x$-coordinate of $P$ is positive.

Coordinate geometry

The curve is given by the equation $y = \sqrt{8x - x^2}$.

Integration

The figure displays a quadrilateral $ABCD$ with $A$ at $(-1, -1)$, $B$ at $(3, 6)$ and $C$ at $(9, 4)$. The diagonals $AC$ and $BD$ meet at $M$. Angle $BMA = 90^{\circ}$ and $BM = MD$.

Coordinate geometry

For the expansion of $(k + \frac{1}{3}x)^5$, the coefficient of $x^2$ is $30$. Determine the value of the constant $k$.

Series

The diagram presents the line $y = x + 1$ together with the curve $y = \sqrt{x + 1}$, which intersect at $(-1, 0)$ and $(0, 1)$.

Integration

The first two terms of a progression are $4$ and $8$ respectively. Determine the sum of the first $10$ terms, given that the progression is

Series

The diagram displays the curve $y = 2x^5 + 3x^3$ together with the line $y = 2x$, and these meet at points $A$, $O$ and $B$.

Coordinate geometry

In the diagram, $ABCD$ forms a parallelogram, with $AB = BD = DC = 10\,\text{cm}$ and $\angle ABD = 0.8$ radians. The arcs $APD$ and $BQC$ are sections of circles centred at $B$ and $D$ respectively.

Coordinate geometry

Starting from $3\sin^2 x - 8\cos x - 7 = 0$, show that, for real values of $x$, $\cos x = \frac{2}{3}$.

Trigonometry

With origin $O$, the position vectors of points $A$ and $B$ are $3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}$ respectively.

Coordinate geometry

A straight line goes through the point $(2, 0)$ and has gradient $m$. State the equation of the line.

Differentiation

The curve $y = f(x)$ passes through the stationary point $P(3, -10)$. It is also stated that $f'(x) = 2x^2 + kx - 12$, with $k$ as a constant.

Differentiation

The functions $f$ and $g$ are given by $f : x \mapsto 2x + 3$ for $x \leq 0$, and $g : x \mapsto x^2 - 6x$ for $x \leq 3$.

Functions

Solve $|4 - 5x| < 3$.

Algebra

Show that $\displaystyle \int_{2}^{6} \frac{2}{4x + 1}\,dx = \ln\frac{5}{3}$.

Integration

The diagram gives the section of the curve $y = \frac{1}{2}\tan 2x$ for $0 \leq x \leq \frac{1}{2}\pi$.

Differentiation

Solve the equation $3^{2x} - 7(3^x) + 10 = 0$, and give your answers correct to $3$ significant figures.

Logarithmic and exponential functions

Let $p(x)$ represent the polynomial $4x^3 + ax^2 + 9x + 9$, where $a$ is a constant. When $p(x)$ is divided by $(2x - 1)$, the remainder is $10$.

Algebra

Verify by calculation that the cubic equation $x^3 - 2x^2 + 5x - 3 = 0$ has a root between $x = 0.7$ and $x = 0.8$.

Numerical solution of equations

A curve is given by the parametric equations $x = e^{3t},\ y = t^2e^t + 3$.

Differentiation

Show that $\cos 3x = 4\cos^3 x - 3\cos x$ by expanding $\cos(2x + x)$ first.

Integration

Solve the inequality $|x + 2| > \left|\tfrac{1}{2}x - 2\right|$ to find the allowed values of $x$.

Algebra

Solve $4^{x+1} = 5^{2x-3}$ using logarithms, and give your answer correct to 3 significant figures.

Logarithmic and exponential functions

The diagram presents the curve $y = x - 2\ln x$ together with its minimum point $M$.

Numerical solution of equations

Determine the exact value of the positive constant $k$ such that $\int_{0}^{k} e^{4x}\,dx = \int_{0}^{2k} e^{x}\,dx$.

Integration

By sketching an appropriate pair of graphs, show that the equation $\frac{1}{x} = \sin x$, where $x$ is measured in radians, has just one root for $0 < x \leq \tfrac{1}{2}\pi$.

Numerical solution of equations

A curve is defined by the parametric equations $x = 1 + 2\sin^2\theta$, $y = 4\tan\theta$.

Differentiation

The polynomial $ax^3 - 3x^2 - 11x + b$, with $a$ and $b$ as constants, is written as $p(x)$. It is stated that $(x + 2)$ divides $p(x)$ exactly, and that the remainder when $p(x)$ is divided by $(x + 1)$ is $12$.

Algebra

Express $5\cos\theta - 3\sin\theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$, giving the exact value of $R$ together with the value of $\alpha$ correct to 2 decimal places.

Trigonometry

Determine the gradient of the curve $y = \ln(5x + 1)$ at the point for which $x = 4$.

Differentiation

Solve for x in the inequality $|2x - 3| \leq |3x|$.

Algebra

Solve $2\ln(x + 3) - \ln x = \ln(2x - 2)$.

Logarithmic and exponential functions

Express $\cos^2 x$ using $\cos 2x$.

Integration

Solve the equation $5\sec^2 2\theta = \tan 2\theta + 9$, giving every solution in the interval $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

We write $p(x)$ for the polynomial $x^4 + ax^3 - x^2 + bx + 2$, with $a$ and $b$ fixed constants.

Algebra

The diagram depicts the curve $y = (x - 4)e^{\frac{1}{2}x}$. At the point $P$, the curve’s gradient is $3$.

Numerical solution of equations

A curve is defined by $2x^2 - 3x - 3y + y^2 = 6$.

Differentiation

By making the substitution $u = e^x$, or by another method, solve the equation

Logarithmic and exponential functions

Showing your working, determine the two square roots of the complex number $1 - 2\sqrt{6}i$. Present your answers in the form $x + iy$, with $x$ and $y$ exact.

Complex numbers

A curve has the following parametric equations

Differentiation

Let $p(x)$ stand for the polynomial $x^4 + 3x^3 + ax + 3$. It is stated that $p(x)$ is divisible by $x^2 - x + 1$.

Algebra

A differential equation connects $x$ and $\theta$, with $0 < \theta < \frac{1}{2}\pi$; moreover, when $\theta = \frac{1}{12}\pi$, the value of $x$ is $0$.

Differential equations

By drawing a suitable pair of graphs, demonstrate that the equation $\sec x = 3 - \frac{1}{2}x^2$, where $x$ is measured in radians, has a root in the interval $0 < x < \frac{1}{2}\pi$.

Numerical solution of equations

Write $\cos x + 3\sin x$ in the form $R\cos(x - \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$, and give the exact value of $R$ together with $\alpha$ correct to $2$ decimal places.

Trigonometry

Relative to the origin $O$, the position vectors of points $A$ and $B$ are $\overrightarrow{OA} = i + 2j + 2k$ and $\overrightarrow{OB} = 3i + 4j$. Point $P$ is on the straight line through $A$ and $B$, with $\overrightarrow{AP} = \lambda \overrightarrow{AB}$.

Vectors

Take $f(x) = \frac{12 + 8x - x^2}{(2 - x)(4 + x^2)}$.

Integration

The sketch displays the curve $y = x^2 \ln x$ together with its minimum point $M$.

Integration

By making the substitution $u = e^x$, or by another method,

Logarithmic and exponential functions

With your working shown, determine the two square roots of the complex number $1 - (2\sqrt{6})i$. Write each answer in the form $x + iy$, with $x$ and $y$ exact.

Complex numbers

The parametric equations for the curve are $x = 3(1 + \sin^2 t)$, $y = 2\cos^3 t$.

Differentiation

The polynomial $x^4 + 3x^3 + ax + 3$ is represented by $p(x)$. It is stated that $p(x)$ is divisible by $x^2 - x + 1$.

Algebra

A differential equation links $x$ and $\theta$ as $\sin 2\theta \, \frac{dx}{d\theta} = (x + 1)\cos 2\theta$, with $0 < \theta < \frac{1}{2}\pi$. The condition $\theta = \frac{1}{12}\pi$ gives $x = 0$.

Differential equations

By sketching a suitable pair of graphs, show that the equation $\sec x = 3 - \tfrac{1}{2}x^2$, where $x$ is in radians, has a root in the interval $0 < x < \tfrac{1}{2}\pi$.

Numerical solution of equations

Write $\cos x + 3\sin x$ in the form $R\cos(x - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$, and give the exact value of $R$ together with the value of $\alpha$ correct to 2 decimal places.

Trigonometry

Relative to the origin $O$, the position vectors of points $A$ and $B$ are $overrightarrow{OA} = i + 2j + 2k$ and $overrightarrow{OB} = 3i + 4j$. Point $P$ is on the straight line through $A$ and $B$, and $overrightarrow{AP} = \lambda \overrightarrow{AB}$.

Vectors

Define $f(x)$ by $\dfrac{12 + 8x - x^2}{(2 - x)(4 + x^2)}$.

Integration

The diagram depicts the curve $y = x^2 \ln x$ together with its minimum point $M$.

Integration

Write the expansion of $\frac{16}{(2 + x)^2}$ in ascending powers of $x$, including terms up to and including $x^2$, and simplify the coefficients.

Algebra

Use the substitution $u = \tan x$ to demonstrate that, for $n \neq -1$, $\int_0^{\frac{\pi}{4}} (\tan^{n+2}x + \tan^n x)\,dx = \frac{1}{n + 1}$.

Integration

A curve is given by $y = \frac{e^{2x}}{1 + e^{2x}}$.

Differentiation

Express $8\cos\theta + 15\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. State $\alpha$ correct to 2 decimal places.

Trigonometry

In the experiment, the number of organisms at time $t$ days is represented by $N$, with $N$ taken to be a continuous variable. It is given that $\frac{dN}{dt} = 1.2e^{-0.02t}N^{0.5}$. At $t = 0$, the number of organisms present is 100.

Differential equations

You are told that $\int_1^a x \ln x\,dx = 22$, with $a$ a constant larger than 1.

Numerical solution of equations

The complex number $w$ is given by $w = -1 + i$.

Complex numbers

The polynomial $p(x)$ is given by $p(x) = ax^3 - x^2 + 4x - a$, where $a$ is a constant. You are told that $(2x - 1)$ divides $p(x)$ exactly.

Algebra

The diagram depicts the curve defined by the parametric equations $x = \sin t + \cos t$, $y = \sin^3 t + \cos^3 t$, for $\frac{\pi}{4} < t < \frac{5\pi}{4}$.

Differentiation

The line $l$ is given by $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix}$, with $a$ as a constant. The plane $p$ is defined by $2x - 2y + z = 10$.

Vectors

A block is fastened to one end of a light inextensible string. The string pulls the block along a horizontal surface at a speed of $2\,\text{m s}^{-1}$. It is inclined at $20^\circ$ to the horizontal, and the tension is $25\,\text{N}$.

Energy, work and power

Particles $A$ and $B$, with masses $0.65\,\text{kg}$ and $0.35\,\text{kg}$ respectively, are fixed to the ends of a light inextensible string that passes over a fixed smooth pulley. $B$ is kept at rest while the string is taut and both straight sections are vertical. The system is released from rest, after which the particles move vertically.

Forces and equilibrium

At point $A$, three coplanar forces with magnitudes $15\,\text{N}$, $12\,\text{N}$ and $12\,\text{N}$ act in the directions indicated in the diagram.

Forces and equilibrium

$A$, $B$ and $C$ are three points on the line of greatest slope of a smooth plane that is inclined at an angle of $\theta^\circ$ to the horizontal. Point $A$ lies above $B$ and $B$ lies above $C$, while the distances $AB$ and $BC$ are $1.76\,\text{m}$ and $2.16\,\text{m}$ respectively. A particle slides down the plane with constant acceleration $a\,\text{m s}^{-2}$. The particle’s speed at $A$ is $u\,\text{m s}^{-1}$. It takes $0.8\,\text{s}$ for the particle to move from $A$ to $B$ and $1.4\,\text{s}$ to move from $A$ to $C$.

Kinematics of motion in a straight line

A block of mass $2\,\text{kg}$ is initially at rest on a horizontal floor. The coefficient of friction between the block and the floor is $\mu$. A force of magnitude $12\,\text{N}$ is applied to the block at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac{3}{4}$. With the applied force directed downwards as in Fig. 1, the block stays at rest.

Forces and equilibrium

$AB$ and $BC$ are straight roads with inclinations of $5^\circ$ to the horizontal and $1^\circ$ to the horizontal respectively. $A$ and $C$ lie at the same horizontal height, while $B$ is $45\,\text{m}$ above the level of $A$ and $C$. A car with mass $1200\,\text{kg}$ moves from $A$ to $C$ via $B$.

Energy, work and power

Particle $P$ departs from point $O$ and travels along a straight line. $P$’s velocity $t\,\text{s}$ after leaving $O$ is $v\,\text{m s}^{-1}$, where $v = 0.16t^{3} - 0.016t^{2}$. $P$ is momentarily at rest at point $A$.

Kinematics of motion in a straight line

A racing cyclist, with a combined mass together with his cycle of $75\,\text{kg}$, produces power at a rate of $720\,\text{W}$ while travelling along a straight horizontal road. The resistive force opposing the cyclist’s motion is constant and has magnitude $R\,\text{N}$.

Newton's laws of motion

A block with mass $6\,\text{kg}$ is moving down the line of greatest slope on a plane that is inclined at $8^{\circ}$ to the horizontal. The coefficient of friction between the block and the plane is $0.2$.

Kinematics of motion in a straight line

A particle $P$ travels along a straight line. It begins at point $O$ on the line with velocity $1.8\,\text{m s}^{-1}$. At time $t\,\text{s}$ after leaving $O$, the acceleration of $P$ is $0.8t^{-0.75}\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A particle $P$ weighs $10\,\text{N}$ and is in limiting equilibrium on a rough horizontal table. The forces drawn in the diagram are the weight of $P$, an applied force of magnitude $4\,\text{N}$ acting on $P$ at $30^{\circ}$ above the horizontal, and the contact force from the table on $P$ (the resultant of the frictional and normal components) with magnitude $C\,\text{N}$.

Forces and equilibrium

Particles $A$ and $B$, with masses $0.9\,\text{kg}$ and $0.6\,\text{kg}$ respectively, are joined by the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is let go from rest while the string is taut, the two straight sections are vertical, and the particles are at the same level above the horizontal floor. In the motion that follows, $B$ does not reach the pulley.

Newton's laws of motion

A lorry with mass $16\,000\,\text{kg}$ travels up the straight hill $ABCD$, which is inclined at angle $\theta$ to the horizontal and satisfies $\sin\theta = \frac{1}{25}$. While moving from $A$ to $B$, the driving force of the lorry does $1200\,\text{kJ}$ of work, and the resistive force remains constant at $1240\,\text{N}$. Its speed is $15\,\text{m s}^{-1}$ at $A$ and $12\,\text{m s}^{-1}$ at $B$.

Energy, work and power

A tractor moves along a straight path from point $A$ to point $B$. At time $t\,\text{s}$ after departing from $A$, its velocity is $v\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A woman travels along a straight line. Her velocity $t$ seconds after she passes a fixed point $A$ on the line is $v\,\text{m s}^{-1}$. The graph of $v$ against $t$ is made up of 4 straight-line segments (see diagram). The woman is at point $B$ when $t = 60$. Find

Kinematics of motion in a straight line

Three coplanar forces with magnitudes $58\,\text{N}$, $31\,\text{N}$ and $26\,\text{N}$ are applied at a point in the directions indicated in the diagram. Since $\tan \alpha = \frac{5}{12}$, determine

Forces and equilibrium

Particles $P$ and $Q$ are connected at the two ends of a light inextensible string that passes over a fixed smooth pulley. The system is released from rest, with the string taut, the straight sections vertical, and both particles $2\,\text{m}$ above horizontal ground. $P$ moves vertically downwards and does not rebound when it reaches the ground. At the moment that $P$ reaches the ground, $Q$ is at point $X$, from where it continues to move vertically upwards without reaching the pulley. Given that $P$ has mass $0.9\,\text{kg}$ and that the tension in the string is $7.2\,\text{N}$ while $P$ is moving, find

Kinematics of motion in a straight line

$ABC$ is a vertical cross-section of a surface. The section of the surface that contains $AB$ is smooth, and $A$ lies $4\,\text{m}$ above $B$. The section of the surface that contains $BC$ is horizontal, and $BC$ has length $5\,\text{m}$ (see diagram). A particle of mass $0.8\,\text{kg}$ is released from rest at $A$ and moves along $ABC$. Find the speed of the particle at $C$ in each of the following cases.

Forces and equilibrium

A particle $P$ travels along a straight line. It is at rest at $A$ when it sets off and it is brought to rest instantaneously at $B$. The velocity of $P$ after $t$ seconds from leaving $A$ is $v\,\text{m s}^{-1}$, where $v = 6t^2 - kt^3$ and $k$ is a constant.

Kinematics of motion in a straight line

The diagram depicts a ring of mass $2\,\text{kg}$ threaded onto a fixed rough vertical rod. A light string is fastened to the ring and is pulled upwards at an angle of $30^\circ$ to the horizontal. The tension in the string is $T\,\text{N}$. The coefficient of friction between the ring and the rod is $0.24$. Find

Forces and equilibrium

A car with mass $600\,\text{kg}$ is moving on a straight horizontal road, having set off from point $A$. The force resisting its motion is $750\,\text{N}$.

Energy, work and power

Rod $AB$ is non-uniform, with length $0.6\,\text{m}$ and weight $9\,\text{N}$, and its centre of mass is $0.4\,\text{m}$ from $A$. The rod’s end $A$ touches a rough vertical wall. It remains in equilibrium, at right angles to the wall, because of a light string fixed to $B$. The string is inclined at $30^{\circ}$ to the horizontal. The tension in the string is $T\,\text{N}$ (see diagram).

Probability

A particle $P$ is fired from point $O$ at $60^{\circ}$ to the horizontal ground. After $0.6\,\text{s}$, the angle of elevation of $P$ from $O$ is $45^{\circ}$ (see diagram).

Representation of data

A light elastic string with natural length $0.4\,\text{m}$ and modulus of elasticity $20\,\text{N}$ has one end fixed at point $O$. Its other end is connected to a particle $P$ of mass $0.25\,\text{kg}$. The particle $P$ hangs at rest in equilibrium below $O$.

Probability

A uniform solid cylinder has radius $0.7\,\text{m}$ and height $h\,\text{m}$. A uniform solid cone has base radius $0.7\,\text{m}$ and height $2.4\,\text{m}$. Both the cylinder and the cone are in equilibrium, each resting with one circular face on a horizontal plane. The plane is then tilted, and its angle of inclination to the horizontal, $\theta^{\circ}$, is increased little by little until the cone is just on the point of toppling.

Representation of data

A ball with mass $0.05\,\text{kg}$ is let go from rest from a point $h\,\text{m}$ above the ground. At time $t\,\text{s}$ after release, its downward velocity is $v\,\text{m s}^{-1}$. A resistive force from the air acts opposite to the motion, with magnitude $0.01v\,\text{N}$.

Representation of data

A smooth bead $B$ of mass $0.3\,\text{kg}$ is fixed to a light inextensible string of length $0.9\,\text{m}$. One end of the string is fastened at the fixed point $A$, while the other end is fastened at the fixed point $C$, which lies vertically beneath $A$. The string has tension $T\,\text{N}$, and the bead moves with angular speed $\omega\,\text{rad s}^{-1}$ in a horizontal circle about the vertical axis through $A$ and $C$.

Representation of data

Rod $AB$, with non-uniform density, has length $0.6\text{ m}$ and weight $9\text{ N}$. Its centre of mass lies $0.4\text{ m}$ from $A$. End $A$ of the rod is touching a rough vertical wall. The rod is in equilibrium, perpendicular to the wall, because a light string is fixed to $B$. The string is at an angle of $30^\circ$ to the horizontal. The tension in the string is $T\text{ N}$ (see diagram).

Probability

From point $O$, particle $P$ is projected at an angle of $60^\circ$ above the horizontal ground. After $0.6\text{ s}$, the angle of elevation of $P$ from $O$ is $45^\circ$ (see diagram).

Representation of data

A light elastic string has natural length $0.4\text{ m}$ and modulus of elasticity $20\text{ N}$; one end is fixed at point $O$, and the other end is connected to a particle $P$ of mass $0.25\text{ kg}$. $P$ hangs in equilibrium beneath $O$.

Probability

A uniform solid cylinder has radius $0.7\text{ m}$ and height $h\text{ m}$. A uniform solid cone has base radius $0.7\text{ m}$ and height $2.4\text{ m}$. Both the cylinder and the cone are in equilibrium, with a circular face resting on a horizontal plane. The plane is then inclined from the horizontal, and the angle of inclination, $\theta^\circ$, is increased slowly until the cone is just on the point of toppling.

Probability

A ball with mass $0.05\text{ kg}$ is let go from rest at a height $h\text{ m}$ above the ground. After $t\text{ s}$ from release, its downward velocity is $v\text{ m s}^{-1}$. The air resistance acts opposite to the motion with a force of magnitude $0.01v\text{ N}$.

Probability

A bead $B$ with a smooth surface and mass $0.3\text{ kg}$ is threaded onto a light inextensible string of length $0.9\text{ m}$. One end of the string is fastened to a fixed point $A$, while the other end is fastened to a fixed point $C$ directly beneath $A$. The string has tension $T\text{ N}$, and the bead moves with angular speed $\omega\text{ rad s}^{-1}$ in a horizontal circle about the vertical axis through $A$ and $C$.

Probability

A particle is fired from a point on horizontal ground with speed $17\text{ m s}^{-1}$ at an angle of $50^\circ$ above the horizontal.

Representation of data

The object is formed from two identical uniform rods $AB$ and $BC$, each measuring $0.6\text{ m}$ in length and weighing $7\text{ N}$. The rods are joined rigidly at $B$ and $angle ABC = 90^\circ$.

Representation of data