May/June 2014

The diagram displays a section of the graph of $y = a + b \sin x$.

Trigonometry

The diagram represents the function $f$ on $-1 \leq x \leq 4$, where $$f(x) = \begin{cases}3x - 2 & \text{for } -1 \leq x \leq 1, \\ \frac{4}{5 - x} & \text{for } 1 < x \leq 4.\end{cases}$$

Functions

A line is given by $y = 2x + c$ and a curve by $y = 8 - 2x - x^2$. When the line is tangent to the curve, determine the constant $c$.

Integration

A curve is defined by $\frac{dy}{dx} = x^2 - x^{\frac{1}{2}}$. The curve goes through the point $(4, \frac{2}{3})$. Find the equation of the curve.

Differentiation

Express $4x^2 - 12x$ in the shape $(2x + a)^2 + b$.

Quadratics

Determine the term independent of $x$ in the expansion of $(4x^3 + \frac{1}{2x})^8$.

Series

A curve is defined by $y = \frac{4}{(3x + 1)^2}$. Find the equation of the tangent to the curve at the point where the line $x = -1$ meets it.

Differentiation

For an arithmetic progression with first term $a$ and common difference $d$, it is stated that the sum of the first $200$ terms is $4$ times the sum of the first $100$ terms.

Series

The diagram represents triangle $ABC$, with $AB$ perpendicular to $BC$. $AB$ measures $4 \text{ cm}$ and angle $CAB$ is $\alpha$ radians. The arc $DE$ has centre $A$ and radius $2 \text{ cm}$, and it cuts $AC$ at $D$ and $AB$ at $E$.

Circular measure

The points $A$ and $B$ have coordinates $(a, 2)$ and $(3, b)$ respectively, where $a$ and $b$ are constants. The distance $AB$ is $\sqrt{125}$ units and the gradient of the line $AB$ is $2$. Determine the possible values of $a$ and $b$.

Coordinate geometry

Using origin $O$ as the reference point, the position vectors for points $A$ and $B$ are $\overrightarrow{OA} = \begin{pmatrix}3p\\4\\p^2\end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix}-p\\-1\\p^2\end{pmatrix}$.

Coordinate geometry

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

Trigonometry

Determine the coordinates of the point where the perpendicular bisector of the line joining $(2, 7)$ to $(10, 3)$ crosses the $x$-axis.

Coordinate geometry

Functions $f$ and $g$ are given by $f : x \mapsto 2x - 3$, $x \in \mathbb{R}$, and $g : x \mapsto x^2 + 4x$, $x \in \mathbb{R}$. The function $h$ is given by $h : x \mapsto x^2 + 4x$ for $x \geq k$, and it is stated that $h$ is invertible.

Functions

Find the coefficient of $x^2$ when $(1 + x^2)\left(\frac{x}{2} - \frac{4}{x}\right)^6$ is expanded.

Series

The reflex angle $\theta$ satisfies $\cos\theta = k$, where $0 \le k \le 1$.

Trigonometry

The diagram represents a sector of a circle with centre O and radius r cm. The chord AB splits the sector into triangle AOB and segment AXB. Angle AOB measures \theta radians.

Circular measure

Prove that $\frac{1}{\cos\theta} - \frac{\cos\theta}{1 + \sin\theta} \equiv \tan\theta$.

Trigonometry

The 1st, 2nd and 3rd terms of a geometric progression are, in order, the 1st, 9th and 21st terms of an arithmetic progression. The 1st term of each progression is $8$, and the common ratio of the geometric progression is $r$, where $r \neq 1$.

Series

The diagram represents trapezium $ABCD$, with $BA$ parallel to $CD$. Relative to an origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}3\\4\\0\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}1\\3\\2\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}4\\5\\6\end{pmatrix}$.

Coordinate geometry

A curve is defined by $\frac{d^2y}{dx^2} = 2x - 1$. It has a minimum point at $(3, -10)$. Find the coordinates of the maximum point.

Differentiation

The diagram displays part of the curve $y = 8 - \sqrt{(4 - x)}$ together with the tangent to the curve at $P\,(3, 7)$.

Integration

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

Series

The curve $y = -x^2 + 12x - 20$ and the line $y = 2x + 1$ are shown in the diagram.

Integration

The diagram illustrates a parallelogram $ABCD$, in which the equation for $AB$ is $y = 3x$ and the equation for $AD$ is $4y = x + 11$. The diagonals $AC$ and $BD$ intersect at the point $E\left(6\frac{1}{2}, 8\frac{1}{2}\right)$.

Coordinate geometry

The progression starts with $36$ as the first term and $32$ as the second term.

Series

The diagram depicts a part of a circle with centre $O$ and radius $6\text{ cm}$. The chord $AB$ is arranged so that angle $AOB = 2.2$ radians.

Circular measure

Prove that $\frac{\tan x + 1}{\sin x \tan x + \cos x} \equiv \sin x + \cos x$.

Trigonometry

The function $f$ is given by $f(x)=\frac{15}{2x+3}$ for $0 \le x \le 6$.

Functions

Consider a curve for which $\frac{dy}{dx} = \frac{12}{\sqrt{4x + a}}$, where $a$ is constant. Point $P(2, 14)$ lies on the curve, and the normal at $P$ has equation $3y + x = 5$.

Integration

The position vectors of points $A$, $B$ and $C$ with respect to an origin $O$ are $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 6 \\ -1 \\ 7 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 2 \\ 4 \\ 7 \end{pmatrix}$.

Trigonometry

Write $2x^2 - 10x + 8$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants, and use your result to state the least value of $2x^2 - 10x + 8$.

Quadratics

The base of a cuboid has side lengths $x\text{ cm}$ and $3x\text{ cm}$. Its volume is $288\text{ cm}^3$.

Differentiation

Solve $|3x - 2| \ge |x + 4|$.

Algebra

Find the gradient of each curve below at the point where $x = 0$.

Differentiation

Determine the quotient when $6x^4 - x^3 - 26x^2 + 4x + 15$ is divided by $(x^2 - 4)$, and check that the remainder is $7$.

Algebra

By sketching an appropriate pair of graphs, demonstrate that the equation $3\ln x = 15 - x^3$ has exactly one real root.

Numerical solution of equations

Prove that $\tan \theta + \cot \theta$ equals $\frac{2}{\sin 2\theta}$.

Trigonometry

Show that $\int_6^{16} \frac{6}{2x - 7} \, dx$ equals $\ln 125$.

Integration

The equation for a curve is $2x^2 + 3xy + y^2 = 3$.

Differentiation

Solve $|x + 2| = |x - 13|$.

Logarithmic and exponential functions

Solve $3\sin 2\theta \tan \theta = 2$ in the interval $0^\circ < \theta < 180^\circ$.

Trigonometry

Find the value of $\int 4\cos\left(\frac{1}{3}x + 2\right)\,dx$.

Integration

A curve is given by the parametric equations $x = 2\ln(t + 1)$ and $y = 4e^t$.

Differentiation

The variables $x$ and $y$ are related by the equation $y = K(2^{px})$, where $K$ and $p$ are constants. The graph of $\ln y$ plotted against $x$ is a straight line that goes through $(1.35, 1.87)$ and $(3.35, 3.81)$, as shown in the diagram.

Numerical solution of equations

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

Algebra

It is stated that $\int_0^a \left(\frac{1}{2}e^{3x} + x^2\right)\,dx = 10$, where $a$ is a positive constant.

Numerical solution of equations

The diagram depicts the curve $y = \tan x \cos 2x$, for $0 \le x \le \frac{\pi}{2}$, together with its maximum point $M$.

Differentiation

Solve $|x + 2| = |x - 13|$.

Logarithmic and exponential functions

Determine the solution to $3 \sin 2\theta \tan \theta = 2$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

Determine $\int 4 \cos \left( \frac{1}{3}x + 2 \right) \, dx$.

Integration

The curve is described parametrically by $x = 2 \ln (t + 1)$ and $y = 4e^t$.

Differentiation

The variables $x$ and $y$ are related by $y = K 2^{px}$, where $K$ and $p$ are constants. The plot of $\ln y$ against $x$ forms a straight line that goes through the points $(1.35, 1.87)$ and $(3.35, 3.81)$, as the diagram shows.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = x^3 + 2x + a$, where $a$ is a constant.

Algebra

You are told that $\int_{0}^{a} \left( \frac{1}{2} e^{3x} + x^2 \right) \, dx = 10$, with $a$ a positive constant.

Numerical solution of equations

The diagram illustrates the curve $y = \tan x \cos 2x$, for $0 \leq x < \frac{1}{2}\pi$, together with its maximum point $M$.

Differentiation

Simplify $\sin 2\alpha \sec \alpha$.

Trigonometry

The diagram displays the curve $y = 10e^{-\frac{1}{4}x} \sin 4x$ for $x \geq 0$. The stationary points are marked $T_1$, $T_2$, $T_3, \ldots$ as illustrated.

Numerical solution of equations

Apply the substitution $u = 1 + 3\tan x$ to determine the exact value of

Integration

A curve is represented by the parametric equations $x = \ln(2t + 3)$, $y = \frac{3t + 2}{2t + 3}$.

Differentiation

The variables $x$ and $y$ satisfy the differential equation $\frac{dy}{dx} = \frac{6y e^{3x}}{2 + e^{3x}}$.

Differential equations

The complex number $z$ is specified by $z = \frac{9\sqrt{3} + 9i}{\sqrt{3} - i}$. Find, showing all your working,

Complex numbers

You are told that $2\ln(4x - 5) + \ln(x + 1) = 3\ln 3$.

Logarithmic and exponential functions

The line $l$ is represented by ${\bf r} = 4\mathbf{i} - \mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k})$. The plane $p$ goes through the point $(4, -1, 2)$ and is at right angles to $l$.

Vectors

By drawing the graphs of $y = \cosec x$ and $y = x(\pi - x)$ for $0 < x < \pi$, demonstrate that the equation $\cosec x = x(\pi - x)$ has exactly two real solutions in the interval $0 < x < \pi$.

Numerical solution of equations

Write $\frac{4 + 12x + x^2}{(3 - x)(1 + 2x)^2}$ in partial fractions.

Algebra

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

Algebra

Using the origin $O$ as the reference point, the position vectors of the points $A$, $B$ and $C$ are $ \overrightarrow{OA} = i + 2j + 3k$, $\overrightarrow{OB} = 2i + 4j + k$ and $\overrightarrow{OC} = 3i + 5j - 3k$.

Vectors

Solve the equation $2\ln(5 - e^{-2x}) = 1$, and give your answer correct to 3 significant figures.

Logarithmic and exponential functions

Find the solutions of $\cos(x + 30^\circ) = 2\cos x$, with all answers taken from the interval $-180^\circ < x < 180^\circ$.

Trigonometry

The curve is described parametrically by $x = t - \tan t$ and $y = \ln(\cos t)$, with $-\frac{1}{2}\pi < t < \frac{1}{2}\pi$.

Differentiation

The polynomial $f(x)$ can be expressed in the form $(x - 2)^2 g(x)$, where $g(x)$ is another polynomial. Show that $(x - 2)$ is a factor of $f'(x)$.

Differentiation

In the diagram, $A$ lies on the circumference of a circle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts the circumference at $B$ and $C$. The angle $OAB$ measures $x$ radians. The shaded region is enclosed by $AB$, $AC$ and the circular arc with centre $A$ joining $B$ to $C$. The perimeter of the shaded region is half the circumference of the circle.

Numerical solution of equations

The equation $z^3 + 2z + a = 0$, where $a$ is real, has root $-1 + (\sqrt{5})i$. Show your working to find $a$, and then write down the other complex root of this equation.

Complex numbers

The curve $y = x\cos\frac{1}{2}x$ is shown in the diagram for $0 \le x \le \pi$.

Integration

At time $t$ years, the population of a country is $N$ millions. It is assumed that, at any instant, the rate at which $N$ increases is proportional to the product of $N$ and $(1 - 0.01N)$. When $t = 0$, $N = 20$ and $\frac{dN}{dt} = 0.32$.

Differential equations

Solve $\log_{10}(x + 9) = 2 + \log_{10} x$.

Logarithmic and exponential functions

The line $l$ is defined by ${\bf r} = {\bf i} + 2{\bf j} - {\bf k} + \lambda(3{\bf i} - 2{\bf j} + 2{\bf k})$ and the plane $p$ is given by $2x + 3y - 5z = 18$.

Vectors

Expand $(1 + 3x)^{-\frac{1}{3}}$ in ascending powers of $x$ up to and including the $x^3$ term, with the coefficients simplified.

Algebra

Demonstrate that the equation $\tan(x - 60^\circ) + \cot x = \sqrt{3}$ can be rearranged into the form $2\tan^2 x + (\sqrt{3})\tan x - 1 = 0$.

Trigonometry

The equation $x = \frac{10}{e^{2x} - 1}$ has a single positive real root, called $\alpha$.

Numerical solution of equations

The variables $x$ and $\theta$ obey the differential equation $2\cos^2\theta\,\frac{dx}{d\theta} = \sqrt{2x + 1}$, and $x$ takes the value 0 when $\theta = \frac{1}{4}\pi$.

Differential equations

The diagram depicts the curve $(x^2 + y^2)^2 = 2(x^2 - y^2)$ together with one of its highest points $M$.

Differentiation

The complex number $\frac{3 - 5i}{1 + 4i}$ is labelled $u$. Show your working and express $u$ in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Suppose $f(x) = \frac{6 + 6x}{(2 - x)(2 + x^2)}$.

Integration

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

Integration

A train is travelling at a constant speed $V\,\text{m s}^{-1}$ on a horizontal straight track. The engine power is $1330\,\text{kW}$, and the total resistance opposing the train’s motion is $28\,\text{kN}$.

Kinematics of motion in a straight line

A rough plane is set at an angle of $\alpha^{\circ}$ to the horizontal. A particle with mass $0.25\,\text{kg}$ remains in equilibrium on the plane. The normal reaction on the particle has magnitude $2.4\,\text{N}$.

Forces and equilibrium

At one point, four coplanar forces are applied. Their magnitudes are $5\,\text{N}$, $4\,\text{N}$, $3\,\text{N}$ and $7\,\text{N}$, and the directions in which they act are indicated in the diagram.

Forces and equilibrium

A particle is launched vertically upward with speed $9\,\text{m s}^{-1}$ from a point $3.15\,\text{m}$ above horizontal ground. It then travels freely under gravity until it reaches the ground.

Kinematics of motion in a straight line

A car with mass $1100\,\text{kg}$ leaves O from rest and travels along the road OAB. OA is a straight stretch of length $1760\,\text{m}$, and it slopes to the horizontal, with A $160\,\text{m}$ above the level of O. AB is a straight horizontal stretch. As the car moves, the driving force is $1800\,\text{N}$ and the resistance to the car’s motion is $700\,\text{N}$. When the car has covered $x\,\text{m}$ from O, its speed is $v\,\text{m s}^{-1}$.

Energy, work and power

Particles $A$ with mass $0.25\,\text{kg}$ and $B$ with mass $0.75\,\text{kg}$ are fixed to the two ends of a light inextensible string that runs over a fixed smooth pulley. The arrangement is kept at rest with the string taut and its straight sections vertical. Each particle is initially at a height of $h\,\text{m}$ above the floor. The system is released from rest, and $0.6\,\text{s}$ later, while both particles are moving, the string snaps. In the later motion, particle $A$ does not reach the pulley.

Newton's laws of motion

Cyclists $P$ and $Q$ move along the straight road $ABC$, setting off together from $A$ and reaching $C$ at the same time. They each pass through $B$ $400\,\text{s}$ after leaving $A$. Cyclist $P$ begins with speed $3\,\text{m s}^{-1}$ and then increases this speed with constant acceleration $0.005\,\text{m s}^{-2}$ until he arrives at $B$.

Kinematics of motion in a straight line

A car with mass $600\text{ kg}$ is moving on a straight, level road. The resistive force opposing the car’s motion remains constant at $R\text{ N}$.

Energy, work and power

Points $A$ and $B$ are $10\,\text{m}$ apart on one horizontal plane. Particle $P$ begins from rest at $A$ and travels straight towards $B$ with constant acceleration $0.5\,\text{m s}^{-2}$. A second particle $Q$ moves straight towards $A$ at constant speed $0.75\,\text{m s}^{-1}$, and it passes through $B$ at the moment that $P$ starts to move. $T$ s after this moment, particles $P$ and $Q$ collide.

Momentum

A and $B$ are two fixed points on a vertical wall, with $A$ directly above $B$. A particle $P$ of mass $0.7\text{ kg}$ is connected to $A$ by a light inextensible string of length $3\text{ m}$. $P$ is also connected to $B$ by a light inextensible string of length $2.5\text{ m}$. $P$ is kept in equilibrium at a distance $2.4\text{ m}$ from the wall by a horizontal force of magnitude $10\text{ N}$ acting on $P$ (see diagram). The two strings are taut, and the $10\text{ N}$ force acts in the plane $APB$, which is perpendicular to the wall.

Forces and equilibrium

Particle $P$ travels along a straight line, beginning from rest at point $O$ on the line. Let $t$ s denote the time since $P$ began moving. The particle then moves along the line with constant acceleration $\frac{1}{4}\,\text{m s}^{-2}$ until it goes through point $A$ when $t = 8$. Once it has passed through $A$, the velocity of $P$ is $\frac{1}{2}t^{3/4}\,\text{m s}^{-1}.

Kinematics of motion in a straight line

A light inextensible rope has block $A$, of mass $5\text{ kg}$, at one end and block $B$, of mass $16\text{ kg}$, at the other. The rope runs over a smooth pulley fixed at the top of a rough plane inclined at an angle of $30^{\circ}$ to the horizontal. Block $A$ is kept at rest at the bottom of the plane, while block $B$ hangs below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $\frac{1}{\sqrt{3}}$. Block $A$ is released from rest and the system begins to move. After each block has moved a distance of $x\text{ m}$, each has speed $v\,\text{m s}^{-1}$.

Energy, work and power

A particle $P$ with mass $0.2\text{ kg}$ is released from rest at a point $7.2\text{ m}$ above the liquid surface in a container. $P$ drops through the air and then enters the liquid. Air resistance is absent and there is no sudden change in speed as $P$ enters the liquid. When $P$ is $0.8\text{ m}$ below the surface of the liquid, $P$’s speed is $6\,\text{m s}^{-1}$. The only force on $P$ from the liquid is a constant resistance to motion of magnitude $R\text{ N}$. The liquid has depth $3.6\text{ m}$ in the container. $P$ is removed from the container and attached to one end of a light inextensible string. $P$ is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of $R\text{ N}$ continues to act. The particle reaches the surface $4\text{ s}$ after leaving the bottom of the container.

Newton's laws of motion

A light inextensible string, of length $5.28\text{ m}$, has particles $A$ and $B$ attached to its ends, with masses $0.25\text{ kg}$ and $0.75\text{ kg}$ respectively. A further particle $P$, of mass $0.5\text{ kg}$, is fixed at the string’s mid-point. Two smooth pulleys $P_1$ and $P_2$ are mounted at opposite ends of a rough horizontal table of length $4\text{ m}$ and height $1\text{ m}$. The string runs over $P_1$ and $P_2$, with particle $A$ kept at rest vertically under $P_1$, the string taut, and $B$ hanging freely below $P_2$. Particle $P$ is touching the table halfway between $P_1$ and $P_2$ (see diagram). The coefficient of friction between $P$ and the table is $0.4$. Particle $A$ is released and the system begins to move with constant acceleration of magnitude $a\,\text{m s}^{-2}$. The tension in the part $AP$ of the string is $T_A\text{ N}$ and the tension in the part $PB$ of the string is $T_B\text{ N}$.

Newton's laws of motion

Block $B$, with mass $7\text{ kg}$, is initially at rest on rough horizontal ground. A force of size $X\text{ N}$ acts on $B$ at an angle of $15^\circ$ to the upward vertical (see diagram).

Forces and equilibrium

A car with mass $1250\text{ kg}$ moves up a straight hill at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.02$. The engine supplies $23\text{ kW}$. The resistance to motion is constant and equals $600\text{ N}$. Find the car’s speed at the moment when its acceleration is $0.5\text{ m s}^{-2}$.

Energy, work and power

A particle $P$ with weight $1.4\text{ N}$ is connected to one end of a light inextensible string $S_1$ of length $1.5\text{ m}$ and also to one end of another light inextensible string $S_2$ of length $1.3\text{ m}$. The opposite end of $S_1$ is fixed to a wall at a point $0.9\text{ m}$ vertically above a point $O$ on the wall. The opposite end of $S_2$ is fixed to the wall at a point $0.5\text{ m}$ vertically below $O$. The particle is maintained in equilibrium at the same horizontal level as $O$ by a horizontal force of magnitude $2.24\text{ N}$ acting away from the wall and perpendicular to it (see diagram).

Forces and equilibrium

A small ball with mass $0.4\text{ kg}$ is let go from rest from a point $5\text{ m}$ above the horizontal ground. The moment the ball reaches the ground, $12.8\text{ J}$ of kinetic energy is lost and it then begins to rise.

Kinematics of motion in a straight line

A lorry with mass $16000\text{ kg}$ moves at steady speed from the base, $O$, to the summit, $A$, of a straight hill. The length $OA$ is $1200\text{ m}$ and $A$ lies $18\text{ m}$ above the level of $O$. The lorry’s driving force is constant and equal to $4500\text{ N}$. When it reaches $A$ the lorry goes on along a straight horizontal road, working against a constant resistance of $2000\text{ N}$. The driving force is then no longer constant, and the speed rises from $9\text{ m s}^{-1}$ at $A$ to $21\text{ m s}^{-1}$ at point $B$ on the road. The distance $AB$ is $2400\text{ m}$.

Energy, work and power

A particle is released from rest at point $O$ and travels along a horizontal straight line. At time $t\text{ s}$ after leaving $O$, its velocity is $v\text{ m s}^{-1}$. When $0 \le t < 60$, the velocity is given by $v = 0.05t - 0.0005t^2$. At $t = 60$, the particle strikes a wall and then changes the direction of its motion. It later comes to rest at point $A$ when $t = 100$, and for $60 < t \le 100$ the velocity is given by $v = 0.025t - 2.5$.

Kinematics of motion in a straight line

A smooth inclined plane with length $160\text{ cm}$ is set so that one end is $40\text{ cm}$ above the other end, and the lower end lies on horizontal ground. Particles $P$ and $Q$, with masses $0.76\text{ kg}$ and $0.49\text{ kg}$ respectively, are joined by the ends of a light inextensible string that passes over a small smooth pulley fixed at the top of the plane. Particle $P$ is initially held at rest on the same line of greatest slope as the pulley, while $Q$ hangs vertically below the pulley at a height of $30\text{ cm}$ above the ground (see diagram). $P$ is released from rest. It moves up the plane and does not reach the pulley.

Newton's laws of motion

A particle is launched at a speed of $12\text{ m s}^{-1}$ from a point on level ground. It returns to the ground $1.6\text{ s}$ later.

Probability

A non-uniform rod $AB$ has weight $6\text{ N}$ and is in limiting equilibrium, with end $A$ touching a rough vertical wall. The length $AB = 1.2\text{ m}$, the centre of mass of the rod is $0.8\text{ m}$ from $A$, and the angle between $AB$ and the downward vertical is $\theta^\circ$. At $B$, a force of magnitude $10\text{ N}$, acting at an angle of $30^\circ$ to the upwards vertical, is applied to the rod (see diagram). The rod and the line of action of the $10\text{ N}$ force both lie in a vertical plane perpendicular to the wall.

Probability

A light elastic string, with natural length $0.8\text{ m}$ and modulus of elasticity $16\text{ N}$, is fixed at one end to the point $O$. A particle $P$ of mass $0.4\text{ kg}$ is attached to the free end, and it hangs in equilibrium directly beneath $O$.

Probability

Particle $P$ is launched from point $O$ on horizontal ground with speed $20\text{ m s}^{-1}$ at an angle of $40^\circ$ above the horizontal.

Probability

A uniform metal frame $OABC$ is formed from a semicircular arc $ABC$ with radius $1.8\text{ m}$ together with a straight rod $AOC$ where $AO = OC = 1.8\text{ m}$ (see diagram). A uniform semicircular lamina of radius $1.8\text{ m}$ weighs $27.5\text{ N}$. A non-uniform object is produced by fixing the frame $OABC$ around the edge of the lamina. The object is hung freely from a fixed point at $A$ and comes to equilibrium. The diameter $AOC$ of the object is inclined at an angle of $22^\circ$ to the vertical.

Representation of data

A particle $P$ with mass $0.6\text{ kg}$ is let go from rest at a point above ground level and moves vertically downward. Its motion is resisted by a force of magnitude $3v\text{ N}$, where $v\text{ m s}^{-1}$ denotes the speed of $P$. Just before $P$ arrives at the ground, $v = 1.95$.

Probability

A bead $B$ of mass $m\text{ kg}$ travels with constant speed round a horizontal circle on a smooth fixed wire. The wire is a circle with centre $O$ and radius $0.4\text{ m}$. One end of a light elastic string, whose natural length is $0.4\text{ m}$ and modulus of elasticity is $42m\text{ N}$, is fastened to $B$. The other end of the string is fixed at point $A$, which is $0.3\text{ m}$ vertically above $O$ (see diagram).

Probability

A particle is launched from a point on horizontal ground with speed $12\,\text{m s}^{-1}$. It reaches the ground again $1.6\,\text{s}$ later.

Probability

A non-uniform rod $AB$ has weight $6\,\text{N}$ and is in limiting equilibrium, with end $A$ touching a rough vertical wall. $AB = 1.2\,\text{m}$, the centre of mass of the rod is $0.8\,\text{m}$ from $A$, and the angle between $AB$ and the downward vertical is $\theta^\circ$. At $B$, a force of magnitude $10\,\text{N}$ is applied at an angle of $30^\circ$ to the upwards vertical (see diagram). The rod and the line of action of the $10\,\text{N}$ force both lie in a vertical plane perpendicular to the wall.

Representation of data

An elastic string of negligible mass has natural length $0.8\,\text{m}$ and modulus of elasticity $16\,\text{N}$. One end is fixed at point $O$, while a particle $P$ of mass $0.4\,\text{kg}$ is attached to the opposite end. In equilibrium, $P$ hangs vertically beneath $O$.

Probability

A particle $P$ is launched with speed $20\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal from point $O$ on level ground.

Probability

A uniform metal frame $OABC$ is constructed from a semicircular arc $ABC$ with radius $1.8\,\text{m}$ and a straight rod $AOC$, where $AO = OC = 1.8\,\text{m}$ (see diagram). A uniform semicircular lamina of radius $1.8\,\text{m}$ has weight $27.5\,\text{N}$. By attaching the frame $OABC$ around the boundary of the lamina, a non-uniform object is produced. The object is then freely hung from a fixed point at $A$ and is in equilibrium. The diameter $AOC$ of the object is at an angle of $22^\circ$ to the vertical.

Representation of data

A particle $P$ of mass $0.6\,\text{kg}$ is let go from rest at a point above ground level and moves vertically downward. The motion of $P$ is resisted by a force of magnitude $3v\,\text{N}$, where $v\,\text{m s}^{-1}$ denotes the speed of $P$. Just before $P$ reaches the ground, $v = 1.95$.

Probability