May/June 2017

The coefficients of $x^2$ and $x^3$ in the expansion of $(3 - 2x)^6$ are $a$ and $b$ respectively.

Series

The diagram displays part of the curve $y = \frac{4}{5 - 3x}.$

Integration

Taking origin $O$, the position vectors of points $A$ and $B$ are given by $\overrightarrow{OA} = \begin{pmatrix} 3 \\ -6 \\ p \end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix} 2 \\ -6 \\ -7 \end{pmatrix}$, and angle $AOB = 90^\circ$. Point $C$ is defined so that $\overrightarrow{OC} = \frac{2}{3}\overrightarrow{OA}$.

Coordinate geometry

Prove that $\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta} = \frac{2}{\sin \theta}$.

Trigonometry

An arithmetic progression begins with $32$, has a 5th term of $22$ and ends with $-28$. Determine the sum of every term in the progression.

Series

A curve is defined by the equation $y = 2\cos x$.

Trigonometry

A solid prism has an equilateral triangle as its horizontal base, with side $x$ cm. The prism’s side faces are vertical. Its height is $h$ cm and its volume is $2000\text{ cm}^3$.

Differentiation

The curve with $\frac{dy}{dx} = 7 - x^2 - 6x$ goes through $(3, -10)$.

Quadratics

The figure shows $OAXB$ as a sector of a circle centred at $O$ with radius $10\,\text{cm}$. The chord $AB$ has length $12\,\text{cm}$. Line $OX$ passes through $M$, the midpoint of $AB$, and $OX$ is perpendicular to $AB$. The shaded part is enclosed by the chord $AB$ and the arc of a circle centred at $X$ with radius $XA$.

Circular measure

The mapping $f$ is defined as $f : x \mapsto \dfrac{2}{3 - 2x}$ for $x \in \mathbb{R}$, $x \neq \dfrac{3}{2}$.

Functions

Determine the coefficient of $x$ in the expansion of $(2x - \frac{1}{x})^5$.

Series

The function $f$ has rule $f(x) = 3 \tan\left(\frac{1}{2}x\right) - 2$, for $-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi$.

Functions

Point $A$ is at $(-2, 6)$. The perpendicular bisector of line $AB$ is given by $2y = 3x + 5$.

Coordinate geometry

Prove the identity $(\frac{1}{\cos \theta} - \tan \theta)^2 \equiv \frac{1 - \sin \theta}{1 + \sin \theta}$.

Trigonometry

The diagram depicts a circle of radius $r$ cm with centre $O$. $A$ and $B$ are points on the circumference, and $ABCD$ is a rectangle. The angle $AOB = 2\theta$ radians, and $AD = r$ cm.

Circular measure

The curve is given by $y = 3 + \frac{12}{2 - x}$.

Differentiation

The diagram displays the straight line $x + y = 5$ crossing the curve $y = \frac{4}{x}$ at the points $A(1, 4)$ and $B(4, 1)$.

Integration

The first two terms of an arithmetic progression are $16$ and $24$. Determine the smallest number of terms from the progression that need to be included so that their sum is greater than $20\,000$.

Series

Using origin $O$, the position vectors of the points $A$, $B$ and $C$ are $ \overrightarrow{OA} = 3\mathbf{i} + p\mathbf{j} - 2p\mathbf{k}$, $\overrightarrow{OB} = 6\mathbf{i} + (p + 4)\mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OC} = (p - 1)\mathbf{i} + 2\mathbf{j} + q\mathbf{k}$, with $p$ and $q$ as constants.

Coordinate geometry

The curve is defined by $y = \frac{8}{\sqrt{x}} - 2x$.

Differentiation

Within the expansion of $(2 + ax)^7$, the coefficients of $x$ and $x^2$ are equal.

Series

Fig. 1 illustrates a section of the curve $y = x^2 - 1$ together with the line $y = h$, where $h$ is a constant.

Integration

The function $f$ is defined for $x \geq 0$. It is given that $f$ reaches its minimum when $x = 2$, and that $f''(x) = (4x + 1)^{-\frac{1}{2}}$.

Differentiation

For a geometric progression with common ratio $r$, the first term is $(r^2 - 3r + 2)$ and the infinite sum is $S$.

Series

Find the coordinates of the points where the curve $y = x^{\frac{2}{3}} - 1$ intersects the curve $y = x^{\frac{1}{3}} + 1$.

Functions

Taking $O$ as the origin, the position vectors of $A$ and $B$ are $\overrightarrow{OA} = \begin{pmatrix}5\\1\\-2\end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix}5\\4\\-3\end{pmatrix}$. The point $P$ is on $AB$ and satisfies $\overrightarrow{AP} = \frac{1}{3}\overrightarrow{AB}$.

Coordinate geometry

Show that the equation $\dfrac{2\sin\theta + \cos\theta}{\sin\theta + \cos\theta} = 2\tan\theta$ can be rearranged into the form $\cos^2\theta = 2\sin^2\theta$.

Trigonometry

The line $3y + x = 25$ is a normal to the curve $y = x^2 - 5x + k$. Determine the value of the constant $k$.

Differentiation

The diagram displays two circles, one centred at $A$ and the other at $B$, with radii $8\text{ cm}$ and $10\text{ cm}$ respectively. They meet at $C$ and $D$, where $CAD$ is a straight line and $AB$ is perpendicular to $CD$.

Circular measure

The points $A(-1, 1)$ and $P(a, b)$ are given, where $a$ and $b$ are constants. The gradient of $AP$ is $2$.

Coordinate geometry

Express $9x^2 - 6x + 6$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are constants.

Functions

From $5^x = 3^{4y}$, apply logarithms to establish that $y = mx$.

Numerical solution of equations

Solve for the values of $x$ in $|4 - x| \leq |3 - 2x|$.

Algebra

It is given that $\int_{0}^{a} 4e^{\frac{1}{2}x + 3} \, dx = 835$.

Numerical solution of equations

The terms generated by the iterative rule $x_{n+1} = \frac{2x_n^2 + x_n + 9}{(x_n + 1)^2}$, beginning with $x_1 = 2$, settle to $\alpha$.

Numerical solution of equations

Express $2 \cos \theta + \sqrt{5} \sin \theta$ in the form $R \cos(\theta - \alpha)$ where $R > 0$ and $0^\circ < \alpha < 90^\circ$, and give the value of $\alpha$ correct to 2 decimal places.

Trigonometry

The diagram depicts the curve $y = \tan 2x$ for $0 \leq x \leq \frac{\pi}{6}$. The shaded area is enclosed by the curve together with the lines $x = \frac{\pi}{6}$ and $y = 0$.

Integration

For the curve, the parametric equations are $x = t^3 + 6t + 1$ and $y = t^4 - 2t^3 + 4t^2 - 12t + 5$.

Differentiation

The diagram illustrates the curve given by $y = 3x^2\ln\left(\frac{1}{6}x\right)$. This curve meets the $x$-axis at $P$ and has a turning point at $M$.

Differentiation

Find the solutions of $|x + a| = |2x - 5a|$, expressing $x$ in terms of the positive constant $a$.

Algebra

Use logarithms to solve $3^{x+4} = 5^{2x}$, and give the answer correct to $3$ significant figures.

Logarithmic and exponential functions

By sketching an appropriate pair of graphs, show that the equation $x^3 = 11 - 2x$ has only one real root.

Numerical solution of equations

Find the equation of the tangent to the curve $y = \frac{e^{4x}}{2x + 3}$ at the point on the curve where $x = 0$. Present your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers.

Differentiation

The variables $x$ and $y$ are related by $y = \frac{K}{a^{2x}}$, where $K$ and $a$ are constants. A plot of $\ln y$ against $x$ is a straight line that passes through the points $(0.6, 1.81)$ and $(1.4, 1.39)$, as the diagram shows.

Numerical solution of equations

Apply the factor theorem to demonstrate that $(x + 2)$ is a factor of $6x^3 + 13x^2 - 33x - 70$, then factorise the expression fully.

Algebra

Find $\int (2\cos\theta - 3)(\cos\theta + 1)\, d\theta$ by integrating the expanded product.

Integration

The sketch represents the curve given by the parametric equations $x = 2 - \cos 2t$, $y = 2\sin^3 t + 3\cos^3 t + 1$ for $0 \le t \le \frac{1}{2}\pi$. Its endpoints are $(1, 4)$ and $(3, 3)$.

Differentiation

Find the solutions to $|x + a| = |2x - 5a|$, giving $x$ in terms of the positive constant $a$.

Algebra

Apply logarithms to solve $3^{x+4} = 5^{2x}$, and give the value of $x$ correct to $3$ significant figures.

Logarithmic and exponential functions

By drawing appropriate graphs, demonstrate that the equation $x^3 = 11 - 2x$ has exactly one real root.

Numerical solution of equations

Determine the equation of the tangent to the curve $y = \frac{e^{4x}}{2x + 3}$ at the point on the curve for which $x = 0$. Give your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers.

Differentiation

The variables $x$ and $y$ are related by $y = \frac{K}{a^{2x}}$, with $K$ and $a$ as constants. The plot of $\ln y$ against $x$ is a straight line that goes through the points $(0.6, 1.81)$ and $(1.4, 1.39)$, as shown in the diagram.

Numerical solution of equations

Use the factor theorem to prove that $(x + 2)$ is a factor of $6x^3 + 13x^2 - 33x - 70$, and then factorise the expression fully.

Algebra

Determine $\int (2\cos \theta - 3)(\cos \theta + 1) \, d\theta$.

Integration

The diagram depicts the curve with parametric equations $x = 2 - \cos 2t$, $y = 2\sin^3 t + 3\cos^3 t + 1$ for $0 \leq t \leq \frac{1}{2}\pi$. Its end-points are $(1,4)$ and $(3,3)$.

Differentiation

Find the solution of the inequality $|2x + 1| < 3|x - 2|$.

Algebra

The diagram illustrates the curve $y = \sin x \cos^2 x$ for $0 \leq x \leq \frac{1}{4}\pi$, together with its maximum point $M$.

Integration

Expand $\dfrac{1}{\sqrt[3]{(1 + 6x)}}$ in ascending powers of $x$, as far as and including the term in $x^3$, with the coefficients simplified.

Algebra

You are given $x = \ln(1 - y) - \ln y$, where $0 \le y \le 1$.

Integration

The curve is given parametrically by $x = \ln(\cos \theta)$ and $y = 3\theta - \tan \theta$, with $0 \leq \theta < \frac{1}{2}\pi$.

Differentiation

A semicircle has centre $O$, radius $r$ and diameter $AB$ as shown. The point $P$ on the circumference is positioned so that the area of the minor segment on $AP$ is equal to half the area of the minor segment on $BP$. The angle $AOP$ is $x$ radians.

Numerical solution of equations

The plane given by the equation $2x + 2y - z = 5$ is labelled $m$. Relative to the origin $O$, the coordinates of $A$ and $B$ are $(3, 4, 0)$ and $(-1, 0, 2)$ respectively.

Vectors

Calculators are not allowed anywhere in this question. The complex numbers $u$ and $w$ are given by $u = -1 + 7i$ and $w = 3 + 4i$.

Complex numbers

Start by expanding $2\sin(x - 30^\circ)$, then rewrite $2\sin(x - 30^\circ) - \cos x$ in the form $R\sin(x - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.

Trigonometry

Write $\frac{1}{x(2x + 3)}$ as a partial fraction decomposition.

Differential equations

Find the solution of the equation $\ln(x^2 + 1) = 1 + 2\ln x$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

The diagram illustrates the curve $y = x^2 \cos 2x$ for $0 \leq x \leq \frac{1}{4}\pi$. On the curve, the highest point is $M$, where $x = p$.

Numerical solution of equations

Solve the inequality given by $|x - 3| < 3x - 4$.

Algebra

Express the equation $\cot \theta - 2\tan \theta = \sin 2\theta$ as $a\cos^4 \theta + b\cos^2 \theta + c = 0$, with constants $a$, $b$ and $c$ to be found.

Trigonometry

The curve is given in parametric form by $x = t^2 + 1$, $y = 4t + \ln(2t - 1)$.

Differentiation

In a certain chemical process, substance $A$ reacts with substance $B$ and reduces it. If the masses of $A$ and $B$ at time $t$ after the process begins are $x$ and $y$ respectively, it is given that $\frac{dy}{dt} = -0.2xy$ and $x = \frac{10}{(1+t)^2}$. At the start of the process, $y = 100$.

Differential equations

For the whole question, calculator use is not allowed. Let the complex number $2 - i$ be called $u$.

Complex numbers

Prove that when $y = \frac{1}{\cos \theta}$, then $\frac{dy}{d\theta} = \sec \theta \tan \theta$.

Integration

Let the function be $f(x) = \dfrac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}$.

Algebra

Measured from the origin $O$, point $A$ has position vector $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$. Line $l$ is defined by $\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})$.

Vectors

Prove that the identity $\frac{\cot x - \tan x}{\cot x + \tan x} = \cos 2x$ holds.

Trigonometry

Points $A$ and $B$ are defined by the position vectors $\overrightarrow{OA} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}$. The line $l$ is described by $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + m\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})$, where $m$ is a constant.

Vectors

You must not use a calculator anywhere in this question.

Complex numbers

Expand $(3 + 2x)^{-3}$ as a series in ascending powers of $x$, including terms up to and including the $x^2$ term, and simplify the coefficients.

Algebra

With $u = e^x$ as the substitution, solve the equation $4e^{-x} = 3e^x + 4$. State your answer correct to 3 significant figures.

Logarithmic and exponential functions

Determine the exact value of $\int_{0}^{\frac{1}{2}\pi} 6\sin\frac{1}{2}\theta\, d\theta$.

Integration

The curve is given by $y = \frac{2}{3}\ln(1 + 3\cos^2 x)$ for $0 \leq x \leq \frac{1}{2}\pi$.

Differentiation

Within $0 < x < \pi$, the equation $\cot x = 1 - x$ has a single root, which is written as $\alpha$.

Numerical solution of equations

The diagram gives a sketch of the curve $y = \frac{e^{1/x}}{x}$ for $x > 0$, together with its minimum point $M$.

Integration

During a particular chemical reaction, compound $A$ is produced from compound $B$. At time $t$ after the reaction begins, the masses of $A$ and $B$ are $x$ and $y$ respectively, and their total mass stays at $50$ for the whole reaction. At any moment, the rate at which the mass of $A$ increases is proportional to the mass of $B$ at that moment.

Differential equations

Define $f(x) = \dfrac{3x^2 - 4}{x^2(3x + 2)}$.

Integration

A particle with mass $0.6\text{ kg}$ is released from rest at a point $8\text{ m}$ above the ground. Just before it reaches the ground, its speed is $10\text{ m s}^{-1}$.

Energy, work and power

A particle with mass $0.8\text{ kg}$ is launched at a speed of $12\text{ m s}^{-1}$ up the line of greatest slope on a rough plane inclined at $10^\circ$ to the horizontal. The coefficient of friction between the particle and the plane is $0.4$.

Kinematics of motion in a straight line

A particle weighing $25\text{ N}$ has two light inextensible strings fixed to it. Each string goes over a smooth fixed pulley and then supports a hanging particle of weight $A\text{ N}$ or $B\text{ N}$ at the lower end. The parts of the strings that slope away make angles of $30^\circ$ and $40^\circ$ respectively with the vertical (see diagram). The system is in equilibrium.

Forces and equilibrium

A car with mass $800\,\text{kg}$ is travelling uphill on a slope making an angle of $\theta^\circ$ with the horizontal, where $\sin\theta = 0.15$. Its initial speed is $8\,\text{m s}^{-1}$. Twelve seconds later, it has gone $120\,\text{m}$ up the slope and its speed is $14\,\text{m s}^{-1}$.

Energy, work and power

A particle $P$ travels along the straight line $ABCD$ with constant deceleration. Its velocities at $A$, $B$ and $C$ are $20\,\text{m s}^{-1}$, $12\,\text{m s}^{-1}$ and $6\,\text{m s}^{-1}$ respectively.

Kinematics of motion in a straight line

A particle $P$ travels along a straight line through a point $O$. When the time is $t\text{ s}$, the velocity of $P$, $v\text{ m s}^{-1}$, is given by $v = qt + rt^2$, where $q$ and $r$ are constants. The particle’s velocity is $4\text{ m s}^{-1}$ at both $t = 1$ and $t = 2$.

Kinematics of motion in a straight line

As the diagram shows, particle $A$ has mass $0.8\text{ kg}$ and rests on a plane inclined at $30^\circ$ to the horizontal, while particle $B$ has mass $1.2\text{ kg}$ and rests on a plane inclined at $60^\circ$ to the horizontal. The particles are linked by a light inextensible string that passes over a small smooth pulley $P$ fixed at the top of the planes. The segments $AP$ and $BP$ of the string are parallel to the lines of greatest slope of the two planes. Both particles are released from rest, with the string taut in both sections.

Forces and equilibrium

A block is attached to one end of a light inextensible string. The string is inclined at an angle of $\theta^\circ$ to the horizontal. The tension in the string is $20\,\text{N}$. The string moves the block across a horizontal surface at a steady speed of $1.5\,\text{m s}^{-1}$ for $12\,\text{s}$. The work done by the tension in the string is $50\,\text{J}$.

Energy, work and power

The diagram depicts a wire $ABCD$ made up of a straight section $AB$ with length $5\,\text{m}$ and a section $BCD$ formed as a semicircle of radius $6\,\text{m}$ with centre $O$. The diameter $BD$ of the semicircle lies horizontally, while $AB$ is vertical. A small ring is placed on the wire and can move along it. It is released from rest at $A$. The section $AB$ of the wire is rough, so the ring has constant acceleration of $2.5\,\text{m s}^{-2}$ from $A$ to $B$.

Energy, work and power

Particle $A$ travels along a straight line at a constant speed of $10\,\text{m s}^{-1}$. Two seconds after $A$ has passed point $O$ on the line, particle $B$ passes through $O$ and moves along the line in the same direction as $A$. At $O$, particle $B$ has speed $16\,\text{m s}^{-1}$ and then undergoes a constant deceleration of $2\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A car with mass $1200\,\text{kg}$ is travelling along a straight road while a steady resisting force of $850\,\text{N}$ acts against its motion.

Energy, work and power

A particle with mass $0.12\,\text{kg}$ rests on a plane that is tilted at $40^\circ$ to the horizontal. A force of size $P\,\text{N}$ acts up the plane at $30^\circ$ above a line of greatest slope, as shown in the diagram, and the particle is in equilibrium. The coefficient of friction between the particle and the plane is $0.32$.

Energy, work and power

The diagram depicts a fixed block with a horizontal upper face and another face that is inclined at an angle of $\theta^\circ$ to the horizontal, where $\sin\theta = \frac{3}{5}$. Particle $A$ of mass $0.3\,\text{kg}$ is at rest on the horizontal face and is connected to one end of a light inextensible string. The string runs over a small smooth pulley $P$ fixed at the block’s edge. The other end is joined to particle $B$ of mass $1.5\,\text{kg}$, which is resting on the sloping face of the block. The system is released from rest with the string taut.

Newton's laws of motion

A man propels a wheelbarrow of mass $25\,\text{kg}$ over a level road using a constant force of magnitude $35\,\text{N}$ applied at $20^\circ$ below the horizontal. A steady resistive force of $15\,\text{N}$ acts against the motion. Starting from rest, the wheelbarrow travels $12\,\text{m}$.

Energy, work and power

The four coplanar forces shown in the diagram are balanced.

Forces and equilibrium

A train runs between stations $A$ and $B$. It leaves $A$ from rest and increases its speed at a constant rate for $T\,\text{s}$ until its speed becomes $25\,\text{m\,s}^{-1}$. It then continues at this same speed before slowing down at a constant rate and stopping at $B$. The size of the train’s deceleration is twice the size of its acceleration. The whole trip takes $180\,\text{s}$.

Kinematics of motion in a straight line

A particle $P$ travels along a straight line from point $O$. After $t$ s has elapsed since it left $O$, its velocity, $v\,\text{m s}^{-1}$, is given by $v = (2t - 5)^3$.

Kinematics of motion in a straight line

One particle is launched vertically upwards from point $O$ at a speed of $12\,\text{m s}^{-1}$. Two seconds afterwards, a second particle is launched vertically upwards from $O$ at a speed of $20\,\text{m s}^{-1}$. At time $t$ s after the second particle is launched, the two particles collide.

Kinematics of motion in a straight line

A car with mass $1200\text{ kg}$ is moving along a level road.

Energy, work and power

Particles $A$ and $B$, with masses $m\text{ kg}$ and $4\text{ kg}$ respectively, are joined by a light inextensible string which runs over a fixed smooth pulley. $A$ rests on a rough fixed slope inclined at $30^{\circ}$ to the horizontal ground. $B$ is suspended vertically beneath the pulley and lies $0.5\text{ m}$ above the ground (see diagram). The coefficient of friction between the slope and $A$ is $0.2$.

Forces and equilibrium

A particle is projected at speed $20\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. Calculate the time after projection when the particle is descending at an angle of $40^\circ$ below the horizontal.

Probability

One end of a light inextensible string is fixed at point $A$. Its other end is fastened to a particle $P$ of mass $m\,\text{kg}$, which hangs vertically beneath $A$. $P$ is also connected to one end of a light elastic string with natural length $0.25\,\text{m}$. The other end of that string is attached to point $B$, which is $0.6\,\text{m}$ from $P$ and lies on the same horizontal level as $P$. A horizontal force of magnitude $7\,\text{N}$ acting on $P$ keeps the system in equilibrium (see Fig. 1).

Representation of data

An object is formed by taking a uniform solid hemisphere of radius $0.56\,\text{m}$ with centre $O$ and then removing from it a hemisphere of radius $0.28\,\text{m}$ with centre $O$. The diagram shows a cross-section of the object through $O$.

Representation of data

A particle is launched from point $O$ on level ground. Its initial velocity has components of $10\,\text{m s}^{-1}$ horizontally and $15\,\text{m s}^{-1}$ vertically upwards. After time $t$ from projection, the horizontal displacement from $O$ is $x\,\text{m}$ and the vertically upwards displacement from $O$ is $y\,\text{m}$.

Representation of data

A semicircular lamina of uniform density has radius $0.7\,\text{m}$ and weight $14\,\text{N}$, with diameter $AB$. It is positioned in a vertical plane and freely pivoted at $A$ about a fixed point. A small smooth peg $P$ above the level of $A$ supports the straight edge $AB$. The line $AB$ is inclined at $30^{\circ}$ to the horizontal, and $AP = 0.9\,\text{m}$ (see diagram).

Representation of data

A particle $P$ with mass $0.15\,\text{kg}$ is attached to one end of a light elastic string whose natural length is $0.4\,\text{m}$ and modulus of elasticity is $12\,\text{N}$. The other end of the string is fixed at point $A$. Particle $P$ travels in a horizontal circle with its centre directly below $A$, while the string makes an angle of $\theta^\circ$ with the vertical and $AP = 0.5\,\text{m}$.

Probability

Particle $P$ has mass $0.5\,\text{kg}$ and is initially at rest at the point $O$ on a rough horizontal surface. At $t = 0$, where $t$ is measured in seconds, a horizontal force in a fixed direction begins to act on $P$. At time $t$ the force has magnitude $0.6t^2\,\text{N}$ and the velocity of $P$ away from $O$ is $v\,\text{m s}^{-1}$. It is stated that $P$ stays at rest at $O$ until $t = 0.5$.

Representation of data

A particle $P$ of mass $0.2\,\mathrm{kg}$ travels with speed $4\,\mathrm{m\,s^{-1}}$ and angular speed $5\,\mathrm{rad\,s^{-1}}$ in a horizontal circle on a smooth surface. $P$ is connected to one end of a light elastic string of natural length $0.6\,\mathrm{m}$. The other end of the string is fixed to the point on the surface that is the centre of the circular motion of $P$.

Probability

The two ends of a pair of light inextensible strings, each of length $0.7\,\mathrm{m}$, are fixed to a particle $P$. Their other ends are fastened to two fixed points $A$ and $B$, which are in the same vertical line with $A$ higher than $B$. The particle $P$ undergoes horizontal circular motion with its centre at the midpoint of $AB$. Both strings make an angle of $60^\circ$ with the vertical. The tension in the string attached to $A$ is $6\,\mathrm{N}$ and the tension in the string attached to $B$ is $4\,\mathrm{N}$ (see diagram).

Probability

A cube-shaped open box with edge length $0.2\,\text{m}$ is set so that its base is horizontal and its four sides are vertical. The base and the four sides are uniform laminas, each weighing $3\,\text{N}$.

Representation of data

An object of mass $0.4\,\text{kg}$ is dropped from rest from a point $8\,\text{m}$ above the ground. It falls vertically, and when its downward displacement from the starting position is $x\,\text{m}$ the speed is $v\,\text{m s}^{-1}$. As the object moves, a force of magnitude $0.2v^2\,\text{N}$ acts opposite to the motion.

Probability

A particle of mass $0.3\,\text{kg}$ is fastened to one end of a light elastic string with natural length $0.8\,\text{m}$ and modulus of elasticity $6\,\text{N}$. The other end of the string is fixed at point $O$. The particle is projected vertically downwards from $O$ with initial speed $2\,\text{m s}^{-1}$.

Probability

At rest on a rough horizontal plane is the end $A$ of a non-uniform rod $AB$ with length $0.6\,\text{m}$ and weight $8\,\text{N}$, where $AB$ is set at $60^\circ$ to the horizontal. The rod is kept in equilibrium by a force of magnitude $3\,\text{N}$ applied at $B$. This force acts $30^\circ$ above the horizontal in the vertical plane containing the rod (see diagram).

Probability

A particle $P$ is launched from a point $O$ with speed $V\ \mathrm{m\,s^{-1}}$. After $t$ s, its horizontal displacement from $O$ is $x$ m and its vertically upward displacement is $y$ m. The trajectory equation for $P$ is $y = 2x - \frac{25x^2}{V^2}$.

Representation of data