Oct/Nov 2017

The curve is described by $y = 2x^{\frac{3}{2}} - 3x - 4x^{\frac{1}{2}} + 4$.

Differentiation

The diagram depicts a segment of the curve $y = \frac{1}{2}(x^4 - 1)$, which is defined for $x \geq 0$.

Integration

Consider the function $f$, given by $f: x \mapsto x^3 - x^2 - 8x + 5$ for $x < a$. It is stated that $f$ increases.

Differentiation

One geometric progression begins with first term $3a$ and has common ratio $r$. A second geometric progression begins with first term $a$ and has common ratio $-2r$. These two progressions have equal sums to infinity. Determine the value of $r$.

Series

A factory machine produces cardboard cones with base radius $r$ cm and vertical height $h$ cm. The cone volume, $V$ cm$^3$, is given by $V = \frac{1}{3}\pi r^2 h$. For the cones made by these machines, $h + r = 18$.

Differentiation

The diagram displays an isosceles triangle $ABC$, with $AC = 16\text{ cm}$ and $AB = BC = 10\text{ cm}$. The circular arcs $BE$ and $BD$ are centred at $A$ and $C$ respectively, and $D$ and $E$ lie on $AC$.

Circular measure

Both $A(1, 1)$ and $B(5, 9)$ are on the curve $6y = 5x^2 - 18x + 19$.

Coordinate geometry

The diagram gives part of the graph of $y = a + b \sin x$. Find the values of the constants $a$ and $b$.

Trigonometry

Taking O as the origin, the position vectors of points P and Q are \\mathbf{p} and \\mathbf{q} respectively. Point R is arranged so that PQR is a straight line, with Q halfway between P and R.

Coordinate geometry

For $x > 3$, the functions $f$ and $g$ are given by $f:x \mapsto \frac{1}{x^2 - 9}$ and $g:x \mapsto 2x - 3$.

Functions

Determine the term independent of $x$ in the expansion of $(2x - \frac{1}{4x^2})^9$.

Series

The diagram depicts a segment of the curve $y = \sqrt{5x - 1}$ together with the normal to the curve at the point $P(2, 3)$. This normal intersects the $x$-axis at $Q$.

Integration

The function $f$ is given by $f : x \mapsto 4 - 5x$ for $x \in \mathbb{R}$.

Functions

The value of a particular rare stamp goes up by $5\%$ of its value at the start of each year. A collector paid $\$10\,000$ for the stamp at the beginning of 2005. Determine its value at the beginning of 2015, correct to the nearest $\$100$.

Series

The diagram depicts a semicircle with centre $O$ and radius $6\text{ cm}$. Radius $OC$ is at right angles to the diameter $AB$. Point $D$ is on $AB$, and $DC$ is an arc from a circle centred at $B$.

Circular measure

Show that the equation $\cos 2x(\tan^2 2x + 3) + 3 = 0$ may be rewritten as $2\cos^2 2x + 3\cos 2x + 1 = 0$.

Trigonometry

The function $f$, given by $f : x \mapsto a + b\sin x$ for $x \in \mathbb{R}$, satisfies $f\left(\frac{\pi}{6}\right) = 4$ and $f\left(\frac{\pi}{2}\right) = 3$.

Functions

The points $A$ and $B$ are located on the curve $y = x^2 - 4x + 7$. Point $A$ is $(4, 7)$, and $B$ is the stationary point on the curve. The equation of line $L$ is $y = mx - 2$, where $m$ is constant.

Coordinate geometry

The curve is defined by $\frac{dy}{dx} = -x^2 + 5x - 4$.

Differentiation

The diagram depicts trapezium $OABC$, with $OA$ parallel to $CB$. The position vectors of $A$ and $B$ from the origin $O$ are $\overrightarrow{OA} = \begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$.

Coordinate geometry

An arithmetic progression starts with first term $-12$ and has common difference $6$. The total of the first $n$ terms is greater than $3000$.

Series

The curve is defined by $y = f(x)$, and you are told that $f'(x) = ax^2 + bx$, with $a$ and $b$ positive constants.

Differentiation

The sketch displays the curve $y = \sqrt{x - 1}$ together with the points $A(1, 0)$ and $B(5, 2)$ on it.

Differentiation

The curve has equation $y = -\frac{2}{x}$, while the straight line is given by $y = ax + 3a$.

Quadratics

Determine the term that is independent of $x$ in the expansion of $\left(\frac{2}{x} - 3x\right)^6$.

Series

The function $f$ is defined by $f(x) = (2x - 1)^{\frac{3}{2}} - 6x$ for $\frac{1}{2} < x < k$, where $k$ is a constant.

Differentiation

Show that the equation $\frac{\cos \theta + 4}{\sin \theta + 1} + 5\sin \theta - 5 = 0$ can be rewritten as $5\cos^2 \theta - \cos \theta - 4 = 0$.

Trigonometry

The functions $f$ and $g$ are given by $f(x) = \dfrac{2}{x^2 - 1}$ for $x < -1$, while $g(x) = x^2 + 1$ for $x > 0$.

Functions

The diagram depicts a rectangle $ABCD$ where $AB = 5$ units and $BC = 3$ units. Point $P$ is on $DC$, and $AP$ is an arc of a circle with centre $B$. Point $Q$ is on $DC$, and $AQ$ is an arc of a circle with centre $D$.

Circular measure

The diagram depicts parts of the graphs of $y = 3 - 2x$ and $y = 4 - 3\sqrt{x}$ meeting at points $A$ and $B$.

Integration

With respect to an origin $O$, the position vectors of points $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix} 8 \\ -6 \\ 5 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} -10 \\ 3 \\ -13 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 2 \\ -3 \\ 7 \end{pmatrix}$. A fourth point, $D$, is such that the magnitudes $|\overrightarrow{AB}|$, $|\overrightarrow{BC}|$ and $|\overrightarrow{CD}|$ are, respectively, the first, second and third terms of a geometric progression.

Coordinate geometry

Solve the equation $\ln(3x + 1) - \ln(x + 2) = 1$, and give the answer in terms of $e$.

Logarithmic and exponential functions

Find the solutions of $5\cos\theta(1 + \cos 2\theta) = 4$ for $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The variable $x$ is such that $1.3^{2x} < 80$ and $|3x - 1| > |3x - 10|$.

Numerical solution of equations

Evaluate $\int \frac{4 + \sin^2 \theta}{1 - \sin^2 \theta} \, d\theta$.

Integration

The polynomial $p(x)$ is given by $p(x) = ax^3 + bx^2 + 37x + 10$, with $a$ and $b$ as constants. You are told that $(x + 2)$ is a factor of $p(x)$. You are also told that the remainder is $40$ when $p(x)$ is divided by $(2x - 1)$.

Algebra

The parametric equations for a curve are $x = 2e^{2t} + 4e^{t}$ and $y = 5te^{2t}$.

Differentiation

The sketch depicts the curve $y = x^2 + 3x + 1 + 5\cos\frac{1}{2}x$. It cuts the $y$-axis at the point $P$, and the gradient of the curve there is $m$. The point $Q$ lies on the curve and has $x$-coordinate $q$, with gradient $-m$ at $Q$.

Numerical solution of equations

Apply logarithms to solve the equation $5^{3x-1} = 2^{4x}$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

It is stated that $x$ satisfies the equation $|x + 1| = 4$.

Algebra

The curve is given by $y = \tan\left(\tfrac{1}{2}x\right) + 3\sin\left(\tfrac{1}{2}x\right)$. It has a stationary point $M$ for values of $x$ in the range $\pi < x < 2\pi$.

Numerical solution of equations

The polynomials $p(x)$ and $q(x)$ are given by $p(x) = x^3 + x^2 + ax - 15$ and $q(x) = 2x^3 + x^2 + bx + 21$, with $a$ and $b$ as constants. It is stated that $(x + 3)$ is a factor of $p(x)$ and also a factor of $q(x)$.

Differentiation

The diagram depicts the curve $y = 4e^{-2x}$ together with a straight line. The curve intersects the $y$-axis at $P$. The straight line intersects the $y$-axis at the point $(0, 9)$, and its gradient matches the gradient of the curve at $P$. The line cuts the curve at two points, including $Q$, as indicated.

Numerical solution of equations

Find the exact value of the integral $\int_{0}^{\frac{\pi}{4}} \sin x\left(4\sin x + 6\cos x\right)\,dx$.

Integration

A curve is defined by $x^2 + 4xy + 2y^2 = 7$.

Differentiation

Solve the equation $\ln(3x + 1) - \ln(x + 2) = 1$, expressing your answer in terms of $e$.

Logarithmic and exponential functions

Solve the equation $5\cos\theta(1 + \cos 2\theta) = 4$ within $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The condition is that $x$ satisfies $1.3^{2x} < 80$ and $|3x - 1| > |3x - 10|$.

Numerical solution of equations

Calculate $\displaystyle \int \frac{4 + \sin^2 \theta}{1 - \sin^2 \theta}\, d\theta$.

Integration

Let $p(x)=ax^3+bx^2+37x+10$, where $a$ and $b$ are constants. It is known that $(x + 2)$ divides $p(x)$. It is also known that the remainder is $40$ when $p(x)$ is divided by $(2x - 1)$.

Algebra

A curve is given by the parametric equations $x = 2e^{2t} + 4e^{t}$, $y = 5te^{2t}$.

Differentiation

The diagram depicts the curve $y = x^2 + 3x + 1 + 5\cos\frac{1}{2}x$. This curve meets the $y$-axis at $P$, where its gradient is $m$. Point $Q$ lies on the curve, has $x$-coordinate $q$, and the gradient there is $-m$.

Numerical solution of equations

Determine the quotient and remainder when $x^4$ is divided by $x^2 + 2x - 1$.

Algebra

The equations for the two lines $l$ and $m$ are $\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$, in that order.

Vectors

It is thought that the two variables $x$ and $y$ obey an equation of the form $y = C(a^x)$, with $C$ and $a$ constant. An experiment gave four matched pairs of $x$ and $y$ values. The table shows the corresponding values of $x$ and $\ln y$: $x = 0.9,\ 1.6,\ 2.4,\ 3.2$ $\ln y = 1.7,\ 1.9,\ 2.3,\ 2.6$.

Logarithmic and exponential functions

The equation $x^3 = 3x + 7$ has a single real root, written as $\alpha$.

Numerical solution of equations

Prove that $\tan(45^\circ + x) + \tan(45^\circ - x) = 2\sec 2x$.

Trigonometry

The curve is defined by the equation $2x^4 + xy^3 + y^4 = 10$.

Differentiation

The variables $x$ and $y$ are linked by the differential equation $\frac{dy}{dx} = 4\cos^2 x\tan x$, for $0 \leq x < \frac{1}{2}\pi$, and $x = 0$ when $y = \frac{1}{4}\pi$.

Differential equations

The complex number $u$ is $u = 8 - 15i$. With full working shown, determine the two square roots of $u$. Express your answers in the form $a + ib$, where $a$ and $b$ are exact real numbers.

Complex numbers

Define $f(x)$ by $\dfrac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}$.

Integration

The diagram depicts the curve $y = (1 + x^2)e^{-\frac{1}{2}x}$ for $x \geq 0$. The shaded region $R$ is bounded by the curve, the $x$-axis, and the lines $x = 0$ and $x = 2$.

Integration

The sketch illustrates the curve $y = \frac{3}{\sqrt{9 - x^3}}$ for $x$ values between $-1.2$ and $1.2$.

Integration

The planes $p$ and $q$ are defined by the equations $x + y + 3z = 8$ and $2x - 2y + z = 3$, respectively.

Vectors

With all essential working shown, solve the equation $2\log_{2}x = 3 + \log_{2}(x + 1)$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

By rewriting the equation $\tan(\theta + 60^\circ) + \tan(\theta - 60^\circ) = \cot\theta$ entirely in terms of $\tan\theta$, Solve the equation for $0^\circ < \theta < 90^\circ$.

Trigonometry

For the curve $y = \dfrac{2 - \sin x}{\cos x}$, there is a single stationary point within $-\tfrac{1}{2}\pi < x < \tfrac{1}{2}\pi$.

Differentiation

The variables $x$ and $y$ obey the differential equation $(x + 1)\frac{dy}{dx} = y(x + 2)$, and the condition $y = 2$ holds when $x = 1$.

Differential equations

The curve is defined by $x^3y - 3xy^3 = 2a^4$, with $a$ as a non-zero constant.

Differentiation

For this question, a calculator must not be used. Write the complex number $1 - \sqrt{3}i$ as $u$.

Complex numbers

Define $f(x)$ by $f(x) = \dfrac{8x^2 + 9x + 8}{(1 - x)(2x + 3)^2}$.

Algebra

It is given that $\int_{1}^{a} x^{\frac{1}{2}} \ln x \, dx = 2$, with $a > 1$.

Numerical solution of equations

Determine the quotient and the remainder when $x^4$ is divided by $x^2 + 2x - 1$.

Algebra

The pair of lines $l$ and $m$ is given by $\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ respectively.

Vectors

It is thought that the two variable quantities $x$ and $y$ obey an equation of the form $y = C a^x$, with $C$ and $a$ as constants. An experiment gave four pairs of $x$ and $y$ values. The table below shows the corresponding values of $x$ and $\nln y$. The values are: $x = 0.9, 1.6, 2.4, 3.2$ $\ln y = 1.7, 1.9, 2.3, 2.6$.

Logarithmic and exponential functions

The equation $x^3 = 3x + 7$ has a single real root, which is represented by $\alpha$.

Numerical solution of equations

Prove that $\tan(45^\circ + x) + \tan(45^\circ - x) = 2\sec 2x$.

Trigonometry

The curve is defined by the equation $2x^4 + xy^3 + y^4 = 10$.

Differentiation

The variables $x$ and $y$ are linked by the differential equation $\frac{dy}{dx} = 4\cos^2 x\tan x$, for $0 \leq x < \frac{1}{2}\pi$, and $x = 0$ when $y = \frac{1}{4}\pi$.

Differential equations

The complex number $u$ is $8 - 15i$. With full working shown, determine the two square roots of $u$. Present your answers in the form $a + ib$, with $a$ and $b$ being exact real values.

Complex numbers

Define $f(x) = \dfrac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}$.

Integration

The diagram depicts the curve $y = (1 + x^2)e^{-\frac{1}{2}x}$ for $x \geq 0$. The shaded region $R$ lies between the curve, the $x$-axis and the straight lines $x = 0$ and $x = 2$.

Integration

A block with mass $3\,\text{kg}$ is at rest at the start on a smooth horizontal floor. A force of $12\,\text{N}$, applied at an angle of $25^\circ$ above the horizontal, acts on the block.

Kinematics of motion in a straight line

A tractor with mass $3700\,\text{kg}$ moves along a straight horizontal road at a steady speed of $12\,\text{m s}^{-1}$. The total resistance opposing motion is $1150\,\text{N}$.

Energy, work and power

A roller-coaster car, passengers included, has a mass of $840\,\text{kg}$. The ride contains a stretch in which the car goes up a straight ramp of length $8\,\text{m}$ set at $30^\circ$ above the horizontal. It then drops straight away down a further ramp of length $10\,\text{m}$ set at $20^\circ$ below the horizontal. A resistance force of $640\,\text{N}$ acts on the car all the way through the motion.

Energy, work and power

The diagram gives the velocity-time graph for a particle moving in a straight line. The graph is made up of $5$ straight line sections. The particle begins from rest at point $A$ when $t = 0$, and at first moves towards point $B$ on the line.

Kinematics of motion in a straight line

A particle leaves point $O$ and travels along a straight line. The velocity of the particle at time $t\,\text{s}$ after leaving $O$ is $v\,\text{m s}^{-1}$, where $v = 1.5 + 0.4t$ for $0 \leq t \leq 5$, and $v = \frac{100}{t^2} - 0.1t$ for $t \geq 5$.

Kinematics of motion in a straight line

At point $P$, the diagram shows coplanar forces with magnitudes $F\,\text{N}$, $3F\,\text{N}$, $G\,\text{N}$ and $50\,\text{N}$ acting together, as illustrated in the diagram.

Forces and equilibrium

Particles $A$ and $B$, with masses $0.9\,\text{kg}$ and $0.4\,\text{kg}$ respectively, are fixed to the two ends of a light inextensible string. The string passes over a fixed smooth pulley attached to the top of two inclined planes. Initially the particles are at rest, with $A$ on a smooth plane inclined at angle $\theta^\circ$ to the horizontal and $B$ on a plane inclined at $25^\circ$ to the horizontal. The string is taut, and the particles can move along the lines of greatest slope of the two planes. A force of magnitude $2.5\,\text{N}$ acts on $B$ down the plane (see diagram).

Forces and equilibrium

A particle with mass $0.2\,\text{kg}$ is in equilibrium at rest on a rough plane that is tilted at $20^\circ$ to the horizontal.

Newton's laws of motion

A block with mass $15\,\text{kg}$ is suspended in equilibrium from a horizontal ceiling by two strings, as the diagram shows. One string makes an angle of $45^\circ$ with the horizontal and has tension $120\,\text{N}$. The second string is at $\theta^\circ$ to the horizontal and carries tension $T\,\text{N}$.

Forces and equilibrium

A car moves along a straight road with uniform acceleration. It goes through points $A$, $B$ and $C$. At point $A$, the car has velocity $14\,\text{m s}^{-1}$. The two sections $AB$ and $BC$ are equal in length. The times needed to travel along $AB$ and $BC$ are $5\,\text{s}$ and $3\,\text{s}$ respectively.

Kinematics of motion in a straight line

Particle $P$ is launched vertically upwards from horizontal ground at a speed of $12\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A cyclist is travelling uphill on a straight slope that makes an angle $\alpha$ with the horizontal, with $\sin \alpha = 0.04$. The combined mass of the bicycle and rider is $80\,\text{kg}$. The cyclist maintains a constant speed of $4\,\text{m s}^{-1}$. A resistive force acts against the motion. The work completed by the cyclist against this resistive force over $25\,\text{m}$ is $600\,\text{J}$.

Energy, work and power

Particles $P$ and $Q$, each of mass $m\,\text{kg}$, are fastened to the two ends of a light inextensible string. That string passes over a fixed smooth pulley attached to the edge of a rough plane. The plane makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac{7}{24}$. Particle $P$ is on the plane and particle $Q$ hangs vertically, as shown in the diagram. The part of the string between $P$ and the pulley is parallel to a line of greatest slope on the plane. The system is in limiting equilibrium.

Newton's laws of motion

A particle begins at rest and travels along a straight line. After $t\,\text{s}$ from the start, its velocity is $v\,\text{m s}^{-1}$, where $v = -0.01t^3 + 0.22t^2 - 0.4t$.

Kinematics of motion in a straight line

At point $P$, three coplanar forces with magnitudes $F$ N, $20$ N and $30$ N act, as shown in the diagram. The resultant of these three forces is directed perpendicular to the force of magnitude $F$ N.

Forces and equilibrium

A lorry with mass $7850\text{ kg}$ is moving on a straight slope that makes an angle of $3^\circ$ with the horizontal. A steady resistive force of $1480\text{ N}$ acts against the motion.

Energy, work and power

A particle is let go from rest and moves down the line of greatest slope on a rough plane tilted at $25^\circ$ to the horizontal. The coefficient of friction between the particle and the plane is $0.4$.

Kinematics of motion in a straight line

Particles $A$ and $B$ have masses $0.35\,\text{kg}$ and $0.45\,\text{kg}$ respectively. They are joined by the ends of a light inextensible string that passes over a small fixed smooth pulley, which is $1\,\text{m}$ above the horizontal ground. Initially, particle $A$ is resting on the ground directly beneath the pulley, with the string taut. Particle $B$ hangs vertically below the pulley, at a height of $0.64\,\text{m}$ above the ground. Particle $A$ is released.

Kinematics of motion in a straight line

A particle is released from a fixed origin with velocity $0.4\,\text{m s}^{-1}$ and travels along a straight line. After $t$ seconds, the acceleration $a\,\text{m s}^{-2}$ is given by $a = k(3t^2 - 12t + 2)$, where $k$ is constant. At $t = 1$, the velocity of $P$ is $0.1\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

The diagram plots the velocity-time graphs of two particles, $P$ and $Q$, moving along the same straight line. $P$'s graph is made up of four straight-line sections, whereas $Q$'s graph has three straight-line sections. Both particles begin at the same starting point $O$ on the line. $Q$ sets off $2$ seconds later than $P$, and each particle is at rest again at time $t = T$. The maximum velocity of $Q$ is $V\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A particle $P$ of mass $0.2\,\text{kg}$ is at rest on a rough plane tilted at $30^\circ$ to the horizontal. The coefficient of friction between $P$ and the plane is $0.3$. A force of magnitude $T\,\text{N}$ acts on $P$ upwards at $15^\circ$ above a line of greatest slope of the plane (see diagram).

Energy, work and power

A rough-walled hollow cylinder has radius $0.5\,\text{m}$. A particle $P$ of mass $0.4\,\text{kg}$ is touching the inside surface of the cylinder. The particle and the cylinder turn together with angular speed $6\,\text{rad s}^{-1}$ about the cylinder’s vertical axis, so the particle travels round in a horizontal circle (see diagram).

Probability

A small ball is launched from a point $1.5\,\text{m}$ above level ground. When it reaches a height of $9\,\text{m}$ above the ground, it is moving at $45^\circ$ above the horizontal with speed $4\,\text{m s}^{-1}$.

Probability

A light inextensible string of length $0.4\,\text{m}$ has one end fixed at point $A$, which lies above a smooth horizontal surface. The other end is attached to a particle $P$ of mass $0.6\,\text{kg}$. With the string taut and making an angle of $60^\circ$ with the horizontal (see diagram), $P$ travels in a circle on the surface at constant speed $v\,\text{m s}^{-1}$.

Probability

A particle $P$ is fired from a point $O$ on horizontal ground with speed $25\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. After $t\,\text{s}$, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical displacement upwards from $O$ is $y\,\text{m}$, respectively.

Representation of data

One end of a light elastic string whose natural length is $0.8\,\text{m}$ and whose modulus of elasticity is $24\,\text{N}$ is fixed at point $O$. The opposite end is connected to a particle $P$ of mass $0.3\,\text{kg}$. From a point $1.2\,\text{m}$ vertically beneath $O$, $P$ is projected vertically upwards at speed $4\,\text{m s}^{-1}$.

Probability

A uniform hemisphere of radius $0.4\,\text{m}$ is joined to a uniform cylinder of radius $0.4\,\text{m}$ so that the two circular edges match exactly. Each part has weight $20\,\text{N}$. The object’s centre of mass is at $O$, the centre of the common circular face.

Representation of data

A particle $P$ with mass $0.2\,\text{kg}$ is released from rest at point $O$ on a rough plane inclined at $60^\circ$ to the horizontal, and moves down the line of greatest slope. The coefficient of friction between $P$ and the plane is $0.3$. A force of magnitude $0.6x\,\text{N}$ acts on $P$ in the direction $PO$, where $x$ is the displacement of $P$ from $O$.

Representation of data

A particle $P$ with mass $0.2\,\text{kg}$ is let go from rest at a point $O$ on a smooth horizontal plane. A horizontal force of magnitude $te^{-v}\,\text{N}$ acting away from $O$ acts on $P$, where $v\,\text{m}\,\text{s}^{-1}$ denotes the velocity of $P$ at time $t$ s after release.

Probability

A uniform solid cone has height $0.6\,\text{m}$ and base radius $0.2\,\text{m}$. A uniform hollow cylinder, open at both ends, has the same dimensions. The cone is placed inside the cylinder so that the cone’s base matches one end of the cylinder (see the diagram, which is a cross-section). The object has total weight $60\,\text{N}$ and its centre of mass lies $0.25\,\text{m}$ from the base of the cone.

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is let go from rest at point $O$ on a smooth plane that is inclined at $30^{\circ}$ to the horizontal. $P$ travels down the line of greatest slope passing through $O$. Its speed is $v\,\text{m}\,\text{s}^{-1}$ when its displacement from $O$ is $x\,\text{m}$. A retarding force of magnitude $0.2v^2\,\text{N}$ acts on $P$ in the direction $PO$.

Probability

A light elastic string has natural length $2\,\text{m}$ and modulus of elasticity $39\,\text{N}$. The string’s ends are fastened to fixed points $A$ and $B$, which lie at the same horizontal height and are $2.4\,\text{m}$ apart. A particle $P$ of mass $m\,\text{kg}$ is joined to the midpoint of the string and is in equilibrium at a point $0.5\,\text{m}$ below $AB$ (see diagram).

Probability

OAB is a uniform lamina in the form of a quadrant of a circle with centre $O$ and radius $0.8\,\text{m}$, and its centre of mass is at $G.$ The lamina is smoothly hinged at $O$ to a fixed point and can rotate freely in a vertical plane. A horizontal force of magnitude $12\,\text{N}$, acting in the plane of the lamina, is applied at $B.$ The lamina is in equilibrium with $AG$ horizontal (see diagram).

Probability

One end of a light elastic string of natural length $0.4\,\text{m}$ and modulus of elasticity $8\,\text{N}$ is fixed at a point $O$ on a smooth horizontal plane. Its other end is joined to a particle $P$ of mass $0.2\,\text{kg}$, which travels round the plane in a circular path centred at $O.$ The speed of $P$ is $v\,\text{m s}^{-1}$ and the string has extension $x\,\text{m}.$

Probability

A small ball $B$ is launched from point $O$, which is $h\,\text{m}$ above a horizontal plane. After $2\,\text{s}$, $B$ is travelling at speed $18\,\text{m s}^{-1}$ in a direction $30^\circ$ above the horizontal.

Representation of data

A particle $P$ of mass $0.4\text{ kg}$ is touching the rough inner wall of a hollow cylinder of radius $0.5\text{ m}$. The particle and cylinder turn together at angular speed $6\text{ rad s}^{-1}$ about the cylinder’s vertical axis, so that $P$ travels in a horizontal circle (see diagram).

Probability

A small ball is launched from a point $1.5\text{ m}$ above horizontal ground. At a height of $9\text{ m}$ above the ground, the ball is moving at $45^\circ$ above the horizontal with velocity $4\text{ m s}^{-1}$. Find the angle at which the ball is projected.

Probability

A light inextensible string of length $0.4\text{ m}$ has one end fixed at point $A$, which lies above a smooth horizontal surface. A particle $P$ of mass $0.6\text{ kg}$ is attached to the free end of the string. With the string taut and inclined at an angle of $60^\circ$ to the horizontal (see diagram), $P$ moves round the surface in a circle with constant speed $v\text{ m s}^{-1}$.

Probability