May/June 2010

The acute angle $x$ radians satisfies $\tan x = k$, with $k$ as a positive constant.

Trigonometry

The diagram depicts parallelogram $OABC$. You are given that $\vec{OA} = i + 3j + 3k$ and $\vec{OC} = 3i - j + k$.

Coordinate geometry

Find the first $3$ terms of the expansion of $(2x - \frac{3}{x})^5$ in descending powers of $x$.

Series

In an arithmetic progression, the 9th term has value $22$, while the sum of the first $4$ terms is $49$. The $n$th term of the progression is $46$.

Series

The diagram depicts the curve $y = 6x - x^2$ together with the line $y = 5$.

Integration

The function $f$ is defined by $f(x) = 2\sin^2 x - 3\cos^2 x$ for $0 \le x \le \pi$.

Trigonometry

A curve is defined by $\frac{dy}{dx} = 3x^2 - 6$, and the point $(9, 2)$ lies on it.

Differentiation

The diagram depicts a section of the curve $y = 2 - \frac{18}{2x + 3}$, which meets the $x$-axis at $A$ and the $y$-axis at $B$. The normal drawn to the curve at $A$ meets the $y$-axis at $C$.

Coordinate geometry

The diagram presents triangle $ABC$, with $A$ at $(3, -2)$ and $B$ at $(15, 22)$. The gradients of $AB$, $AC$ and $BC$ are $2m$, $-2m$ and $m$ respectively, where $m$ is a positive constant.

Coordinate geometry

For $x \in \mathbb{R}$, the function $f$ maps $x$ to $2x^2 - 12x + 7$. For the same domain, $g$ maps $x$ to $2x + k$.

Quadratics

Show that the equation $3(2\sin x - \cos x) = 2(\sin x - 3\cos x)$ may be rearranged into the form $\tan x = -\tfrac{3}{4}$.

Trigonometry

The curve is defined by $y = \tfrac{1}{6}(2x-3)^3 - 4x$.

Differentiation

Consider the function $f: x \mapsto 4 - 3\sin x$, defined over the domain $0 \leq x \leq 2\pi$.

Functions

The diagram shows a segment of the curve $y = \tfrac{a}{x}$, where $a$ is a positive constant.

Integration

For $x \in \mathbb{R}$, the functions $f$ and $g$ are given by $f: x \mapsto 4x - 2x^2$, $g: x \mapsto 5x + 3$.

Functions

From the diagram, $A$ has the coordinates $(-1,3)$ and $B$ has the coordinates $(3,1)$. Line $L_1$ goes through $A$ and is parallel to $OB$. Line $L_2$ goes through $B$ and is perpendicular to $AB$. The lines $L_1$ and $L_2$ intersect at $C$.

Coordinate geometry

With respect to an origin $O$, the position vectors of the points $A$ and $B$ are $\vec{OA} = \begin{pmatrix}-2 \\ 3 \\ 1\end{pmatrix}$ and $\vec{OB} = \begin{pmatrix}4 \\ 1 \\ p\end{pmatrix}$.

Coordinate geometry

Find the first $3$ terms of the expansion of $(1+ax)^5$ in ascending powers of $x$.

Series

Find the total of all the multiples of $5$ from $100$ to $300$ inclusive.

Series

A solid rectangular block is built with a square base of side $x$ cm. Its height is $h$ cm, and the block’s total surface area is $96\text{ cm}^2$.

Differentiation

The diagram shows the curve $y = (x-2)^2$ and the line $y + 2x = 7$, which meet at points $A$ and $B$.

Integration

The initial term in a geometric progression is 12, and the next term is $-6$.

Series

The domain of the function $f : x \mapsto 2x^2 - 8x + 14$ is $x \in \mathbb{R}$.

Differentiation

State the first three terms, arranged from highest to lowest power of $x$, in the expansion of $(x - \frac{2}{x})^6$.

Series

The function $f : x \mapsto a + b\cos x$ is defined for $0 \leq x \leq 2\pi$. It is given that $f(0) = 10$ and that $f(\frac{2}{3}\pi) = 1$.

Trigonometry

Demonstrate that the equation $2\sin x \tan x + 3 = 0$ can be rewritten as $2\cos^2 x - 3\cos x - 2 = 0$.

Trigonometry

The curve is described by the differential equation $\frac{dy}{dx} = \frac{6}{\sqrt{3x - 2}}$, and it goes through the point $P(2, 11)$.

Differentiation

With respect to an origin $O$, the position vectors of points $A$, $B$ and $C$ are $overrightarrow{OA} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$, $\overrightarrow{OB} = 3\mathbf{i} + 2\mathbf{j} + 8\mathbf{k}$, and $\overrightarrow{OC} = -\mathbf{i} - 2\mathbf{j} + 10\mathbf{k}$.

Coordinate geometry

The diagram illustrates a metal plate $ABCDEF$ formed by removing the two shaded regions from a circle of radius $10$ cm and centre $O$. The parallel sides $AB$ and $ED$ each have length $12$ cm.

Circular measure

The diagram depicts a rhombus $ABCD$ with $A$ at $(-1, 2)$, $C$ at $(5, 4)$, and $B$ located on the $y$-axis.

Coordinate geometry

The diagram displays part of the curve $y = x + \frac{4}{x}$, and this curve has a minimum point at $M$. The line $y = 5$ cuts the curve at the points $A$ and $B$.

Integration

Solve the inequality $|2x - 3| > 5$.

Algebra

Show that the value of $\int_0^6 \frac{1}{x + 2}\,dx$ is $2\ln 2$.

Integration

Demonstrate that the equation $\tan(x + 45^\circ) = 6\tan x$ may be rewritten as $6\tan^2 x - 5\tan x + 1 = 0$.

Trigonometry

The polynomial $x^3 + 3x^2 + 4x + 2$ is represented by $f(x)$.

Algebra

Given that $y = 2^x$, show that $2^x + 3(2^{-x}) = 4$ can be expressed as $y^2 - 4y + 3 = 0$.

Logarithmic and exponential functions

The curve is given by $x^2y + y^2 = 6x$.

Differentiation

Show by drawing a suitable pair of graphs that the equation $e^{2x} = 2 - x$ has only one root.

Numerical solution of equations

Differentiate $\frac{\cos x}{\sin x}$ to demonstrate that, when $y = \cot x$, $\frac{dy}{dx} = -\csc^2 x$.

Integration

Use logarithms on $13^x = (2.8)^y$ to show that $y = kx$, then determine $k$ correct to $3$ significant figures.

Logarithmic and exponential functions

The diagram illustrates a section of the curve $y = xe^{-x}$. The shaded region $R$ is enclosed by the curve and the lines $x = 2$, $x = 3$ and $y = 0$.

Integration

Solve the inequality $|2x - 1| < |x + 4|$.

Algebra

Show that $\int_0^{\frac{1}{4}\pi} \cos 2x\,dx = \frac{1}{2}$.

Integration

A curve has equation $y = x^3e^{-x}$.

Differentiation

By drawing a suitable pair of graphs, demonstrate that the equation $\ln x = 2 - x^2$ has only one root.

Numerical solution of equations

The polynomial $2x^3 + ax^2 + bx + 6$, with $a$ and $b$ as constants, is written as $p(x)$. It is stated that dividing $p(x)$ by $(x - 3)$ leaves a remainder of $30$, and dividing $p(x)$ by $(x + 1)$ leaves a remainder of $18$.

Algebra

Prove that the identity $\sin(x - 30^\circ) + \cos(x - 60^\circ) \equiv (\sqrt{3})\sin x$ is true.

Trigonometry

Starting from $13^x = (2.8)^y$, apply logarithms to demonstrate that $y = kx$ and determine the value of $k$ correct to $3$ significant figures.

Logarithmic and exponential functions

The diagram depicts a section of the curve $y = xe^{-x}$. The shaded region $R$ is enclosed by the curve together with the lines $x = 2$, $x = 3$ and $y = 0$.

Integration

Solve $|2x - 1| < |x + 4|$.

Algebra

Show that the value of $\int_0^{\pi} \cos 2x \, dx$ is $\frac{1}{2}$.

Integration

The curve is defined by $y = x^3e^{-x}$.

Differentiation

By drawing an appropriate pair of graphs, show that the equation $\ln x = 2 - x^2$ has just one root.

Numerical solution of equations

Let $p(x)$ denote the polynomial $2x^3 + ax^2 + bx + 6$, where $a$ and $b$ are constants. It is stated that dividing $p(x)$ by $(x - 3)$ leaves a remainder of $30$, and dividing $p(x)$ by $(x + 1)$ leaves a remainder of $18$.

Algebra

Prove the identity that $\sin(x - 30^{\circ}) + \cos(x - 60^{\circ}) = (\sqrt{3})\sin x$.

Trigonometry

Find the solution of the inequality $|x + 3a| > 2|x - 2a|$, with $a$ a positive constant.

Algebra

The vector equations of the lines $l$ and $m$ are $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + s(\mathbf{i} - \mathbf{j} + 2\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + \mathbf{k} + t(2\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ respectively.

Vectors

Determine all solutions of $\sin \theta = 2\cos 2\theta + 1$ for $0^{\circ} \leq \theta \leq 360^{\circ}$.

Trigonometry

The variables $x$ and $y$ are linked by the equation $x^n y = C$, where $n$ and $C$ are constants. For $x = 1.10$, $y = 5.20$, and for $x = 3.20$, $y = 1.05$.

Logarithmic and exponential functions

By expanding $\cos(3x - x)$ and $\cos(3x + x)$, prove that $\frac{1}{2}(\cos 2x - \cos 4x) = \sin 3x \sin x$.

Trigonometry

If $y = 0$ when $x = 1$.

Differential equations

The diagram depicts a semicircle $ACB$ with centre $O$ and radius $r$. The angle $BOC$ is $x$ radians. The shaded segment has an area equal to one quarter of the semicircle’s area.

Numerical solution of equations

The complex number $2 + 2i$ is represented by $u$.

Complex numbers

Rewrite $\frac{2}{(x + 1)(x + 3)}$ as partial fractions.

Integration

The diagram depicts the curve $y = \sqrt{\frac{1 - x}{1 + x}}$.

Differentiation

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

Logarithmic and exponential functions

Determine the constants $A$, $B$, $C$ and $D$ so that $\frac{2x^3 - 1}{x^2(2x - 1)} \equiv A + \frac{B}{x} + \frac{C}{x^2} + \frac{D}{2x - 1}$.

Integration

Show that $\int_0^{\pi} x^2 \sin x \, dx = \pi^2 - 4$.

Integration

You are told that $\cos a = \frac{3}{5}$, with $0^\circ < a < 90^\circ$. Show your working, and without using a calculator to evaluate $a$,

Trigonometry

The sketch depicts the curve $y = \frac{\sin x}{x}$ for $0 < x \leq 2\pi$, together with its minimum point $M$.

Numerical solution of equations

Define $p(x)$ as the polynomial $2x^3 + 5x^2 + ax + b$, with $a$ and $b$ as constants. It is stated that $(2x + 1)$ is a factor of $p(x)$, and that the remainder on dividing $p(x)$ by $(x + 2)$ is $9$.

Algebra

The curve is described by the equation $x \ln y = 2x + 1$.

Differentiation

The variables $x$ and $t$ satisfy the differential equation $e^{2t}\frac{dx}{dt} = \cos^2 x$, with $t \geq 0$. When $t = 0$, $x = 0$.

Differential equations

The complex variable z is defined by z = 1 + \cos 2\theta + i \sin 2\theta, and \theta can take any value in the interval -\frac{1}{2}\pi < \theta < \frac{1}{2}\pi.

Complex numbers

Plane $p$ is given by $3x + 2y + 4z = 13$. A further plane $q$, with equation $ax + y + z = 4$, is perpendicular to $p$, where $a$ is a constant.

Vectors

Solve the inequality $|x - 3| > 2|x + 1|$ and determine all values of $x$.

Algebra

The line $l$ is defined by $\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$. The plane $p$ has equation $3x - y + 2z = 9$. The point at which $l$ meets the plane $p$ is $A$.

Vectors

The variables $x$ and $y$ are related by $y^3 = A e^{2x}$, with $A$ constant. A plot of $\ln y$ against $x$ is linear.

Logarithmic and exponential functions

Solve the equation $\tan(45^\circ - x) = 2\tan x$, and give all solutions for $0^\circ < x < 180^\circ$.

Trigonometry

Suppose that $x = 1$ when $t = 0$,

Differential equations

The curve drawn in the diagram is $y = e^{-x} - e^{-2x}$, and its turning point of maximum value is $M$. The $x$-coordinate of $M$ is written as $p$.

Integration

The curve $y = \frac{\ln x}{x + 1}$ has a single stationary point.

Numerical solution of equations

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

Trigonometry

The equation $2x^3 - x^2 + 2x + 12 = 0$ has one real root and two complex roots. By showing your working, confirm that $1 + i\sqrt{3}$ is one of the complex roots. State the other complex root.

Complex numbers

Rewrite $\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}$ as a partial fraction decomposition.

Algebra

The car has mass $1150 \text{ kg}$ and moves uphill along a straight slope that makes an angle of $1.2^{\circ}$ with the horizontal. The force resisting the motion is $975 \text{ N}$.

Energy, work and power

The diagram presents the velocity-time graph for a machine cutting tool’s motion. It is made up of five straight-line sections. The tool travels forward for $8 \text{ s}$ while cutting, then needs $3 \text{ s}$ to get back to its starting point.

Kinematics of motion in a straight line

A ring of mass $0.8 \text{ kg}$, small in size, is placed on a rough rod fixed horizontally. The ring is in equilibrium and experiences a force of magnitude $7 \text{ N}$ acting upward at $45^{\circ}$ to the horizontal.

Forces and equilibrium

Three coplanar forces with magnitudes $250 \text{ N}$, $160 \text{ N}$ and $370 \text{ N}$ are applied at a point $O$ in the directions indicated in the diagram, with angle $\alpha$ defined by $\sin \alpha = 0.28$ and $\cos \alpha = 0.96$.

Forces and equilibrium

$P$ and $Q$ are two fixed points on the line of greatest slope of an inclined plane. Point $Q$ is $0.45 \text{ m}$ above the level of $P$. A particle with mass $0.3 \text{ kg}$ moves in the upward direction along $PQ$.

Energy, work and power

Particles $A$ and $B$, with masses $0.2 \text{ kg}$ and $0.45 \text{ kg}$ respectively, are linked by a light inextensible string of length $2.8 \text{ m}$. The string runs over a small smooth pulley at the edge of a rough horizontal surface, which is $2 \text{ m}$ above the floor. Particle $A$ is kept in contact with the surface at a distance of $2.1 \text{ m}$ from the pulley, while particle $B$ hangs freely. The coefficient of friction between $A$ and the surface is $0.3$. When particle $A$ is released, the system starts to move.

Newton's laws of motion

A vehicle travels along a straight line. Its velocity $v \text{ m s}^{-1}$ at time $t \text{ s}$ after it begins is given by $$v = A(t - 0.05t^{2}) \text{ for } 0 \le t \le 15,$$ $$v = \frac{B}{t^{2}} \text{ for } t > 15,$$ where $A$ and $B$ are constants. The distance the vehicle covers from $t = 0$ to $t = 15$ is $225 \text{ m}$.

Kinematics of motion in a straight line

A car with mass $1150\,\text{kg}$ moves along a straight hill that is tilted at $1.2^\circ$ above the horizontal. The resistive force acting on the car is $975\,\text{N}$.

Energy, work and power

The diagram presents a velocity-time graph for the motion of a machine’s cutting tool. It is made up of five straight-line sections. The tool travels forwards for $8\,\text{s}$ while it is cutting and then needs $3\,\text{s}$ to get back to its starting point.

Kinematics of motion in a straight line

A rough horizontal rod carries a small ring of mass $0.8\,\text{kg}$, and the ring is held in equilibrium while a force of magnitude $7\,\text{N}$ acts on it upwards at $45^\circ$ to the horizontal.

Forces and equilibrium

Three coplanar forces with magnitudes $250\,\text{N}$, $160\,\text{N}$ and $370\,\text{N}$ act at point $O$ in the directions shown in the diagram, where the angle $\alpha$ satisfies $\sin\alpha = 0.28$ and $\cos\alpha = 0.96$.

Forces and equilibrium

$P$ and $Q$ are two fixed points lying on the line of greatest slope of an inclined plane. The point $Q$ is $0.45\,\text{m}$ higher than the level of $P$. A particle with mass $0.3\,\text{kg}$ travels upwards along $PQ$.

Energy, work and power

Particles $A$ and $B$, with masses $0.2\,\text{kg}$ and $0.45\,\text{kg}$ respectively, are joined by a light inextensible string of length $2.8\,\text{m}$. The string goes over a small smooth pulley at the edge of a rough horizontal surface, which is $2\,\text{m}$ above the floor. Particle $A$ is kept in contact with the surface at a distance of $2.1\,\text{m}$ from the pulley, while particle $B$ hangs freely. The coefficient of friction between $A$ and the surface is $0.3$. Particle $A$ is released and the system starts to move.

Newton's laws of motion

A vehicle travels along a straight line. Its velocity $v\,\text{m s}^{-1}$, measured at time $t\,\text{s}$ after the start of motion, is defined by $v = A(t - 0.05t^2)$ for $0 \le t \le 15$, and by $v = \frac{B}{t^2}$ for $t > 15$, where $A$ and $B$ are constants. The distance covered by the vehicle from $t = 0$ to $t = 15$ is $225\,\text{m}$.

Kinematics of motion in a straight line

Three forces acting in one plane are applied at a single point. Their magnitudes are $5.5\,\text{N}$, $6.8\,\text{N}$ and $7.3\,\text{N}$, and the directions in which they act are shown in the diagram.

Forces and equilibrium

A particle is set off from point $O$ and travels along a straight line. Its velocity $t\,\text{s}$ after leaving $O$ is $(1.2t - 0.12t^2)\,\text{m s}^{-1}$. Find the displacement of the particle from $O$ when its acceleration is $0.6\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A load is moved along a horizontal straight track from $A$ to $B$ by a force of magnitude $P\,\text{N}$ acting at $30^\circ$ above the horizontal. The distance $AB$ is $80\,\text{m}$. As the load travels from $A$ to the midpoint $M$ of $AB$, its speed stays constant at $1.2\,\text{m s}^{-1}$.

Energy, work and power

The sketch depicts a vertical cross-section through a triangular prism that is secured so that two of its faces make an angle of $60^\circ$ with the horizontal. One face is smooth, while the other is rough. Particles $A$ and $B$, with masses $0.36\,\text{kg}$ and $0.24\,\text{kg}$ respectively, are joined to the ends of a light inextensible string that passes over a small smooth pulley fixed at the highest point of the cross-section. $B$ is kept at rest at a point on the rough face of the cross-section, and $A$ hangs freely in contact with the smooth face (see diagram). When $B$ is released, it moves up the face with acceleration $0.25\,\text{m s}^{-2}$.

Newton's laws of motion

A ball travels along the horizontal surface of a billiards table, slowing down with constant deceleration of magnitude $d\,\text{m s}^{-2}$. It leaves $A$ at speed $1.4\,\text{m s}^{-1}$ and, $1.2\,\text{s}$ later, arrives at the table edge at $B$ with speed $1.1\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

Particles $P$ and $Q$ travel along the line of greatest slope on a smooth inclined plane. $P$ starts from rest at point $O$ on the line, and after $2\,\text{s}$ it passes point $A$ with speed $3.5\,\text{m s}^{-1}$.

Newton's laws of motion

Two rectangular boxes $A$ and $B$ have the same dimensions. They are initially stationary on a rough horizontal floor, with $A$ placed on top of $B$. Box $A$ has mass $200\,\text{kg}$ and box $B$ has mass $250\,\text{kg}$. A horizontal force of magnitude $P\,\text{N}$ acts on $B$ (see diagram). The boxes stay at rest when $P \le 3150$ and begin to move when $P > 3150$.

Newton's laws of motion

A frame is made from a uniform semicircular wire with radius $20\,\text{cm}$ and mass $2\,\text{kg}$, together with a uniform straight wire of length $40\,\text{cm}$ and mass $0.9\,\text{kg}$. The semicircular wire’s ends are fixed to the two ends of the straight wire (see diagram).

Representation of data

A solid cone of uniform density has height $30\,\text{cm}$ and base radius $r\,\text{cm}$. It is positioned with its axis vertical on a rough horizontal plane. The plane is then slowly tilted, and the cone remains in equilibrium until the angle of inclination reaches $35^\circ$, when it topples. The diagram shows a cross-section of the cone.

Representation of data

A particle with mass $0.24\,\text{kg}$ is connected to one end of a light inextensible string of length $2\,\text{m}$. The string’s other end is fixed at a point. The particle travels at constant speed around a horizontal circle. The string is inclined at angle $\theta$ to the vertical (see diagram), and the tension in the string is $T\,\text{N}$. The particle has acceleration of magnitude $7.5\,\text{m s}^{-2}$.

Probability

A lamina of uniform density and weight $15\,\text{N}$ is shaped as trapezium $ABCD$, with the measurements shown in the diagram. It is freely hinged at $A$ to a fixed point. One end of a light inextensible string is fastened to the lamina at $B$. The lamina is in equilibrium with $AB$ horizontal; the string is taut, lies in the same vertical plane as the lamina, and is inclined at an angle of $30^\circ$ above the horizontal (see diagram).

Probability

A particle is fired from a point $O$ on level ground. The initial speed is $20\,\text{m s}^{-1}$, and it is directed upwards at an angle $\theta$ to the horizontal. The particle goes through the point that is $7\,\text{m}$ above the ground and $16\,\text{m}$ horizontally from $O$, and lands on the ground at $A$.

Representation of data

A particle $P$ with mass $0.35\,\text{kg}$ is fixed to the midpoint of a light elastic string whose natural length is $4\,\text{m}$. The two ends of the string are fastened to fixed points $A$ and $B$, which are $4.8\,\text{m}$ apart and at the same horizontal level. $P$ is in equilibrium at a point $0.7\,\text{m}$ vertically beneath the midpoint $M$ of $AB$ (see diagram).

Representation of data

A particle $P$ with mass $0.25\,\text{kg}$ travels in a straight line on a smooth horizontal surface. $P$ begins at $O$ with speed $10\,\text{m s}^{-1}$ and travels towards a fixed point $A$ on the line. At time $t\,\text{s}$, the displacement of $P$ from $O$ is $x\,\text{m}$ and its velocity is $v\,\text{m s}^{-1}$. A resistive force of magnitude $(5 - x)\,\text{N}$ acts on $P$ in the direction towards $O$.

Representation of data

The frame is formed from a uniform semicircular wire of radius $20\text{ cm}$ and mass $2\text{ kg}$, together with a uniform straight wire of length $40\text{ cm}$ and mass $0.9\text{ kg}$. The two ends of the semicircular wire are joined to the two ends of the straight wire (see diagram).

Representation of data

A uniform solid cone of height $30\text{ cm}$ and base radius $r\text{ cm}$ stands with its axis vertical on a rough horizontal plane. The plane is then gradually tilted, and the cone stays in equilibrium until the plane’s angle of inclination reaches $35^\circ$, at which point the cone topples. The diagram shows a cross-section through the cone.

Probability

A particle with mass $0.24\text{ kg}$ is fixed to one end of a light inextensible string of length $2\text{ m}$. The opposite end of the string is secured at a fixed point. The particle travels at constant speed in a horizontal circle. The string is inclined at an angle $\theta$ to the vertical (see diagram), and the tension in the string is $T\text{ N}$. The particle’s acceleration has magnitude $7.5\text{ m s}^{-2}$.

Probability

A uniform lamina weighing $15\text{ N}$ is shaped as a trapezium $ABCD$, with the measurements shown in the diagram. The lamina is hinged freely at $A$ to a fixed point. One end of a light inextensible string is fastened to the lamina at $B$. The lamina is in equilibrium with $AB$ horizontal; the string is taut, lies in the same vertical plane as the lamina, and is inclined at an angle of $30^\circ$ above the horizontal (see diagram).

Probability

A particle is fired from a point $O$ on level ground. Its initial speed is $20\text{ m s}^{-1}$ and it is directed upwards at an angle $\theta$ to the horizontal. The particle goes through the point that is $7\text{ m}$ above the ground and $16\text{ m}$ horizontally from $O$, and then lands on the ground at $A$.

Representation of data

A particle $P$ with mass $0.35\text{ kg}$ is fastened to the midpoint of a light elastic string whose natural length is $4\text{ m}$. The two ends of the string are fixed at points $A$ and $B$, which are $4.8\text{ m}$ apart and lie at the same horizontal height. In equilibrium, $P$ hangs at a point $0.7\text{ m}$ vertically beneath the midpoint $M$ of $AB$ (see diagram).

Probability

A particle $P$ with mass $0.25\text{ kg}$ travels in a straight line along a smooth horizontal surface. It begins at the point $O$ with speed $10\text{ m s}^{-1}$ and travels towards a fixed point $A$ on the line. After $t\text{ s}$, the displacement of $P$ from $O$ is $x\text{ m}$ and its velocity is $v\text{ m s}^{-1}$. A resistive force of magnitude $(5 - x)\text{ N}$ acts on $P$ in the direction towards $O$.

Representation of data