May/June 2016

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

Series

With origin $O$, the position vectors of $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 5 \\ -1 \\ k \end{pmatrix}$ and $\vec{OC} = \begin{pmatrix} 2 \\ 6 \\ -3 \end{pmatrix}$ respectively, where $k$ is a constant.

Coordinate geometry

For $-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi$, the function $f$ is specified by $f : x \mapsto 4\sin x - 1$.

Functions

Solve the equation $3\sin^2\theta = 4\cos\theta - 1$ for values of $\theta$ in the interval $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The diagram illustrates a section of the curve $x = \frac{12}{y^2} - 2$. The shaded area is enclosed by the curve, the $y$-axis, and the lines $y = 1$ and $y = 2$.

Integration

For the curve, $\frac{dy}{dx} = 2 - 8(3x + 4)^{-\frac{1}{2}}$.

Differentiation

A farmer splits a rectangular plot of land into $8$ equal rectangular sheep pens, as the diagram shows. Each pen has dimensions $x$ m by $y$ m and is completely surrounded by metal fencing. Altogether, the farmer uses $480$ m of fencing.

Differentiation

Determine the values of the constant $m$ for which the line $y = mx$ is tangent to the curve $y = 2x^2 - 4x + 8$.

Quadratics

In the diagram, $AOB$ makes a quarter circle with centre $O$ and radius $r$. Point $C$ is on arc $AB$, and point $D$ is on $OB$. The line $CD$ runs parallel to $AO$, and angle $AOC = \theta$ radians.

Circular measure

The curve is given by $y = 3x - \frac{4}{x}$ and it passes through $A(1, -1)$ and $B(4, 11)$. At the points $C$ and $D$ on the curve, the tangent is parallel to $AB$. Find the equation of the perpendicular bisector of $CD$.

Differentiation

In a geometric progression where every term is positive, the first term is $50$ and the third term is $32$. Find the sum to infinity of this progression.

Series

The functions $f$ and $g$ are given by $f : x \mapsto 10 - 3x,\ x \in \mathbb{R}$, and $g : x \mapsto \frac{10}{3 - 2x},\ x \in \mathbb{R},\ x \ne \frac{3}{2}$.

Functions

The diagram shows the section of the curve $y = \frac{8}{x} + 2x$ for $x > 0$, together with the minimum point $M$.

Integration

The function $f$ is given by $f : x \mapsto 6x - x^2 - 5$ for all $x \in \mathbb{R}$.

Functions

The curve satisfies $\frac{dy}{dx} = \frac{8}{(5 - 2x)^2}$. Since it goes through $(2, 7)$,

Integration

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

Coordinate geometry

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

Series

From the diagram, triangle $ABC$ is right-angled at $C$, and $M$ is the midpoint of $BC$. You are told that angle $ABC = \frac{\pi}{3}$ radians and angle $BAM = \theta$ radians. If the lengths of $BM$ and $MC$ are each called $x$,

Trigonometry

The figure depicts a circle of radius $r$ cm with centre $O$. $PT$ is tangent to the circle at $P$, and $POT = \alpha$ radians. The line $OT$ intersects the circle at $Q$.

Circular measure

Prove that the identity $\frac{1 + \cos\theta}{1 - \cos\theta} - \frac{1 - \cos\theta}{1 + \cos\theta} = \frac{4}{\sin\theta \tan\theta}$ holds.

Trigonometry

Three points are given by the coordinates $A(0, 7)$, $B(8, 3)$ and $C(3k, k)$. Find the value of the constant $k$ for which

Coordinate geometry

A full water tank contains $2000$ litres. A tiny hole in the bottom is enlarging slowly, so the quantity of water escaping each day is increasing.

Series

Determine the coefficient of $x$ in the expansion of $(\frac{1}{x} + 3x^2)^5$.

Series

The function $f$ is defined by $f(x) = 2x + 3$ when $x \geq 0$. The function $g$ is defined by $g(x) = ax^2 + b$ for $x \leq q$, where $a$, $b$ and $q$ are constants. The composite function $fg$ satisfies $fg(x) = 6x^2 - 21$ for $x \leq q$.

Functions

The vertices of triangle $ABC$ are $A(-2, -1)$, $B(4, 6)$ and $C(6, -3)$.

Coordinate geometry

The diagram illustrates a section of the curve $y = \sqrt{x^3 + 1}$ together with the point $P(2, 3)$ on the curve.

Integration

The curve satisfies $\frac{dy}{dx}=6x^2+\frac{k}{x^3}$ and it passes through the point $P(1,9)$. The gradient at $P$ is $2$.

Integration

The $1$st, $3$rd and $13$th terms in an arithmetic progression are also, in that same order, the $1$st, $2$nd and $3$rd terms of a geometric progression. In both progressions, the first term is $3$.

Series

The curve is given by $y = 8x + (2x - 1)^{-1}$. Find the $x$-values where the curve has stationary points and determine the type of each stationary point, giving reasons for your answers.

Differentiation

The figure depicts triangle $ABC$, with $AB = 5$ cm, $AC = 4$ cm and $BC = 3$ cm. There are three circles centred at $A$, $B$ and $C$ with radii $3$ cm, $2$ cm and $1$ cm, respectively. The circles are tangent to one another at points $E$, $F$ and $G$, which lie on $AB$, $AC$ and $BC$ respectively.

Integration

Point $P(x, y)$ moves along the curve $y = x^2 - \frac{10}{3}x^3 + 5x$ so that the rate at which $y$ changes stays constant.

Differentiation

Show that $3 \sin x \tan x - \cos x + 1 = 0$ may be transformed into a quadratic equation in $\cos x$ and hence solve $3 \sin x \tan x - \cos x + 1 = 0$ for $0 \leq x \leq \pi$.

Trigonometry

The position vectors of $A$, $B$ and $C$ from the origin $O$ are $\vec{OA} = \begin{pmatrix}2\\3\\-4\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}1\\5\\p\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}5\\0\\2\end{pmatrix}$, with $p$ as a constant.

Coordinate geometry

Determine the gradient of the curve $y = 3e^{4x} - 6\ln(2x + 3)$ at the point where $x = 0$.

Differentiation

Find the values of $\theta$ that satisfy the equation $5\tan 2\theta = 4\cot \theta$ for $0^\circ < \theta < 180^\circ$.

Trigonometry

If $3e^x + 8e^{-x} = 14$.

Logarithmic and exponential functions

The polynomial $p(x)$ is given by $p(x) = 8x^3 + 30x^2 + 13x - 25$.

Algebra

The curve is given by the parametric equations $x = 2\tan \theta$, $y = 3\sin 2\theta$, where $0 \leq \theta \leq \frac{1}{2}\pi$.

Differentiation

The curve is defined by $y = \frac{3x^2}{x^2 + 4}$. At the point on the curve where the $x$-coordinate is positive and equal to $p$, the gradient is $\tfrac{1}{2}$.

Numerical solution of equations

Find the value of $\displaystyle \int \frac{1 + \cos^4 2x}{\cos^2 2x}\,dx$.

Integration

Because $5^{3x} = 7^{4y}$.

Logarithmic and exponential functions

Determine the quotient and remainder when $2x^3 - 7x^2 - 9x + 3$ is divided by $x^2 - 2x + 5$.

Algebra

Solve the equation given by $|3u + 1| = |2u - 5|$.

Trigonometry

Show that $\sin(\theta + 60^{\circ}) + \sin(\theta + 120^{\circ})$ is equal to $(\sqrt{3})\cos \theta$.

Trigonometry

The curve is defined by $y = 6xe^{\frac{1}{3}x}$. At the point on it where the $x$-coordinate is $p$, the gradient is $40$.

Numerical solution of equations

Find the value of $\int \frac{4 + e^x}{2e^{2x}}\, dx$.

Integration

The diagram displays the curve defined by the parametric equations $x = 2 - \cos t$, $y = 1 + 3\cos 2t$, for $0 < t < \pi$. The minimum point is $M$, and the curve intersects the $x$-axis at $P$ and $Q$.

Differentiation

Suppose that $5^{3x} = 7^{4y}$.

Logarithmic and exponential functions

Find the quotient together with the remainder when $2x^3 - 7x^2 - 9x + 3$ is divided by $x^2 - 2x + 5$.

Algebra

Solve $|3u + 1| = |2u - 5|$.

Trigonometry

Show that the expression $\sin(\theta + 60^\circ) + \sin(\theta + 120^\circ)$ is equal to $\sqrt{3} \cos \theta$.

Trigonometry

The curve is given by $y = 6x e^{\frac{1}{3}x}$. At the point on the curve where the $x$-coordinate is $p$, the gradient of the curve equals $40$.

Numerical solution of equations

Calculate $\int \frac{4 + e^x}{2e^{2x}}\, dx$.

Integration

The diagram presents the curve defined by the parametric equations $x = 2 - \cos t$, $y = 1 + 3 \cos 2t$, for $0 < t < \pi$. The lowest point is $M$, and the curve intersects the $x$-axis at $P$ and $Q$.

Differentiation

Find the values of $x$ from the equation $2|x - 1| = 3|x|$.

Logarithmic and exponential functions

Without using a calculator and showing all your working, determine the square roots of the complex number $7 - 6\sqrt{2}i$. Present your answers in the form $x + iy$, where $x$ and $y$ are real and exact.

Complex numbers

Determine the exact value of $\int_0^{\frac{1}{2}} x e^{-2x}\,dx$.

Integration

Rewrite $\cosec\,\theta = 3\sin\theta + \cot\theta$ in terms of $\cos\theta$ only, then solve it for $0^\circ < \theta < 180^\circ$.

Trigonometry

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

Differential equations

The curve defined by the equation $y = \sin x \cos 2x$ has a single stationary point in the interval $0 < x < \frac{1}{2}\pi$.

Differentiation

By drawing a suitable pair of graphs, show that the equation $5e^{-x} = \sqrt{x}$ has one root.

Numerical solution of equations

The curve has equation $x^3 - 3x^2y + y^3 = 3$.

Differentiation

Define $f(x) = \frac{4x^2 + 12}{(x + 1)(x - 3)^2}$.

Algebra

Relative to origin $O$, the position vectors of $A$, $B$, $C$, $D$ are $\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$, $\overrightarrow{OB} = 2\mathbf{i} + \mathbf{j} - \mathbf{k}$, $\overrightarrow{OC} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k}$, $\overrightarrow{OD} = -3\mathbf{i} + \mathbf{j} + 2\mathbf{k}$.

Vectors

Apply logarithms to solve the equation $4^{3x-1} = 3(5^x)$, and give the answer correct to 3 decimal places.

Logarithmic and exponential functions

Show all working, and solve the equation $iz^2 + 2z - 3i = 0$, writing your answers in the form $x + iy$, where $x$ and $y$ are exact real numbers.

Complex numbers

Write $\frac{1}{\sqrt{1-2x}}$ as an expansion in ascending powers of $x$, including terms up to and including $x^3$, and simplify the coefficients.

Algebra

Find the exact value of $\int_{0}^{\frac{1}{2}\pi} x^2 \sin 2x \, dx$.

Integration

The curve given by $y = \frac{(\ln x)^2}{x}$ has two stationary points.

Vectors

Prove that $\cos 4\theta - 4\cos 2\theta = 8\sin^4 \theta - 3$.

Trigonometry

The variables $x$ and $\theta$ are linked by the differential equation $(3 + \cos 2\theta)\,\frac{dx}{d\theta} = x \sin 2\theta$, and the condition is that $x = 3$ when $\theta = \frac{1}{4}\pi$.

Differential equations

Define $f(x) = \frac{4x^2 + 7x + 4}{(2x + 1)(x + 2)}$.

Integration

The diagram displays the curve $y = \cosec x$ for $0 < x < \pi$ together with part of the curve $y = e^{-x}$. When $x = a$, the tangents to the two curves are parallel.

Numerical solution of equations

Points $A$, $B$ and $C$ have position vectors relative to the origin $O$, namely $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\overrightarrow{OB} = 4\mathbf{j} + \mathbf{k}$ and $\overrightarrow{OC} = 2\mathbf{i} + 5\mathbf{j} - \mathbf{k}$. A further point $D$ is chosen so that quadrilateral $ABCD$ is a parallelogram.

Vectors

Find the solution to the inequality $2|x - 2| > |3x + 1|$.

Algebra

Define $f(x) = \frac{10x - 2x^2}{(x + 3)(x - 1)^2}$.

Algebra

The variables $x$ and $y$ are linked by $3^y = 4^{2 - x}$.

Logarithmic and exponential functions

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

Trigonometry

A curve is defined parametrically by $x = t + \cos t$, $y = \ln(1 + \sin t)$, with $-\frac{\pi}{2} < t < \frac{\pi}{2}$.

Differentiation

The variables $x$ and $y$ obey the differential equation $\frac{dy}{dx} = e^{-2y}\tan^2 x$, for $0 \le x < \frac{\pi}{2}$, and you are told that $y = 0$ when $x = 0$.

Differential equations

The curve defined by $y = x^2 \cos\left(\tfrac{1}{2}x\right)$ possesses a stationary point at $x = p$ for values of $x$ in the range $0 < x < \pi$.

Numerical solution of equations

Define $I = \int_0^1 \frac{x^5}{(1 + x^2)^3}\,dx$.

Integration

The position vectors of points $A$ and $B$, measured from the origin $O$, are $\vec{OA} = \mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\vec{OB} = 2\mathbf{i} + 3\mathbf{k}$. The line $l$ is given by $\mathbf{r} = 2\mathbf{i} - 2\mathbf{j} - \mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$.

Vectors

For this question, calculators must not be used. Let the complex numbers $-1 + 3i$ and $2 - i$ be represented by $u$ and $v$ respectively. On an Argand diagram with origin $O$, points $A$, $B$ and $C$ stand for the numbers $u$, $v$ and $u + v$ respectively.

Complex numbers

A lift starts from rest and increases its speed upwards at $0.9\,\text{m s}^{-2}$ for $3\,\text{s}$. It then continues for $6\,\text{s}$ at a steady speed and, after that, slows with a constant deceleration until it stops in another $4\,\text{s}$.

Kinematics of motion in a straight line

A box with mass $25\,\text{kg}$ is dragged a distance of $36\,\text{m}$ up a rough plane that is inclined at $20^\circ$ to the horizontal, moving at constant speed. It travels along the line of greatest slope while a steady frictional force of $40\,\text{N}$ acts against it. The pulling force is parallel to the line of greatest slope.

Energy, work and power

A car with mass $1000\,\text{kg}$ moves in a straight line on a level road while experiencing total resistive forces of $300\,\text{N}$.

Energy, work and power

Four coplanar forces with magnitudes $50\,\text{N}$, $48\,\text{N}$, $14\,\text{N}$ and $P\,\text{N}$ act at a point in the directions shown in the diagram. The system is in equilibrium. Given that $\tan\alpha = \frac{7}{24}$, determine the values of $P$ and $\theta$.

Forces and equilibrium

Two particles, with masses $5\,\text{kg}$ and $10\,\text{kg}$, are joined by a light inextensible string passing over a fixed smooth pulley. The $5\,\text{kg}$ particle lies on a rough fixed slope inclined at an angle of $\alpha$ to the horizontal, where $\tan\alpha = \frac{3}{4}$. The $10\,\text{kg}$ particle hangs beneath the pulley (see diagram). The coefficient of friction between the slope and the $5\,\text{kg}$ particle is $\frac{1}{2}$. The particles are released from rest.

Newton's laws of motion

Particle $P$ travels along a straight line. It begins at point $O$ on the line and, $t\,\text{s}$ after leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 6t^2 - 30t + 24$.

Kinematics of motion in a straight line

A particle of mass $30\,\text{kg}$ rests on a plane that is inclined at $20^\circ$ to the horizontal. It is initially at rest and is then pulled up the plane by a force of magnitude $200\,\text{N}$ acting parallel to the line of greatest slope.

Newton's laws of motion

Coplanar forces with magnitudes $7\,\text{N}$, $6\,\text{N}$ and $8\,\text{N}$ act at a point in the directions illustrated in the diagram. If $\sin \alpha = \frac{3}{5}$, find the magnitude and direction of the resultant of the three forces.

Forces and equilibrium

Particle $P$ travels along a straight line, beginning at point $O$. For time $t$ after it has left $O$, the velocity of $P$, in $\text{m s}^{-1}$, is $v = 4t^2 - 8t + 3$.

Kinematics of motion in a straight line

A particle of mass $8\,\text{kg}$ is launched at a speed of $5\,\text{m s}^{-1}$ up the line of greatest slope on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac{5}{13}$. Its motion is opposed by a constant friction force of magnitude $15\,\text{N}$. After moving a distance $x\,\text{m}$ up the plane, the particle is brought to instantaneous rest.

Energy, work and power

A sprinter takes part in a $400\,\text{m}$ race, and his total running time is $52\,\text{s}$. The diagram gives the velocity-time graph for the sprinter’s motion. He begins from rest and increases his speed uniformly to $8.2\,\text{m s}^{-1}$ in $6\,\text{s}$. He then keeps a speed of $8.2\,\text{m s}^{-1}$ for $36\,\text{s}$ before slowing down uniformly to a speed of $V\,\text{m s}^{-1}$ at the finish.

Kinematics of motion in a straight line

A block of mass $2.5\,\text{kg}$ rests on a plane inclined at $30^\circ$ to the horizontal. A light string, which is $20^\circ$ above the line of greatest slope, keeps the block in equilibrium. The tension in the string is $T\,\text{N}$, as shown in the diagram. The coefficient of friction between the block and plane is $\frac{1}{4}$. The block is in limiting equilibrium and is on the point of moving up the plane.

Forces and equilibrium

A car with mass $1100\,\text{kg}$ is travelling along a road while a steady resisting force of $1550\,\text{N}$ acts opposite to the motion.

Energy, work and power

Particle $A$, with mass $1.6\,\text{kg}$, is at rest on a horizontal table and is linked to one end of a light inextensible string. This string runs over a small smooth pulley $P$ at the table’s edge. The free end is connected to particle $B$, of mass $2.4\,\text{kg}$, which hangs vertically beneath the pulley. The system is let go from rest, with the string taut and $B$ positioned $0.5\,\text{m}$ above the ground, as the diagram shows. During the resulting motion, $A$ does not reach $P$ before $B$ hits the ground.

Newton's laws of motion

A particle with mass $8\,\text{kg}$ is drawn, at constant speed, a distance of $20\,\text{m}$ along a rough plane that is inclined at an angle of $30^\circ$ to the horizontal by a force acting in the line of greatest slope.

Energy, work and power

Alan begins moving from point $O$ at a steady speed of $4\,\text{m s}^{-1}$, travelling along a horizontal path. Ben travels on the same path and also sets off from $O$. Ben remains at rest for $5\,\text{s}$ after Alan begins, then accelerates at $1.2\,\text{m s}^{-2}$ for $5\,\text{s}$. After that, Ben keeps moving at a constant speed until he reaches the same point, $P$, as Alan.

Kinematics of motion in a straight line

The coplanar forces in the diagram are balanced.

Forces and equilibrium

A particle of mass $15\,\text{kg}$ rests stationary on a rough plane inclined at an angle of $20^\circ$ to the horizontal. The coefficient of friction between the particle and the plane is $0.2$. A force of magnitude $X\,\text{N}$, applied parallel to a line of greatest slope of the plane, is then used to keep the particle in equilibrium.

Forces and equilibrium

A constant force of magnitude $650\,\text{N}$ opposes the motion of a car with mass $1400\,\text{kg}$.

Energy, work and power

Two particles with masses $1.3\,\text{kg}$ and $0.7\,\text{kg}$ are joined by a light inextensible string passing over a fixed smooth pulley. The particles are initially at the same vertical level, with the string taut. Each particle is $2\,\text{m}$ above a horizontal plane, and each is $4\,\text{m}$ beneath the pulley. The particles are then released from rest.

Newton's laws of motion

A particle $P$ travels along a straight line. At time $t\,\text{s}$, its displacement from $O$ is $s\,\text{m}$, and its acceleration is $a\,\text{m s}^{-2}$, where $a = 6t - 2$. When $t = 1$, $s = 7$ and when $t = 3$, $s = 29$.

Kinematics of motion in a straight line

A small ball is fired from a point on horizontal ground with speed $16\text{ m s}^{-1}$ at an angle of $45^{\circ}$ above the horizontal.

Representation of data

A uniform wire is bent into a semicircular arc, and the diameter $AB$ has length $0.8\text{ m}$. The wire is joined to a vertical wall by a smooth hinge at $A$. It is in equilibrium with $AB$ making an angle of $70^{\circ}$ to the upward vertical, supported by a light string fixed to $B$. The other end of the string is fixed at point $C$ on the wall, $0.8\text{ m}$ vertically above $A$. The tension in the string is $15\text{ N}$ (see diagram).

Representation of data

A particle $P$ with mass $0.4\text{ kg}$ is released from rest at point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. If the displacement of $P$ from $O$ is $x\text{ m}$ down the plane, its velocity is $v\text{ m s}^{-1}$. A force of magnitude $0.8\mathrm{e}^{-x}\text{ N}$ acts on $P$ up the plane along the line of greatest slope through $O$.

Probability

A uniform solid cone has base radius $0.4\text{ m}$ and height $4.4\text{ m}$. A uniform solid cylinder has radius $0.4\text{ m}$ and weight equal to the weight of the cone. The cone and cylinder are joined together so that the cone’s base and one circular end of the cylinder touch, with their circumferences matching exactly. The combined body is in equilibrium with its circular base resting on a plane that is inclined at $20^{\circ}$ to the horizontal (see diagram).

Representation of data

A particle is launched at an angle of $\theta^{\circ}$ below the horizontal from the top of a vertical cliff that is $26\text{ m}$ high. It lands on horizontal ground $8\text{ m}$ from the foot of the cliff $2\text{ s}$ after it is projected.

Probability

A light inextensible string is threaded through a small smooth bead $B$ of mass $0.4\text{ kg}$. One end of the string is fastened to a fixed point $A$ $0.4\text{ m}$ above a fixed point $O$ on a smooth horizontal surface, and the other end is fastened to a fixed point $C$ which lies vertically below $A$ and $0.3\text{ m}$ above the surface. The bead moves at constant speed on the surface in a circle of centre $O$ and radius $0.3\text{ m}$ (see diagram).

Probability

A particle $P$ is fastened to one end of a light elastic string whose natural length is $1.2\text{ m}$ and modulus of elasticity is $12\text{ N}$. The other end of the string is fixed at point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. $P$ is in equilibrium on the plane, $1.6\text{ m}$ from $O$.

Probability

A small ball $B$ is launched from a point $O$ on horizontal ground with speed $12\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. After $0.8\,\text{s}$, $B$ is $0.5\,\text{m}$ vertically above the top of a vertical post.

Representation of data

A light elastic string with natural length $0.4\,\text{m}$ has one end fastened to the fixed point $O$. Its other end is attached to a particle of weight $5\,\text{N}$, and the particle is in equilibrium $0.6\,\text{m}$ vertically below $O$.

Probability

The point $O$ lies $8\,\text{m}$ above a horizontal plane. A particle $P$ is projected from $O$. Once projected, the horizontal displacement of $P$ from $O$ is $x\,\text{m}$ and its vertically upward displacement from $O$ is $y\,\text{m}$. The path of $P$ is given by $y = 2x - x^2$.

Representation of data

A uniform body is obtained by boring a cylindrical hole right through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section $ABCD$ that passes through the centre of mass of the body. $AB = CD = 0.7\,\text{m}$, $BC = AD = 0.4\,\text{m}$, and the centre of the hole is $0.1\,\text{m}$ from $AB$ and $0.2\,\text{m}$ from $AD$ (see diagram). The hole has cross-sectional area $0.03\,\text{m}^2$.

Representation of data

Particle P, of mass $0.4\,\text{kg}$, is initially at rest at point A on a rough horizontal surface. A horizontal force of magnitude $0.6t\,\text{N}$ acts on P in the direction away from A, where $t\,\text{s}$ denotes the time after P is placed at A. The coefficient of friction between A and the surface is $0.3$, and at time $t\,\text{s}$ the displacement of P from A is $x\,\text{m}$.

Representation of data

$OA$ is a rod that turns in a horizontal circle about a vertical axis passing through $O$. A particle $P$ with mass $0.2\,\text{kg}$ is fixed to the midpoint of a light inextensible string. One end of the string is fastened to the rod at $A$, and the other end is fastened to a point $B$ on the axis. It is given that $OA = OB$, angle $OAP =$ angle $OBP = 30^\circ$, and $P$ is $0.4\,\text{m}$ from the axis. The rod and the particle rotate together about the axis, with $P$ lying in the plane $OAB$ (see diagram).

Representation of data

A small ball is launched at a speed of $16\,\text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal from a point on level ground.

Representation of data

A uniform wire is shaped into a semicircular arc, and its diameter AB has a length of $0.8\,\text{m}$. It is fixed to a vertical wall by a smooth hinge at $A$. The wire is in equilibrium, with $AB$ inclined at $70^\circ$ to the upward vertical, and it is supported by a light string attached at $B$. The other end of the string is fastened to the point $C$ on the wall, $0.8\,\text{m}$ vertically above $A$. The tension in the string is $15\,\text{N}$ (see diagram).

Representation of data