May/June 2012

Solve the equation $\sin 2x = 2\cos 2x$, for $0^\circ \leq x \leq 180^\circ$.

Trigonometry

It is stated that the curve is described by $y = f(x)$, where $f(x) = x^3 - 2x^2 + x$.

Differentiation

The diagram includes the line $y = 1$ together with a section of the curve $y = \frac{2}{\sqrt{x+1}}$.

Integration

Find the coefficient of $x^6$ in the expansion of $(2x^3 - \frac{1}{x^2})^7$.

Series

The diagram shows an equilateral triangle $ABC$ with side $2\text{ cm}$. Point $Q$ lies at the midpoint of $BC$. A circular arc with centre $A$ is tangent to $BC$ at $Q$, and it intersects $AB$ at $P$ and $AC$ at $R$.

Circular measure

A watermelon is taken to have a spherical shape while it grows. Its mass, $M$ kg, and radius, $r$ cm, are linked by $M = kr^3$, where $k$ is constant. The radius is also assumed to increase at a steady rate of $0.1$ centimetres per day. On one particular day, the radius is $10$ cm and the mass is $3.2$ kg.

Differentiation

The diagram illustrates the curve $y = 7\sqrt{x}$ together with the line $y = 6x + k$, where $k$ is a constant. The curve and the line cross at the points $A$ and $B$.

Differentiation

The vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u} = \begin{pmatrix} p^2 \\ -2 \\ 6 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ p - 1 \\ 2p + 1 \end{pmatrix}$, with $p$ as a constant.

Coordinate geometry

The first two terms of an arithmetic progression are $1$ and $\cos^2 x$ respectively. Show that the total of the first ten terms can be written in the form $a - b\sin^2 x$, where the constants $a$ and $b$ are to be determined.

Series

The function $f : x \mapsto x^2 - 4x + k$ is given on the domain $x \geq p$, with $k$ and $p$ as constants.

Functions

Point $A$ has coordinates $(-3, 2)$, while point $C$ has coordinates $(5, 6)$. The midpoint of $AC$ is $M$, and the perpendicular bisector of $AC$ meets the $x$-axis at $B$.

Coordinate geometry

The diagram displays the area bounded by the curve $y = \frac{6}{2x - 3}$, the $x$-axis and the vertical lines $x = 2$ and $x = 3$.

Integration

The functions $f$ and $g$ are specified by $f : x \mapsto 2x + 5$ for $x \in \mathbb{R}$, and $g : x \mapsto \frac{8}{x - 3}$ for $x \in \mathbb{R}, x \neq 3$.

Functions

A curve has equation $y = 4\sqrt{x} + \frac{2}{\sqrt{x}}$.

Differentiation

The coefficient of $x^3$ in $(a + x)^5 + (2 - x)^6$ is $90$. Determine the value of the positive constant $a$.

Series

Point $A$ is at $(-1, -5)$ and point $B$ is at $(7, 1)$. The perpendicular bisector of $AB$ crosses the $x$-axis at $C$ and the $y$-axis at $D$. Find the length of $CD$.

Coordinate geometry

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

Trigonometry

The diagram depicts a metal plate formed by cutting out a segment from a circle with centre $O$ and radius $8$ cm. The line $AB$ is a chord of the circle, and the angle $AOB = 2.4$ radians.

Coordinate geometry

For an arithmetic progression with the sum of the first $n$ terms, $S_n$, given by $S_n = n^2 + 8n$, determine the first term and the common difference.

Series

Find the angle between the vectors $3\mathbf{i} - 4\mathbf{k}$ and $2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$.

Coordinate geometry

The diagram illustrates a section of the curve $y = -x^2 + 8x - 10$ that goes through the points $A$ and $B$. The curve reaches a maximum at $A$, and the gradient of line $BA$ is $2$.

Integration

Prove the identity $\tan^2\theta - \sin^2\theta = \tan^2\theta\sin^2\theta$.

Trigonometry

The line is given by the equation $2y + x = k$, where $k$ is constant, and the curve is described by $xy = 6$.

Coordinate geometry

The function $f$ is given by $f(x) = 8 - (x - 2)^2$, for $x \in \mathbb{R}$. The function $g$ is given by $g(x) = 8 - (x - 2)^2$, for $k \leq x \leq 4$, where $k$ is a constant.

Differentiation

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

Coordinate geometry

When $(1 - 2x)^2(1 + ax)^6$ is expanded in ascending powers of $x$, the first three terms are $1 - x + bx^2$.

Series

Solve $\sin 2x + 3\cos 2x = 0$ within the interval $0^\circ \leq x \leq 360^\circ$.

Trigonometry

The diagram illustrates a section of the curve $x = \frac{8}{y^2} - 2$, which meets the $y$-axis at the point $A$. The point $B(6,1)$ lies on the curve. The shaded region is enclosed by the curve, the $y$-axis and the line $y = 1$.

Integration

An arithmetic progression has first term $12$, and the total of its first $9$ terms is $135$.

Series

The curve $y = \frac{10}{2x + 1} - 2$ cuts the $x$-axis at $A$, and the tangent at $A$ meets the $y$-axis at $C$.

Coordinate geometry

In the figure, $AB$ is a circular arc with centre $O$ and radius $r$. The segment $XB$ touches the circle at $B$, and $A$ is the midpoint of $OX$.

Circular measure

The curve is defined by $\frac{d^2y}{dx^2} = -4x$. It reaches a maximum at $(2,12)$.

Differentiation

Find the solution of $|x^3 - 14| = 13$, and include every step in your working.

Algebra

The variables $x$ and $y$ are linked by $y = A(b^x)$, where $A$ and $b$ are constants. A graph of $\ln y$ against $x$ is a straight line that passes through $(0, 2.14)$ and $(5, 4.49)$, as shown in the diagram.

Logarithmic and exponential functions

The polynomial $p(x)$ is given by $p(x) = ax^3 - 3x^2 - 5x + a + 4$, with $a$ a constant.

Algebra

If $35 + \sec^2\theta = 12\tan\theta$, determine the value of $\tan\theta$.

Trigonometry

The diagram depicts the curve $y = 4\mathrm{e}^{\frac{1}{2}x} - 6x + 3$ together with its minimum point $M$.

Integration

A curve is defined by the parametric equations $x = \dfrac{1}{(2t + 1)^2}$, $y = \sqrt{t + 2}$. The point $P$ on the curve is associated with parameter $p$, and the gradient at $P$ is stated to be $-1$.

Numerical solution of equations

Show that $(2\sin x + \cos x)^2$ may be expressed in the form $\dfrac{5}{2} + 2\sin 2x - \dfrac{3}{2}\cos 2x$.

Integration

Determine the values of $x$ for which the inequality $|x + 3| < |2x + 1|$ holds.

Algebra

If $5^{2x} + 5^x = 12$, determine the value of $5^x$.

Logarithmic and exponential functions

Determine the quotient when $8x^3 - 4x^2 - 18x + 13$ is divided by $4x^2 + 4x - 3$, and show that the remainder is $4$.

Algebra

Rewrite $9\sin\theta - 12\cos\theta$ as $R\sin(\theta - \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the value of $\alpha$ correct to $2$ decimal places.

Trigonometry

The parametric equations for the curve are $x = \ln(t + 1)$ and $y = e^{2t} + 2t$.

Differentiation

The diagram depicts the curve $y = \frac{\sin 2x}{x + 2}$ for $0 \leq x \leq \frac{1}{2}\pi$. The $x$-coordinate of the highest point $M$ is called $\alpha$.

Numerical solution of equations

Demonstrate that $\tan^2 x + \cos^2 x = \sec^2 x + \tfrac{1}{2}\cos 2x - \tfrac{1}{2}$ and hence determine the exact value of $\int_{0}^{\frac{1}{4}\pi} (\tan^2 x + \cos^2 x) \, dx$.

Integration

Solve $|x^3 - 14| = 13$, showing all of your working.

Algebra

The variables $x$ and $y$ are related by the equation $y = A(b^x)$, where $A$ and $b$ are constants. The graph of $\ln y$ plotted against $x$ forms a straight line and passes through the points $(0, 2.14)$ and $(5, 4.49)$, as shown in the diagram.

Logarithmic and exponential functions

The polynomial $p(x)$ is specified as $p(x) = ax^3 - 3x^2 - 5x + a + 4$, where $a$ is constant.

Algebra

If $35 + \sec^2 \theta = 12 \tan \theta$, determine the value of $\tan \theta$.

Trigonometry

The diagram depicts the curve $y = 4e^{\frac{1}{2}x} - 6x + 3$ together with its minimum point $M$.

Integration

A curve is defined by parametric equations $x = \frac{1}{(2t + 1)^2}$ and $y = \sqrt{t + 2}$. The point $P$ on the curve is associated with parameter $p$, and the gradient at $P$ is stated to be $-1$.

Numerical solution of equations

Show that $(2 \sin x + \cos x)^2$ may be expressed as $\frac{5}{2} + 2 \sin 2x - \frac{3}{2} \cos 2x$.

Integration

Solve the equation $|4 - 2^x| = 10$, and give your answer correct to 3 significant figures.

Logarithmic and exponential functions

Given that $2\tan 2x + 5\tan^2 x = 0$, let $\tan x=t$ and form an equation in $t$; hence show that either $t = 0$ or $t = \sqrt[3]{(t + 0.8)}$.

Numerical solution of equations

Expand $\frac{1}{\sqrt{1 - 4x}}$ as a series in ascending powers of $x$, including terms up to and including $x^2$, and simplify the coefficients.

Algebra

The polynomial $p(x)$ is given by $p(x) = x^3 - 3ax + 4a$, with $a$ a constant.

Algebra

The complex number $u$ is specified by $u = \frac{(1 + 2i)^2}{2 + i}$.

Complex numbers

The diagram depicts the curve $y = 8\sin\left(\frac{1}{2}x\right) - \tan\left(\frac{1}{2}x\right)$ for $0 \le x < \pi$. Its maximum point has $x$-coordinate $\alpha$, and the shaded region is bounded by the curve together with the lines $x = \alpha$ and $y = 0$.

Trigonometry

The curve is given by the equation $3x^2 - 4xy + y^2 = 45$.

Differentiation

The variables $x$ and $y$ are linked by the differential equation $\frac{dy}{dx} = \frac{6xe^{3x}}{y^2}$. You are told that $y = 2$ when $x = 0$.

Differential equations

Point $P$ is at coordinates $(-1, 4, 11)$, while line $l$ is given by $\mathbf{r} = \begin{pmatrix}1 \\ 3 \\ -4\end{pmatrix} + \lambda \begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix}$.

Vectors

By rewriting $\frac{4x^2 + 5x + 3}{2x^2 + 5x + 2}$ as a partial fractions expansion first, prove that $\int_0^4 \frac{4x^2 + 5x + 3}{2x^2 + 5x + 2}\,dx = 8 - \ln 9$.

Integration

Solve the equation $\ln(3x + 4) = 2\ln(x + 1)$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

The plane equations for $m$ and $n$ are $x + 2y - 2z = 1$ and $2x - 2y + z = 7$ respectively, and the line $l$ is given by $\mathbf{r} = \mathbf{i} + \mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

Vectors

The diagram shows triangle $ABC$, where angle $ABC$ is a right angle and $BC = a$. A circular arc, centred at $C$ and with radius $a$, connects $B$ to point $M$ on $AC$. The angle $ACB$ is $\theta$ radians. The area of sector $CMB$ is one third of the area of triangle $ABC$.

Numerical solution of equations

Expand $\sqrt{\left(\frac{1 - x}{1 + x}\right)}$ as a series in ascending powers of $x$, including terms up to $x^2$, and simplify the coefficients.

Algebra

Solve the equation $\cosec 2\theta = \sec \theta + \cot \theta$, and state every solution in the interval $0^\circ < \theta < 360^\circ$.

Trigonometry

The variables $x$ and $y$ satisfy the differential equation $\frac{dy}{dx} = e^{2x+y}$, with $y = 0$ when $x = 0$.

Differential equations

The curve is given by $y = 3\sin x + 4\cos^3 x$.

Differentiation

For this question, you must not use a calculator. The complex number $u$ is given by $u = \frac{1 + 2i}{1 - 3i}$.

Complex numbers

Take $I = \int_{2}^{5} \frac{5}{x + \sqrt{6 - x}} \, dx$.

Integration

The diagram depicts the curve $y = x^{\frac{1}{2}} \ln x$. The shaded area bounded by the curve, the $x$-axis and the straight line $x = e$ is labelled $R$.

Integration

Express $\frac{1}{\sqrt{4 + 3x}}$ as a series in ascending powers of $x$, including terms up to $x^2$, and simplify the coefficients.

Algebra

The complex numbers $u$ and $w$ satisfy the equations $u - w = 4i$ and $uw = 5$. Find $u$ and $w$, giving every answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Solve the equation $\ln(2x + 3) = 2 \ln x + \ln 3$, with your answer correct to $3$ significant figures.

Logarithmic and exponential functions

The curve is given parametrically by $x = \sin 2\theta - \theta$, $y = \cos 2\theta + 2 \sin \theta$.

Differentiation

The curve given by $y = \frac{e^{2x}}{x^3}$ contains one stationary point.

Differentiation

In one chemical reaction, substance $A$ reacts with substance $B$. At $t$ seconds after the process begins, the masses in grams of $A$ and $B$ are $x$ and $y$ respectively. It is stated that $\frac{dy}{dt} = -0.6xy$ and $x = 5e^{-3t}$. When $t = 0$, $y = 70$.

Differential equations

Given that $\tan 3x = k \tan x$, with $k$ as a constant and $\tan x \neq 0$.

Trigonometry

The diagram depicts a section of the curve $y = \cos(\sqrt{x})$ for $x \geq 0$, with $x$ measured in radians. The shaded area bounded by the curve, the coordinate axes and the line $x = p^2$, where $p > 0$, is called $R$. The area of $R$ is $1$.

Numerical solution of equations

Define $f(x)$ by $f(x) = \frac{4x^2 - 7x - 1}{(x + 1)(2x - 3)}$.

Integration

The equations of the lines $l$ and $m$ are $\mathbf{r} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} - \mathbf{k})$, respectively, with $a$ and $b$ as constants.

Vectors

A car with mass $880\,\text{kg}$ is moving on a level straight road, and its engine is delivering power at a steady rate of $P\,\text{W}$. The resistive force opposing the motion is $700\,\text{N}$. When the car’s speed is $16\,\text{m s}^{-1}$, its acceleration is $0.625\,\text{m s}^{-2}$.

Energy, work and power

At point $O$, forces of magnitudes $13\,\text{N}$ and $14\,\text{N}$ act in the directions indicated in the diagram. Their resultant has magnitude $15\,\text{N}$.

Forces and equilibrium

A load with mass $160\,\text{kg}$ is lifted vertically upward from rest at a fixed point $O$ on the ground by means of a winding drum. When the load reaches point $A$, which is $20\,\text{m}$ above $O$, its speed is $1.25\,\text{m s}^{-1}$ (see diagram). Find, for the motion from $O$ to $A$,

Energy, work and power

A particle $P$ begins at $O$ and moves along a straight line. After $t$ seconds from leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 0.75t^2 - 0.0625t^3$.

Kinematics of motion in a straight line

The diagram gives the vertical cross-section $OAB$ of a slide. The straight section $AB$ is tangent to the curve $OA$ at $A$. The line $AB$ makes an angle $\alpha$ with the horizontal, where $\sin \alpha = 0.28$. Point $O$ is $10\,\text{m}$ above $B$, and $AB$ is $10\,\text{m}$ long (see diagram). The section of the slide containing the curve $OA$ is smooth, whereas the section containing $AB$ is rough. A particle $P$ of mass $2\,\text{kg}$ is released from rest at $O$ and slides down the slide.

Newton's laws of motion

Particles $P$ and $Q$, with masses $0.6\,\text{kg}$ and $0.4\,\text{kg}$ respectively, are attached to the two ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a vertical cross-section of a triangular prism. The prism has its base on horizontal ground, and both sloping faces are smooth. Each sloping face is inclined at angle $\theta$ to the ground, where $\sin \theta = 0.8$. At the beginning, the particles are held stationary on the sloping faces, with the string taut (see diagram). They are then released and travel along lines of greatest slope.

Newton's laws of motion

A small ring of mass $0.2\,\text{kg}$ is placed on a fixed vertical rod. The end $A$ of a light inextensible string is fastened to the ring. The other end $C$ of the string is fixed to a point on the rod above $A$. A horizontal force of magnitude $8\,\text{N}$ acts at the point $B$ of the string, where $AB = 1.5\,\text{m}$ and $BC = 2\,\text{m}$. The system is in equilibrium, the string is taut and $AB$ is at right angles to $BC$ (see diagram).

Forces and equilibrium

A block is drawn across level ground in a straight path by a force of fixed magnitude that acts at an angle of $60^{\circ}$ above the horizontal. The work done by the force in moving the block through $5\,\text{m}$ is $75\,\text{J}$.

Forces and equilibrium

At point $P$, three coplanar forces with magnitudes $F\,\text{N}$, $12\,\text{N}$ and $15\,\text{N}$ act in equilibrium in the directions indicated in the diagram.

Forces and equilibrium

A particle $P$ travels along a straight line, beginning at $O$ with velocity $2\,\text{m s}^{-1}$. The acceleration of $P$ at time $t\,\text{s}$ after leaving $O$ is $2t^{\frac{2}{3}}\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A ring of mass $4\,\text{kg}$ is fastened to one end of a light string. The ring is threaded onto a fixed horizontal rod, and the string is pulled at an angle of $25^{\circ}$ below the horizontal (see diagram). When the tension in the string is $T\,\text{N}$, the ring is in equilibrium. The coefficient of friction between the ring and the rod is $0.4$.

Forces and equilibrium

Block $A$ has mass $3\,\text{kg}$ and is tied to one end of a light inextensible string $S_1$. At the far end of $S_1$ is block $B$ of mass $2\,\text{kg}$, and $B$ is also joined to one end of another light inextensible string $S_2$. The other end of $S_2$ is fastened to a fixed point $O$, so the blocks hang in equilibrium beneath $O$ (see diagram). When $S_2$ breaks, the particles fall. The air resistance on $A$ is $1.6\,\text{N}$ and the air resistance on $B$ is $4\,\text{N}$.

Newton's laws of motion

A car with mass $1250\,\text{kg}$ moves from the foot to the top of a straight hill of length $400\,\text{m}$, which makes an angle $\alpha$ to the horizontal where $\sin\alpha = 0.125$. The resistive force acting on the car is $800\,\text{N}$. Determine the work done by the car’s engine in each case below.

Energy, work and power

A frictional force of magnitude $0.12\,\text{N}$ acts on a small block of mass $0.15\,\text{kg}$ while it is travelling over a horizontal surface. The block is launched from a point $X$ on the surface with speed $3\,\text{m s}^{-1}$. After $2\,\text{s}$ it reaches a vertical wall at point $Y$ on the surface. It rebounds from the wall and then travels straight back towards $X$ until it stops at point $Z$ (see diagram). When the block strikes the wall, its kinetic energy decreases by $0.072\,\text{J}$. At time $t\,\text{s}$ after leaving $X$, the velocity of the block in the direction from $X$ to $Y$ is $v\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A ring passes over a fixed horizontal bar. One end of a light inextensible string is fastened to the ring and is used to draw the ring along the bar at a steady speed of $0.5\,\text{m s}^{-1}$. The string stays at a constant angle of $24^{\circ}$ to the bar, and the tension in the string is $6\,\text{N}$ (see diagram).

Energy, work and power

A smooth ring $R$ of mass $0.16\,\text{kg}$ is passed onto a light inextensible string. The string ends are fixed at points $A$ and $B$. A horizontal force of magnitude $11.2\,\text{N}$ acts on $R$, in the same vertical plane as $A$ and $B$. The ring is in equilibrium. The string is taut, with angle $ARB = 90^{\circ}$, and the segment $AR$ of the string is inclined at an angle of $\theta^{\circ}$ to the horizontal (see diagram). The tension in the string is $T\,\text{N}$.

Forces and equilibrium

A particle $P$ moves from point $O$ in a straight line and reaches instantaneous rest at point $A$. The velocity of $P$ after time $t$ since leaving $O$ is $v\,\text{m s}^{-1}$, where $v = 0.027(10t^2 - t^3)$.

Kinematics of motion in a straight line

A car with mass $1230\,\text{kg}$ speeds up from $4\,\text{m s}^{-1}$ to $21\,\text{m s}^{-1}$ over $24.5\,\text{s}$. The table beneath gives matching values of time $t\,\text{s}$ and speed $v\,\text{m s}^{-1}$.

Energy, work and power

A lorry with mass $16000\,\text{kg}$ travels along a straight hillside that makes an angle $\alpha^{\circ}$ with the horizontal. The slope has length $500\,\text{m}$.

Energy, work and power

A block weighing $6.1\,\text{N}$ is stationary on a plane inclined at angle $\alpha$ to the horizontal, with $\tan \alpha = \frac{11}{60}$. The coefficient of friction between the block and the plane is $\mu$. A force of magnitude $5.9\,\text{N}$, applied parallel to the line of greatest slope, acts on the block.

Forces and equilibrium

Particles $A$ and $B$ have masses $0.12\,\text{kg}$ and $0.38\,\text{kg}$ respectively. They are connected by the ends of a light inextensible string, which goes over a fixed smooth pulley. $A$ is kept stationary with the string taut, and both sections of the string are vertical. Each particle is $0.65\,\text{m}$ above the horizontal ground (see diagram). $A$ is then released, and $B$ moves downwards.

Kinematics of motion in a straight line

The rod $AB$ has length $1.2\,\mathrm{m}$, with end $A$ freely pivoted at a fixed point. It turns about $A$ in a vertical plane.

Representation of data

The diagram represents a circular shape assembled from a uniform semicircular lamina of weight $11\,\mathrm{N}$ together with a uniform semicircular arc of weight $9\,\mathrm{N}$. Both parts have centre $O$ and radius $0.7\,\mathrm{m}$, and they are connected at the endpoints of their shared diameter $AB$.

Representation of data

A sphere $S$ of mass $m\,\mathrm{kg}$ is inside a smooth hollow bowl with a vertical axis and a sloping face that makes an angle of $60^\circ$ to the horizontal. $S$ travels at constant speed round a horizontal circle of radius $0.6\,\mathrm{m}$ (see Fig. 1). $S$ touches both the flat base and the sloping side of the bowl (see Fig. 2).

Probability

A light elastic string has a natural length of $2.4\,\mathrm{m}$ and an elasticity modulus of $21\,\mathrm{N}$. A particle $P$, with mass $m\,\mathrm{kg}$, is fixed to the midpoint of the string. The two ends of the string are fastened to fixed points $A$ and $B$, which are $2.4\,\mathrm{m}$ apart and lie at the same horizontal height. $P$ is projected vertically upward from the midpoint of $AB$ with speed $12\,\mathrm{m\,s^{-1}}$. In the ensuing motion, $P$ is momentarily at rest at a point $1.6\,\mathrm{m}$ above $AB$.

Probability

A particle $P$ with mass $0.4\,\mathrm{kg}$ starts from rest at the top of a smooth plane that is tilted at $30^\circ$ to the horizontal. As $P$ moves down the slope, its motion is resisted by a force of magnitude $0.6x\,\mathrm{N}$, where $x\,\mathrm{m}$ denotes the distance $P$ has moved down the slope. Before it reaches the bottom of the slope, $P$ comes to rest.

Probability

The diagram gives the cross-section $OABCDE$ passing through the centre of mass of a uniform prism. The interior angles of the cross-section at $O$, $A$, $C$, $D$ and $E$ are each right angles. $OA = 0.4\,\mathrm{m}$, $AB = 0.5\,\mathrm{m}$ and $BC = CD = 1\,\mathrm{m}$. The prism has weight $120\,\mathrm{N}$. A force of magnitude $F\,\mathrm{N}$ acting along $DE$ keeps the prism in equilibrium while $OA$ lies on a rough horizontal surface.

Representation of data

A light ball $B$ is launched from point $O$ with speed $15\,\mathrm{m\,s^{-1}}$ at $41^\circ$ above the horizontal. Point $O$ is $1.6\,\mathrm{m}$ above level ground. After $t\,\mathrm{s}$ from launch, the horizontal displacement of $B$ from $O$ is $x\,\mathrm{m}$ and its vertical upward displacement from $O$ is $y\,\mathrm{m}$. A vertical fence stands $1.5\,\mathrm{m}$ from $O$ and is perpendicular to the plane of motion of $B$. The ball just clears the fence and then lands on the ground at $A$.

Representation of data

A particle $P$ of mass $0.6\,\text{kg}$ is projected horizontally from point $O$ on a smooth horizontal surface with velocity $2\,\text{m s}^{-1}$. A horizontal force of magnitude $0.3x\,\text{N}$ acts on $P$ in the direction $OP$, where $x$ m gives the distance of $P$ from $O$.

Representation of data

A uniform hemispherical shell of weight $8\,\text{N}$ and a uniform solid hemisphere of weight $12\,\text{N}$ are joined at their circumferences to make a non-uniform sphere of radius $0.2\,\text{m}$. The sphere is resting on a horizontal surface, with its axis of symmetry horizontal. It is kept in equilibrium by a force of magnitude $F\,\text{N}$ acting parallel to the axis of symmetry and applied at the highest point of the sphere.

Probability

The natural length of a light elastic string is $2.2\,\text{m}$, and its modulus of elasticity is $14.3\,\text{N}$. A particle $P$ of mass $m\,\text{kg}$ is fixed to the midpoint of the string. The string ends are fixed at points $A$ and $B$, which are $2.4\,\text{m}$ apart and lie at the same horizontal level. $P$ is released from rest at the midpoint of $AB$. During the motion that follows, $P$ reaches its maximum speed at a point $0.5\,\text{m}$ below $AB$.

Probability

A particle $P$ with mass $0.25\,\text{kg}$ travels along a straight line on a smooth horizontal surface. At time $t$ s, its velocity is $v\,\text{m s}^{-1}$. A variable force of magnitude $3t\,\text{N}$ acts in the opposite direction to the motion of $P$.

Representation of data

A ball is launched with velocity $25\,\text{m s}^{-1}$ at an angle of $70^{\circ}$ above the horizontal from a point $O$ on level ground. It then rebounds once on the ground at a point $P$ before coming to rest at a point $Q$. The distance $PQ$ is $17.1\,\text{m}$.

Representation of data

The diagram depicts a uniform lamina $ABCDEF$ made by taking a semicircle with centre $O$ and radius $1\,\text{m}$ and cutting away a semicircular part with centre $O$ and radius $r\,\text{m}$. The centre of mass of the lamina is on the arc $ABC$. When the lamina is freely suspended at $F$, it hangs in equilibrium.

Representation of data

Particles $P$ and $Q$, with masses $0.8\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string that passes through a small hole in a smooth horizontal table of negligible thickness. $P$ travels on the upper surface of the table in a circular path with constant angular speed $6.25\,\text{rad s}^{-1}$.

Probability

A particle $P$ is fired with speed $25\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal from point $O$ on horizontal ground. Calculate the distance $OP$ after $2\,\text{s}$.

Representation of data

The diagram represents a uniform object $ABC$ with weight $3\,\text{N}$, shaped as an arc of a circle with centre $O$ and radius $0.7\,\text{m}$. The angle $AOC$ is $2$ radians. The object is in equilibrium with $A$ resting on a horizontal surface and $C$ positioned vertically above $A$. This equilibrium is kept by a horizontal force of magnitude $F\,\text{N}$ acting at $C$ in the plane of the object.

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