May/June 2015

If $\theta$ is an obtuse angle in radians and $\sin\theta = k$, determine, in terms of $k$, an expression for

Trigonometry

The diagram presents a section of the curve $y = \frac{8}{\sqrt{3x + 4}}$. It crosses the $y$-axis at $A(0,4)$. The normal drawn to the curve at $A$ meets the line $x = 4$ at $B$.

Integration

The curve $y = 2x^2$ is drawn with the points $X(-2,0)$ and $P(p,0)$ marked on it. Point $Q$ is on the curve, and $PQ$ is parallel to the $y$-axis.

Differentiation

Find the first three terms, in ascending powers of $x$, of the expansion of $(1 - x)^6$.

Series

Measured from the origin $O$, the position vectors of points $A$ and $B$ are $ \overrightarrow{OA} = \begin{pmatrix}3 \\ 0 \\ -4\end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix}6 \\ -3 \\ 2\end{pmatrix}$. The position vector of $C$ is $\overrightarrow{OC} = \begin{pmatrix}k \\ -2k \\ 2k - 3\end{pmatrix}$.

Coordinate geometry

A wire of length $24$ cm is shaped to make the perimeter of a sector of a circle with radius $r$ cm.

Quadratics

The straight line with gradient $-2$ that passes through $P(3t, 2t)$ cuts the $x$-axis at $A$ and the $y$-axis at $B$. The line through $P$ that is perpendicular to $AB$ meets the $x$-axis at $C$.

Coordinate geometry

A geometric progression has third term $\frac{1}{3}$ and fourth term $\frac{2}{9}$ respectively. Find the sum to infinity of the progression.

Series

The function $f : x \mapsto 5 + 3\cos\left(\frac{1}{2}x\right)$ is defined over the interval $0 \le x \le 2\pi$.

Functions

The curve is given by $y = x^3 + px^2$, where $p$ is a positive constant. A second curve is given by $y = x^3 + px^2 + px$.

Differentiation

The function $f$ satisfies $f'(x) = 5 - 2x^2$, and the point $(3, 5)$ lies on the graph of $y = f(x)$.

Functions

The curve is given by the equation $y = \frac{4}{2x - 1}$.

Integration

The mapping $f$ is defined, for $x \in \mathbb{R}$, by $f: x \mapsto 2x^2 - 6x + 5$.

Functions

The diagram shows $AYB$ as a semicircle with $AB$ as its diameter, while $OAXB$ is a sector of a circle centred at $O$ with radius $r$. The angle $AOB$ is $2\theta$ radians.

Circular measure

Find the coefficients of $x^2$ and $x^3$ within the expansion of $(2 - x)^6$.

Series

The variables $u$, $x$ and $y$ satisfy $u = 2x(y - x)$ together with $x + 3y = 12$.

Differentiation

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

Trigonometry

A city-centre tourist attraction is a large vertical wheel that passengers can travel on. Its motion is modelled by the formula $h = 60(1 - \cos kt)$, where $h$ m is the passenger’s height above the ground, $k$ is a constant, $t$ is the number of minutes since the passenger began the ride at ground level, and $kt$ is measured in radians.

Trigonometry

The point $C$ is located on the perpendicular bisector of the segment joining $A(4, 6)$ and $B(10, 2)$. $C$ is also on the line through $(3, 11)$ that runs parallel to $AB$.

Coordinate geometry

In an arithmetic progression, the first term, second term and final term are $56$, $53$ and $-22$ respectively. Find the sum of all the terms in the progression.

Series

With $O$ taken as the origin, the position vectors of $A$ and $B$ are $\overrightarrow{OA} = 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}$ and $\overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + 4\mathbf{k}$. Point $C$ is defined so that $\overrightarrow{AB} = \overrightarrow{BC}$.

Coordinate geometry

Write $2x^2 - 12x + 7$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

Quadratics

The points $A(2, 9)$ and $B(3, 0)$ are on the curve $y = 9 + 6x - 3x^2$, as the diagram shows. The tangent at $A$ meets the $x$-axis at $C$.

Differentiation

The diagram shows $OAB$ as a sector of a circle centred at $O$ with radius $r$. Point $C$ lies on $OB$ so that the angle $ACO$ is a right angle. The angle $AOB$ is $\alpha$ radians, and $AC$ splits the sector into two parts of equal area.

Circular measure

The curve is defined by $\frac{dy}{dx} = \sqrt{2x + 1}$, and the point $(4, 7)$ is on it.

Integration

Give the first 4 terms of the expansion of $(a - x)^5$, arranged in ascending powers of $x$.

Series

Write $3\sin\theta = \cos\theta$ in the form $\tan\theta = k$ and find the solutions when $0^{\circ} < \theta < 180^{\circ}$.

Trigonometry

Taking $O$ as the origin, the position vectors of the points $A$, $B$ and $C$ are $overrightarrow{OA} = \begin{pmatrix} 3 \\ 2 \\ -3 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 5 \\ -1 \\ -2 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 6 \\ 1 \\ 2 \end{pmatrix}$.

Coordinate geometry

The diagram presents the graph of $y = f^{-1}(x)$, where $f^{-1}$ is given by $f^{-1}(x) = \frac{1 - 5x}{2x}$ for $0 < x \leq 2$.

Functions

Point $A$ is at $(p, 1)$, while point $B$ is at $(9, 3p + 1)$, with $p$ fixed as a constant.

Coordinate geometry

The function $f$ takes the form $f(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}$ for $x > -1$.

Differentiation

The initial term of an arithmetic progression is $-2222$ and the common difference is $17$. Find the value of the first positive term.

Series

Solve the equation $|3x + 4| = |3x - 11|$.

Logarithmic and exponential functions

The variables $x$ and $y$ obey $y = A e^{p(x-1)}$, with $A$ and $p$ as constants. The graph of $\ln y$ plotted against $x$ is a straight line that goes through the points $(2, 1.60)$ and $(5, 2.92)$, as the diagram indicates.

Logarithmic and exponential functions

The curve is defined by the equation $y = 6 \sin x - 2 \cos 2x$.

Differentiation

The polynomials $f(x)$ and $g(x)$ are given by $f(x) = x^3 + ax^2 + b$ and $g(x) = x^3 + bx^2 - a$, where $a$ and $b$ are constants. It is known that $(x + 2)$ is a factor of $f(x)$. It is also known that, after $g(x)$ is divided by $(x + 1)$, the remainder is $-18$.

Differentiation

If $\int_0^a (3e^{\frac{1}{2}x} + 1)\,dx = 10$, demonstrate that the positive constant $a$ obeys $a = 2 \ln\left(\frac{16 - a}{6}\right)$.

Numerical solution of equations

Prove that $2 \cosec 2\theta \tan \theta \equiv \sec^2 \theta$.

Trigonometry

The curve is given by $y^3 + 4xy = 16$.

Differentiation

Apply logarithms to find the solution of the equation $2^x = 20^5$, with the final answer correct to $3$ significant figures.

Logarithmic and exponential functions

Since $(x + 2)$ is a factor of $4x^3 + ax^2 - (a + 1)x - 18$, determine the value of the constant $a$.

Algebra

It is stated that $\theta$ is an acute angle in degrees, and that $2\sec^2\theta + 3\tan\theta = 22$.

Trigonometry

The diagram displays the curve $y = e^x + 4e^{-2x}$ together with its minimum point $M$.

Integration

Use sketches of two suitable graphs to show that the equation $|3x| = 16 - x^4$ has two real roots.

Numerical solution of equations

The diagram displays a section of the curve with equation $y = 4\sin^2 x + 8\sin x + 3$ and the point of intersection $P$ on the $x$-axis.

Integration

Determine the gradient of the curve $3\ln x + 4\ln y + 6xy = 6$ at the point $(1, 1)$.

Differentiation

Solve the equation $2^x = 20^5$ by using logarithms, and give your answer correct to 3 significant figures.

Logarithmic and exponential functions

Find the value of the constant $a$ if $(x + 2)$ is a factor of $4x^3 + ax^2 - (a + 1)x - 18$.

Algebra

You are told that $\theta$ is an acute angle, measured in degrees, and that $2\sec^2\theta + 3\tan\theta = 22$.

Trigonometry

The graph depicts the curve $y = e^x + 4e^{-2x}$ together with its minimum point $M$.

Integration

By sketching an appropriate pair of graphs, show that the equation $|3x| = 16 - x^4$ has two real roots.

Numerical solution of equations

The diagram displays a section of the curve with equation $y = 4\sin^2 x + 8\sin x + 3$ together with its point of intersection $P$ on the $x$-axis.

Integration

Find the gradient, at the point $(1, 1)$, of the curve $3\ln x + 4\ln y + 6xy = 6$.

Differentiation

Employ logarithms to solve the equation $2^{5x} = 3^{2x+1}$, and give the result correct to $3$ significant figures.

Logarithmic and exponential functions

The diagram displays part of the curve with parametric equations $x = 2\ln(t + 2)$, $y = t^3 + 2t + 3$.

Numerical solution of equations

With three intervals, use the trapezium rule to approximate $\int_{0}^{3} |3^x - 10| \, dx$.

Numerical solution of equations

Show that, for sufficiently small $x^2$, $(1 - 2x^2)^{-2} - (1 + 6x^2)^{\frac{2}{3}} \approx kx^4$, where the constant $k$ is still to be found.

Algebra

The curve is defined by $y = 3\cos 2x + 7\sin x + 2$.

Differentiation

Find the value of $\int (4 + \tan^2 2x) \, dx$.

Integration

The straight line $l_1$ goes through the points $(0, 1, 5)$ and $(2, -2, 1)$. The straight line $l_2$ is given by $\mathbf{r} = 7\mathbf{i} + \mathbf{j} + k + \mu(\mathbf{i} + 2\mathbf{j} + 5\mathbf{k})$.

Vectors

Knowing that $y = 1$ when $x = 0$, solve the differential equation

Differential equations

The complex number $w$ is given by $w = \frac{22 + 4i}{(2 - i)^2}$.

Complex numbers

The diagram displays the curve $y = x^2 e^{2-x}$ together with its maximum point $M$.

Integration

Apply the trapezium rule using three intervals to estimate $\int_{0}^{2\pi} \ln(1 + \sin x)\,dx$, and present the answer correct to 2 decimal places.

Numerical solution of equations

The points $A$ and $B$ are given by the position vectors $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OB} = \mathbf{i} + \mathbf{j} + 5\mathbf{k}$. The line $l$ is described by $\mathbf{r} = \mathbf{i} + \mathbf{j} + 2\mathbf{k} + \mu(3\mathbf{i} + \mathbf{j} - \mathbf{k})$.

Vectors

By making the substitution $u = 4^x$, solve the equation $4^x + 4^2 = 4^{x+2}$, and give your answer correct to 3 significant figures.

Logarithmic and exponential functions

The curve is given by $y = \cos x \cos 2x$. Find the $x$-coordinate of the stationary point on the curve for $0 < x < \tfrac{1}{2}\pi$, and give your answer accurate to 3 significant figures.

Differentiation

Rewrite $3\sin\theta + 2\cos\theta$ as $R\sin(\theta + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$; give the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.

Trigonometry

The diagram depicts a circle whose centre is $O$ and whose radius is $r$. The tangents to the circle at $A$ and $B$ intersect at $T$, while the angle $AOB$ is $2x$ radians. The shaded part is enclosed by the tangents $AT$ and $BT$, together with the minor arc $AB$. Its perimeter is the same as the circle’s circumference.

Numerical solution of equations

Define $I = \int_{0}^{1} \frac{\sqrt{x}}{2 - \sqrt{x}}\,dx$.

Integration

The complex number $u$ is defined as $u = -1 + (4\sqrt{3})i$.

Complex numbers

Consider $f(x) = \frac{5x^2 + x + 6}{(3 - 2x)(x^2 + 4)}$.

Algebra

Let $x$ denote the population size at time $t$. If $x$ is treated as a continuous variable, it satisfies the differential equation $\frac{dx}{dt} = \frac{xe^{-t}}{k + e^{-t}}$, where $k$ is a positive constant.

Differential equations

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

Logarithmic and exponential functions

Define $f(x)$ by $f(x) = \frac{11x + 7}{(2x - 1)(x + 2)^2}$.

Integration

Solve for $x$ in the inequality $|x - 2| > 2x - 3$.

Algebra

Find the values of $x$ that satisfy $\cot 2x + \cot x = 3$ for $0^\circ < x < 180^\circ$.

Trigonometry

The graph of $y = \frac{e^{2x}}{4 + e^{3x}}$ includes a single stationary point.

Vectors

For a curve, the parametric equations are $x = a\cos^4 t$ and $y = a\sin^4 t$, with $a$ a positive constant.

Differentiation

We are told that $\int_0^a x\cos x\,dx = 0.5$, with $0 < a < \frac{1}{2}\pi$.

Numerical solution of equations

Let the number of micro-organisms in a population at time $t$ be represented by $M$. At any instant, the change in $M$ is assumed to obey the differential equation $\frac{dM}{dt} = k(\sqrt{M})\cos(0.02t)$, where $k$ is a constant and $M$ is treated as a continuous variable. It is given that when $t = 0$, $M = 100$.

Differential equations

Let $u$ represent the complex number $1 - i$.

Complex numbers

The equations of two planes are $x + 3y - 2z = 4$ and $2x + y + 3z = 5$. Their intersection is the straight line $l$.

Vectors

A block $B$ with mass $2.7\,\text{kg}$ is being pulled at a steady speed in a straight path across a rough horizontal floor. The pulling force is $25\,\text{N}$ and is applied at an angle of $\theta$ above the horizontal. The normal component of the contact force on $B$ is $20\,\text{N}$.

Energy, work and power

At point O, which is the origin of the x-axis and y-axis, three horizontal forces of magnitudes $F\,\text{N}$, $63\,\text{N}$ and $25\,\text{N}$ are acting. These forces are in equilibrium. The force with magnitude $F\,\text{N}$ is inclined at an angle $\theta$ anticlockwise to the positive $x$-axis. The force of magnitude $63\,\text{N}$ is directed along the negative $y$-axis. The force of magnitude $25\,\text{N}$ is directed at $\tan^{-1}0.75$ clockwise from the negative $x$-axis (see diagram).

Forces and equilibrium

A block weighing $6.1\,\text{N}$ moves down a slope that makes angle $\tan^{-1}\left(\frac{11}{60}\right)$ with the horizontal. The coefficient of friction between the block and the slope is $\frac{1}{4}$. As the block goes through point $A$, its speed is $2\,\text{m s}^{-1}$.

Energy, work and power

A lorry with mass $14\,000\,\text{kg}$ travels along a road, beginning at rest at point $O$. It gets to point $A$, and then continues to point $B$, where it has a speed of $24\,\text{m s}^{-1}$. The stretch $OA$ is straight, horizontal and $400\,\text{m}$ long. The stretch $AB$ is straight, slopes downwards at an angle of $\theta^\circ$ to the horizontal, and has length $300\,\text{m}$.

Energy, work and power

A cyclist together with her bicycle has a combined mass of $84\,\text{kg}$. She delivers power at a steady rate of $P\,\text{W}$ while travelling along a straight road that is sloping at an angle $\theta$ to the horizontal, where $\sin\theta = 0.1$. When she is going uphill, the cyclist’s acceleration is $1.25\,\text{m s}^{-2}$ at the moment when her speed is $3\,\text{m s}^{-1}$. When she is going downhill, the cyclist’s acceleration is $1.25\,\text{m s}^{-2}$ at the moment when her speed is $10\,\text{m s}^{-1}$. The resistive force opposing the cyclist’s motion, whether she is travelling uphill or downhill, is $R\,\text{N}$.

Energy, work and power

Particles $A$ and $B$ start moving at the same moment from point $O$. They travel in the same direction along one straight line. At time $t\,\text{s}$ after the motion begins, the acceleration of $A$ is $a\,\text{m s}^{-2}$, where $a = 0.05 - 0.0002t$.

Kinematics of motion in a straight line

Particles $A$ and $B$, with masses $0.3\,\text{kg}$ and $0.7\,\text{kg}$ respectively, are joined to the two ends of a light inextensible string. Particle $A$ is initially at rest on a rough horizontal table, and the string passes over a smooth pulley fixed at the table edge. The coefficient of friction between $A$ and the table is $0.2$. Particle $B$ hangs vertically below the pulley, with its height $0.5\,\text{m}$ above the floor (see diagram). The system is released from rest, and $0.25\,\text{s}$ later the string breaks. In the later motion, $A$ does not reach the pulley.

Newton's laws of motion

A block is fastened to one end of a light inextensible string. The string is inclined at $60^\circ$ above the horizontal and pulls the block along a horizontal floor in a straight line with acceleration $0.5\,\text{m s}^{-2}$. The tension in the string is $8\,\text{N}$. The block starts moving with speed $0.3\,\text{m s}^{-1}$. During the first $5\,\text{s}$ of the motion, find

Energy, work and power

The combined mass of the cyclist and the cycle is $80\,\text{kg}$. There is no resistance to motion.

Energy, work and power

A plane makes an angle of $\sin^{-1}\!\left(\tfrac{1}{8}\right)$ with the horizontal. $A$ and $B$ are two points on the same line of steepest descent, with $A$ lying above $B$. The length $AB$ is $12\,\text{m}$. A small object $P$ of mass $8\,\text{kg}$ is released from rest at $A$ and slides down the plane, reaching $B$ with speed $4.5\,\text{m s}^{-1}$. For the motion of $P$ from $A$ to $B$, determine

Energy, work and power

A particle $P$ travels along a straight line. $t$ seconds after it starts from rest at the point $O$ on the line, its acceleration is $a\,\text{m s}^{-2}$, where $a = 0.075t^{2} - 1.5t + 5$.

Kinematics of motion in a straight line

Particle $P$ begins from rest at point $O$ on a horizontal straight line. $P$ travels along the line with constant acceleration and arrives at point $A$ on the line with a speed of $30\,\text{m s}^{-1}$. At the moment that $P$ departs from $O$, particle $Q$ is projected vertically upwards from point $A$ with a speed of $20\,\text{m s}^{-1}$. Later, $P$ and $Q$ collide at $A$. Find

Kinematics of motion in a straight line

Particles $P$ and $Q$ have masses $m\,\text{kg}$ and $(1-m)\,\text{kg}$ respectively. They are connected to the ends of a light inextensible string that passes over a smooth fixed pulley. At the start, $P$ is kept at rest with the string taut and the two straight sections of string vertical. Each particle is at a height of $h\,\text{m}$ above horizontal ground. When $P$ is released, $Q$ moves downwards. After that, $Q$ reaches the ground and stops. Fig. 2 gives the velocity-time graph for $P$ during the time that $Q$ is moving downwards.

Newton's laws of motion

A small ring $R$ is fastened to one end of a light inextensible string of length $70\,\text{cm}$. The string passes through a fixed rough vertical wire. Its other end is tied to a point $A$ on the wire, vertically above $R$. A horizontal force of magnitude $5.6\,\text{N}$ acts at the point $J$ of the string, which is $30\,\text{cm}$ from $A$ and $40\,\text{cm}$ from $R$. The system is in equilibrium, with both parts $AJ$ and $JR$ taut and angle $AJR$ equal to $90^\circ$ (see diagram).

Forces and equilibrium

A block is being pulled across a level floor by a rope that is horizontal. The rope tension is $500\,\text{N}$, and the block travels at a steady speed of $2.75\,\text{m s}^{-1}$.

Energy, work and power

Particles $A$ and $B$, with masses of $0.35\,\text{kg}$ and $0.15\,\text{kg}$ respectively, are joined by a light inextensible string at their ends. $A$ starts from rest on a smooth horizontal surface, while the string passes over a small smooth pulley fixed at the edge of the surface. $B$ hangs vertically below the pulley, at a height of $h\,\text{m}$ above the floor (see diagram). $A$ is released and the particles move. $B$ reaches the floor and $A$ then arrives at the pulley with a speed of $3\,\text{m s}^{-1}$.

Energy, work and power

A car with mass $860\,\text{kg}$ moves on a level straight road. The engine supplies power $P\,\text{W}$, while the resistive force opposing the motion is $R\,\text{N}$. At one point, the car has speed $4.5\,\text{m s}^{-1}$ and acceleration $4\,\text{m s}^{-2}$. At another point, its speed is $22.5\,\text{m s}^{-1}$ and its acceleration is $0.3\,\text{m s}^{-2}$.

Forces and equilibrium

A lorry with mass $12\,000\,\text{kg}$ travels up a straight hill that is $500\,\text{m}$ long, beginning at the foot of the hill with speed $24\,\text{m s}^{-1}$ and arriving at the top with speed $16\,\text{m s}^{-1}$. The summit of the hill is $25\,\text{m}$ higher than the level at the bottom. The resistance to motion of the lorry is $7500\,\text{N}$.

Newton's laws of motion

At a point, four coplanar forces with magnitudes $4\,\text{N}$, $8\,\text{N}$, $12\,\text{N}$ and $16\,\text{N}$ are applied. Their directions are illustrated in Fig. 1.

Forces and equilibrium

A small box of mass $5\,\text{kg}$ is dragged at a steady speed of $2.5\,\text{m s}^{-1}$ down the line of greatest slope on a rough plane inclined at $10^{\circ}$ to the horizontal. The pulling force is $20\,\text{N}$ and acts downward, parallel to the plane’s line of greatest slope.

Forces and equilibrium

A particle $P$ travels along a straight line. It begins at a point $O$ on the line and is back at $O\,100\,\text{s}$ later. The velocity of $P$ is $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ after leaving $O$, where $v = 0.0001t^{3} - 0.015t^{2} + 0.5t$.

Kinematics of motion in a straight line

One end of a light elastic string with natural length $0.7\,\text{m}$ is fixed at point $A$ on a smooth horizontal surface. The other end is fastened to a particle $P$ of mass $0.3\,\text{kg}$, which is initially held at point $B$ on the horizontal surface, where $AB = 1.2\,\text{m}$. It is given that $P$ is released from rest at $B$ and that when $AP = 0.9\,\text{m}$, the particle has speed $4\,\text{m s}^{-1}$.

Representation of data

From a point $O$ on horizontal ground, a stone is projected. Its trajectory is described by $y = 1.2x - 0.15x^2$, where $x\,\text{m}$ and $y\,\text{m}$ represent respectively the stone's horizontal displacement from $O$ and its upward vertical displacement from $O$.

Representation of data

One end of a light inextensible string is fastened to a fixed point $A$, while the other end is joined to a particle $P$. Particle $P$ travels at constant angular speed $5\,\text{rad s}^{-1}$ in a horizontal circle with centre $O$ directly beneath $A$. The string is at angle $\theta$ to the vertical (see diagram). The tension in the string is three times the weight of $P$.

Representation of data

A small ball $B$ is launched from a point $O$ above horizontal ground, with initial speed $15\,\text{m s}^{-1}$ at an angle of projection of $30^\circ$ above the horizontal (see diagram). The ball hits the ground $3\,\text{s}$ after projection.

Representation of data

A particle $P$ of mass $0.3\,\text{kg}$ is connected to one end of a light elastic string of natural length $0.9\,\text{m}$ and modulus of elasticity $18\,\text{N}$. The other end of the string is secured to a fixed point $O$, which is $3\,\text{m}$ above the ground.

Probability

A particle $P$ of mass $0.1\,\text{kg}$ travels in a straight line on a smooth horizontal surface, with its speed steadily reducing. A horizontal resisting force of magnitude $0.2e^{-x}\,\text{N}$ acts on $P$, where $x\,\text{m}$ is the displacement of $P$ from a fixed point $O$ on the line. If the displacement from $O$ is $x\,\text{m}$, the velocity of $P$ is $v\,\text{m s}^{-1}$. $P$ passes through $O$ with velocity $2.2\,\text{m s}^{-1}$.

Probability

The diagram presents the cross-section $OABCDE$ passing through the centre of mass of a uniform prism resting on a rough inclined plane. The section $ADEO$ forms a rectangle with $AD = OE = 0.6\,\text{m}$ and $DE = AO = 0.8\,\text{m}$. The section $BCD$ is an isosceles triangle in which angle $BCD$ is a right angle, and $A$ is the mid-point of $BD$. The plane is at $45^\circ$ to the horizontal, $BC$ is along a line of greatest slope of the plane and $DE$ is horizontal. The prism weighs $21\,\text{N}$, and equilibrium is maintained by a horizontal force of magnitude $P\,\text{N}$ acting along $ED$.

Representation of data

A particle $P$ with mass $0.6\,\text{kg}$ lies on the rough upper surface of a horizontal disc centred at $O$. The distance $OP$ is $0.4\,\text{m}$. The disc and $P$ rotate at angular speed $3\,\text{rad s}^{-1}$ about a vertical axis passing through $O$.

Probability

A light elastic string has natural length $0.5\,\text{m}$ and modulus of elasticity $30\,\text{N}$. One end is fixed at point $O$. The other end is attached to a particle $P$, which hangs in equilibrium vertically beneath $O$, with $OP = 0.8\,\text{m}$.

Probability

A triangular frame $ABC$ is formed by two uniform rigid rods, each of length $0.8\,\text{m}$ and weight $3\,\text{N}$, together with a longer uniform rod of weight $4\,\text{N}$. In the frame, $AB = BC$, and $\angle BAC = \angle BCA = 30\degree$.

Representation of data

One end of a light inextensible string with length $0.5\,\text{m}$ is fixed to point $A$. The string’s other end is joined to a particle $P$ whose weight is $6\,\text{N}$. A second light inextensible string of length $0.5\,\text{m}$ links $P$ to a fixed point $B$, and $B$ is $0.8\,\text{m}$ vertically beneath $A$. Particle $P$ moves at constant speed in a horizontal circle centred at the midpoint of $AB$. Both strings are taut.

Representation of data

A uniform solid cube with edges of length $0.4\,\text{m}$ is in equilibrium on a rough plane that is inclined at $30\degree$ to the horizontal. $ABCD$ shows a cross-section through the cube’s centre of mass, with $AB$ lying along a line of greatest slope. $B$ lies below $A$. One end of a light elastic string, of modulus of elasticity $12\,\text{N}$ and natural length $0.4\,\text{m}$, is fixed to $C$. The other end is fixed to a point on the same line of greatest slope below $B$, so that the string makes an angle of $30\degree$ with the plane (see diagram). The cube is about to topple.

Probability

A small ball $B$ is launched from $O$ with speed $U\,\text{m s}^{-1}$ at an angle of $\theta\degree$ above the horizontal. Two seconds later, $B$ hits a smooth wall inclined at $60\degree$ to the horizontal. At that instant, the speed of $B$ is $18\,\text{m s}^{-1}$ and its direction of motion is perpendicular to the wall (see Fig. 1). $B$ rebounds from the wall with speed $V\,\text{m s}^{-1}$, again moving perpendicular to the wall. After $0.8\,\text{s}$, $B$ meets the wall again at a lower point $A$ (see Fig. 2).

Representation of data

A force with magnitude $0.4t\,\text{N}$, acting at an angle of $30\degree$ above the horizontal, is applied to a particle $P$, where $t\,\text{s}$ is the time elapsed since the force begins to act. $P$ is stationary on rough horizontal ground when $t = 0$. The mass of $P$ is $0.2\,\text{kg}$ and the coefficient of friction between $P$ and the ground is $\mu$.

Probability

The diameter $AB$ of a uniform semicircular lamina is $0.8\text{ m}$. The lamina is placed in a vertical plane, with point $B$ touching a rough horizontal surface and $A$ positioned vertically above $B$. A force of magnitude $6\text{ N}$ acting in the plane of the lamina is applied at $A$ at an angle of $20^\circ$ below the horizontal, and this keeps the lamina in equilibrium.

Representation of data

Starting from point $O$ on horizontal ground, a particle $P$ is projected with speed $V\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. $1.5\text{ s}$ after projection, $P$ is travelling at an angle of $45^\circ$ above the horizontal.

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