Oct/Nov 2013

Determine the first three terms in ascending powers of $x$ when $(2 + 3x)^6$ is expanded.

Series

The diagram depicts the curve $y = (3 - 2x)^3$ together with the tangent drawn at the point $(\frac{1}{2}, 8)$.

Differentiation

A curve is described by the equation $y = f(x)$. It is stated that $f'(x) = \frac{1}{\sqrt{x + 6}} + \frac{6}{x^2}$ and that $f(3) = 1$.

Functions

The diagram depicts a pyramid $OABCD$ with the vertical edge $OD$ measuring 3 units. Point $E$ lies at the centre of the horizontal rectangular base $OABC$. The edges $OA$ and $AB$ are 6 units and 4 units long respectively. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively.

Coordinate geometry

Solve for $x$ in the equation $4\sin^2 x + 8\cos x - 7 = 0$ where $0^\circ \leq x \leq 360^\circ$.

Trigonometry

The function $f$ is given by $f: x \mapsto x^2 + 1$ for $x \geq 0$.

Functions

The diagram illustrates a metal plate formed by fixing together two pieces, $OABCD$ (shaded) and $OAED$ (unshaded). The piece $OABCD$ is a minor sector of a circle with centre $O$ and radius $2r$. The piece $OAED$ is a major sector of a circle with centre $O$ and radius $r$. Angle $AOD$ is $\alpha$ radians. Simplify your answers where possible and find, in terms of $\alpha$, $\pi$ and $r$.

Circular measure

Point $A$ is at $(-1, 6)$, while point $B$ is at $(7, 2)$.

Coordinate geometry

The inner lane of a school running track is made up of two straight parts, each of length $x$ metres, together with two semicircular parts, each of radius $r$ metres, as shown in the diagram. The straight parts are at right angles to the diameters of the semicircular parts. The perimeter of the inner lane is 400 metres.

Differentiation

In an arithmetic progression, the total of the first ten terms is 400 and the total of the next ten terms is 1000. Determine the common difference and the first term.

Series

Given that $\cos x = p$, where $x$ is an acute angle in degrees, find the expression in terms of $p$.

Trigonometry

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

Quadratics

Fig. 1 illustrates a hollow cone without a base, constructed from paper. Its radius is $6\text{ cm}$ and its height is $8\text{ cm}$. The paper is sliced from $A$ to $O$ and then unfolded to create the sector shown in Fig. 2. The circular lower edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The sector angle is $\theta$ radians. Calculate

Circular measure

A curve is given by $y = \frac{2}{\sqrt{5x - 6}}$.

Integration

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

Coordinate geometry

The diagram is of a rectangle $ABCD$ with point $A$ at $(0, 8)$ and point $B$ at $(4, 0)$. The diagonal $AC$ is given by $8y + x = 64$. Calculate the coordinates of $C$ and $D$.

Coordinate geometry

The diagram shows $S$ at $(0, 12)$ and $T$ at $(16, 0)$. Point $Q$ lies on $ST$, between $S$ and $T$, with coordinates $(x, y)$. Points $P$ and $R$ are on the $x$-axis and $y$-axis respectively, and $OPQR$ forms a rectangle.

Differentiation

A marathon runner covers the opening mile in $5$ minutes. After that, each mile takes $12$ seconds more than the one before it. If the $n$th mile lasts $9$ minutes, calculate the value of $n$.

Series

For $0 \leq x \leq 2\pi$, the function $f$ is defined by $f : x \mapsto 3\cos x - 2$.

Trigonometry

The diagram presents a segment of the curve $y = \frac{8}{x} + 2x$ together with three points $A$, $B$ and $C$ on the curve, whose $x$-coordinates are $1$, $2$ and $5$ respectively.

Integration

Solve for the values of $x$ in the inequality $x^2 - x - 2 > 0$.

Quadratics

The function $f$ is given by $f : x \mapsto x^2 + 4x$ for $x \geq c$, where $c$ is a constant. It is stated that $f$ is a one-one function.

Functions

The diagram illustrates the curve $y = \sqrt{x^4 + 4x + 4}$.

Integration

The curve is described by the equation $y = f(x)$. It is known that $f'(x) = x^{-\frac{3}{2}} + 1$ and that $f(4) = 5$.

Integration

Point $A$ is located at $(3, 1)$ and point $B$ is located at $(-21, 11)$. Point $C$ is the midpoint of $AB$.

Coordinate geometry

The diagram depicts a pyramid $OABC$ with edge $OC$ running vertically. Its horizontal base $OAB$ is a triangle that is right-angled at $O$, and $D$ is the mid-point of $AB$. The edges $OA$, $OB$ and $OC$ measure $8$ units, $6$ units and $10$ units respectively. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OC$ respectively.

Coordinate geometry

For a geometric progression, the sum to infinity equals eight times the first term. Determine the common ratio.

Series

The diagram represents sector $OAB$, with centre $O$ and radius $11\,\text{cm}$. The angle $AOB$ is $\alpha$ radians. Points $C$ and $D$ are on $OA$ and $OB$ respectively. Arc $CD$ is centred at $O$ and has radius $5\,\text{cm}$.

Circular measure

Find the possible values of $x$ satisfying $\sin^{-1}(x^2 - 1) = \frac{1}{3}\pi$, and give each answer correct to $3$ decimal places.

Trigonometry

Find the coefficient of $x^8$ in the expansion of $(x + 3x^2)^4$.

Series

A curve is given by $y = \dfrac{kx^2}{x + 2} + x$, where $k$ is a positive constant.

Differentiation

Solve the inequality $|x + 1| < |3x + 5|$ for $x$.

Algebra

The diagram presents the curve $y = x^4 + 2x - 9$. This curve crosses the positive $x$-axis at the point $P$.

Numerical solution of equations

The curve has equation $y = \frac{1}{2}e^{2x} - 5e^x + 4x$. Determine the exact $x$-coordinate of every stationary point on the curve, and identify the nature of each stationary point.

Differentiation

The polynomial $x^3 + ax^2 + bx + 8$, with $a$ and $b$ as constants, is called $p(x)$. It is stated that the remainder on dividing $p(x)$ by $(x - 3)$ is $14$, and that the remainder on dividing $p(x)$ by $(x + 2)$ is $24$. Determine the values of $a$ and $b$.

Algebra

A curve has parametric equations $x = \cos 2\theta - \cos \theta$, $y = 4\sin^2 \theta$, for $0 \leq \theta \leq \pi$.

Differentiation

Find $\int \frac{e^{2x} + 6}{e^{2x}}\, dx$.

Integration

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

Trigonometry

Find the value of $\int \frac{2}{4x - 1} \, dx$.

Integration

The curve $y = \frac{e^{3x-1}}{2x}$ has a single stationary point.

Differentiation

Find all solutions of $2\cot^2\theta - 5\cosec\theta = 10$ for $0^\circ \leq \\theta \leq 360^\circ$.

Trigonometry

Define p(x) to be the polynomial ax^3 + bx^2 - 25x - 6, where a and b are constants.

Algebra

A curve is described parametrically by $x = 1 + \sqrt{t}$ and $y = 3 \ln t$.

Differentiation

Evaluate $\int (\sin x - \cos x)^2 \, dx$.

Integration

The diagram depicts part of the curve $y = 8x + \frac{1}{2}e^x$. The shaded region $R$ is enclosed by the curve and the lines $x = 0$, $y = 0$ and $x = a$, with $a$ positive. The area of $R$ is $\frac{1}{2}$.

Numerical solution of equations

Solve $|x + 1| < |3x + 5|$.

Algebra

The diagram displays the curve $y = x^4 + 2x - 9$. This curve meets the positive $x$-axis at point $P$.

Numerical solution of equations

The curve is given by $y = \frac{1}{2}e^{2x} - 5e^x + 4x$.

Differentiation

The polynomial $x^3 + ax^2 + bx + 8$, in which $a$ and $b$ are constants, is represented by $p(x)$. It is given that when $p(x)$ is divided by $(x - 3)$ the remainder is $14$, and that when $p(x)$ is divided by $(x + 2)$ the remainder is $24$. Find the values of $a$ and $b$.

Algebra

A curve is defined parametrically by $x = \cos 2\theta - \cos \theta$, $y = 4\sin^2\theta$, with $0 \leq \theta \leq \pi$.

Differentiation

Find $\int \frac{e^{2x} + 6}{e^{2x}} \, dx$.

Integration

Express $3\cos\theta + \sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$, and state the exact value of $R$ together with the value of $\alpha$ correct to $2$ decimal places.

Trigonometry

For $x > -\frac{1}{2}$, the curve is defined by $y = \frac{1 + x}{1 + 2x}$.

Differentiation

A water-filled tank is shaped as a cone with vertex $C$. Its axis is vertical, and the semi-vertical angle is $60^\circ$, as the diagram shows. When $t = 0$, the tank is completely full and the water depth is $H$. At that moment, a tap at $C$ is turned on and water starts draining out. The volume of water in the tank falls at a rate proportional to $\sqrt{h}$, where $h$ denotes the water depth at time $t$. The tank is empty when $t = 60$.

Differential equations

Solve $2|3^x - 1| = 3^x$, and give the answers correct to 3 significant figures.

Logarithmic and exponential functions

Determine the exact value of $\int_1^4 \frac{\ln x}{\sqrt{x}} \, dx$.

Integration

A curve is given by the parametric equations $x = e^{-t} \cos t$, $y = e^{-t} \sin t$.

Differentiation

Prove that $\cot \theta + \tan \theta = 2 \cosec 2\theta$.

Integration

The diagram shows $A$ lying on the circumference of a circle whose centre is $O$ and whose radius is $r$. An arc of a circle centred at $A$ intersects the circumference at $B$ and $C$. The angle $OAB$ is $\theta$ radians. The shaded region is enclosed by the circle’s circumference and the arc centred at $A$ that joins $B$ and $C$. Its area is the same as half the area of the circle.

Numerical solution of equations

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

Algebra

Calculators are not allowed anywhere in this question.

Complex numbers

The diagram depicts three points $A$, $B$ and $C$, with position vectors relative to the origin $O$ given by $\vec{OA} = \begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}0 \\ 3 \\ 1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3 \\ 0 \\ 4\end{pmatrix}$. Point $D$ is situated on $BC$, between $B$ and $C$, and satisfies $CD = 2DB$.

Vectors

The curve is defined by $y = \frac{1 + x}{1 + 2x}$, with $x > -\frac{1}{2}$.

Differentiation

The water is held in a conical tank with vertex $C$. Its axis is vertical, and the semi-vertical angle is $60^\circ$, as the diagram indicates. When $t = 0$, the tank is completely filled and the water depth is $H$. At this moment, a tap at $C$ is opened so that water starts to leave the tank. The volume of water in the tank falls at a rate proportional to $\sqrt{h}$, where $h$ is the water depth at time $t$. The tank is empty at $t = 60$.

Differential equations

Solve the equation $2|3^x - 1| = 3^x$, and give the solutions to 3 significant figures.

Logarithmic and exponential functions

Determine the exact value of $\int_{1}^{4} \frac{\ln x}{\sqrt{x}}\,dx$.

Integration

A curve is described by the parametric equations $x = e^{-t}\\cos t$, $y = e^{-t}\\sin t$.

Differentiation

Prove that $\cot \theta + \tan \theta = 2\cosec 2\theta$ by simplifying the left-hand side.

Integration

The diagram shows point $A$ lying on the circumference of a circle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts the circumference at $B$ and $C$. The angle $OAB$ is $\theta$ radians. The shaded region is enclosed by the circle’s circumference and the arc centred at $A$ joining $B$ to $C$. Its area is equal to half the area of the circle.

Numerical solution of equations

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

Algebra

A calculator must not be used anywhere in this question.

Complex numbers

The diagram depicts three points $A$, $B$ and $C$ whose position vectors relative to the origin $O$ are $\vec{OA} = \begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, with $B$ and $C$ as the end points, and satisfies $CD = 2DB$.

Vectors

If $2\ln(x + 4) - \ln x = \ln(x + a)$, find $x$ in terms of $a$.

Logarithmic and exponential functions

One particular solution of the differential equation $3y^2 \frac{dy}{dx} = 4(y^3 + 1)\cos^2 x$ satisfies $y = 2$ when $x = 0$. The diagram gives a sketch of the graph of this solution for $0 \leq x \leq 2\pi$; stationary points are marked at $A$ and $B$.

Differential equations

Apply the substitution $u = 3x + 1$ in order to evaluate $\int \frac{3x}{3x + 1} \, dx$.

Integration

The polynomial $f(x)$ is given by $f(x) = x^3 + ax^2 - ax + 14$, with $a$ a constant. It is stated that $(x + 2)$ is a factor of $f(x)$.

Differentiation

The curve is defined by $3e^{2x}y + e^{x}y^3 = 14$. Find the gradient of the curve at the point $(0, 2)$.

Differentiation

You are told that $\int_0^p 4x e^{-\frac{1}{2}x} \, dx = 9$, with $p$ being a positive constant.

Numerical solution of equations

The two planes are given by $3x - y + 2z = 9$ and $x + y - 4z = -1$.

Vectors

If $\sec \theta + 2\cosec \theta = 3\cosec 2\theta$, prove that $2\sin \theta + 4\cos \theta = 3$.

Trigonometry

Write $\frac{7x^2 + 8}{(1 + x)^2(2 - 3x)}$ as partial fractions.

Algebra

Without a calculator, use the quadratic formula to solve $(2 - i)z^2 + 2z + 2 + i = 0$. Write the solutions in the form $a + bi$.

Complex numbers

Particle $P$ has mass $0.3\,\mathrm{kg}$ and is joined to one end of a light inextensible string. The free end of the string is fixed at point $X$. A horizontal force of magnitude $F\,\mathrm{N}$ acts on the particle, and it is in equilibrium when the string makes an angle $\alpha$ with the vertical, where $\tan\alpha = \frac{8}{15}$ (see diagram).

Forces and equilibrium

A block $B$ rests on a rough horizontal plane. As shown in the diagram, two horizontal forces of magnitudes $30\,\mathrm{N}$ and $40\,\mathrm{N}$ act on $B$; these make angles $\alpha$ and $\beta$ respectively with the $x$-direction. The block is travelling in the $x$-direction at constant speed. It is given that $\cos\alpha = 0.6$ and $\cos\beta = 0.8$.

Energy, work and power

A cyclist is pushing with a steady driving force of magnitude $F\,\mathrm{N}$ as he travels up a straight hill that makes an angle $\alpha$ with the horizontal, where $\sin\alpha = \frac{36}{325}$. The cyclist experiences a constant resistive force of $32\,\mathrm{N}$ opposite the motion. The combined weight of the cyclist and his bicycle is $780\,\mathrm{N}$. The cyclist’s acceleration is $-0.2\,\mathrm{m\,s^{-2}}$.

Energy, work and power

Particles $P$ and $Q$ move along a straight line on a rough horizontal plane, and friction is the only horizontal force acting on the particles.

Newton's laws of motion

A lorry with mass $15\,000\,\mathrm{kg}$ moves up a straight hill of length $1440\,\mathrm{m}$ from the bottom to the top at a steady speed of $15\,\mathrm{m\,s^{-1}}$. The summit is $16\,\mathrm{m}$ above the level of the foot of the hill. The resistance to motion is constant at $1800\,\mathrm{N}$.

Energy, work and power

Particles $A$ and $B$, with masses of $0.3\,\mathrm{kg}$ and $0.7\,\mathrm{kg}$ respectively, are fixed to the two ends of a light inextensible string. This string runs over a fixed smooth pulley. $A$ is initially held at rest and $B$ is hanging freely, while the two vertical sections of the string are straight and both particles are $0.52\,\mathrm{m}$ above the floor (see diagram). $A$ is then released, so both particles begin to move.

Newton's laws of motion

A particle $P$ begins at rest at point $O$ and travels along a straight path. For time $t$ seconds after leaving $O$, $P$ has acceleration $0.6t\,\mathrm{m\,s^{-2}}$ until $t = 10$.

Kinematics of motion in a straight line

A small block with weight $5.1\,\text{N}$ is at rest on a smooth plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{8}{17}$. It is maintained in equilibrium by a light inextensible string. The string is inclined at an angle $\beta$ above the line of greatest slope on which the block rests, where $\sin\beta = \frac{7}{25}$ (see diagram).

Forces and equilibrium

A box with mass $25\,\text{kg}$ is dragged in a straight line over a horizontal floor. It starts from rest at point $A$ and reaches point $B$ with speed $3\,\text{m s}^{-1}$. The distance $AB$ is $15\,\text{m}$. The pulling force has magnitude $220\,\text{N}$ and acts at an angle $\alpha$ above the horizontal. As the box moves from $A$ to $B$, the work done against the resistance to motion on the box is $3000\,\text{J}$.

Forces and equilibrium

The force resisting a runner of mass $70\,\text{kg}$ is $kv\,\text{N}$, where $v\,\text{m s}^{-1}$ represents the runner’s speed and $k$ is a constant. The maximum power the runner can produce is $100\,\text{W}$. The runner’s highest steady speed on horizontal ground is $4\,\text{m s}^{-1}$.

Forces and equilibrium

A rough plane is set at an angle $\alpha$ to the horizontal, with $\tan\alpha = 2.4$. A small block of mass $0.6\,\text{kg}$ is kept at rest on the plane by a horizontal force of magnitude $P\,\text{N}$. This force lies in a vertical plane passing through a line of greatest slope (see diagram). The coefficient of friction between the block and the plane is $0.4$. The block is just about to slide down the plane.

Forces and equilibrium

A particle $P$ moves along a straight line. $P$ begins from rest at $O$ and reaches $A$, where it comes to rest again, in $50$ seconds. The speed of $P$ at time $t$ seconds after leaving $O$ is $v\,\text{m s}^{-1}$, where $v$ is defined as follows: For $0 \leq t \leq 5$, $v = t - 0.1t^2$, for $5 \leq t \leq 45$, $v$ is constant, for $45 \leq t \leq 50$, $v = 9t - 0.1t^2 - 200$.

Kinematics of motion in a straight line

Particles $A$ and $B$ have masses of $0.4\,\text{kg}$ and $1.6\,\text{kg}$ respectively. They are connected by a light inextensible string passing over a fixed smooth pulley. Particle $A$ is kept at rest while $B$ hangs freely, with the two vertical sections of the string straight and each particle initially $1.2\,\text{m}$ above the floor (see diagram). $A$ is let go, and the two particles begin to move.

Energy, work and power

A cable pulls an elevator straight upward. The velocity-time graph for this motion is shown above.

Newton's laws of motion

A particle travels upwards along the line of greatest slope on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = 0.28$. The coefficient of friction between the particle and the plane is $\frac{1}{3}$.

Kinematics of motion in a straight line

Particle $A$, with mass $0.2\,\text{kg}$, and particle $B$, with mass $0.6\,\text{kg}$, are joined to the two ends of a light inextensible string. The string passes over a fixed smooth pulley. $B$ is initially at rest $1.6\,\text{m}$ above the floor. $A$ hangs freely at a height of $h\,\text{m}$ above the floor. Both straight sections of the string are vertical (see diagram). $B$ is released and both particles begin to move. When $B$ reaches the floor it stays at rest, but $A$ continues moving vertically upwards until it is $3\,\text{m}$ above the floor.

Kinematics of motion in a straight line

A particle $P$ with mass $1.05\,\text{kg}$ is attached to one end of each of two light inextensible strings, whose lengths are $2.6\,\text{m}$ and $1.25\,\text{m}$. The opposite ends of the strings are fixed to points $A$ and $B$, which lie at the same horizontal level. $P$ is held in equilibrium at a point $1\,\text{m}$ below the level of $A$ and $B$ (see diagram).

Forces and equilibrium

A box of mass $30\,\text{kg}$ is initially at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = 0.1$, and a force of magnitude $40\,\text{N}$ acts on it. This force is directed upwards, parallel to a line of greatest slope of the plane. The box is on the verge of sliding up the plane.

Forces and equilibrium

A car moves in a straight line from $A$ to $B$, covering $12\,\text{km}$ in $552$ seconds. It begins at rest at $A$ and speeds up for $T_1\,\text{s}$ with acceleration $0.3\,\text{m s}^{-2}$, reaching speed $V\,\text{m s}^{-1}$. It then travels at constant speed $V\,\text{m s}^{-1}$ for $T_2\,\text{s}$. After that, it slows down for $T_3\,\text{s}$ at $1\,\text{m s}^{-2}$, stopping at $B$.

Kinematics of motion in a straight line

A lorry with mass $12\,500\,\text{kg}$ moves along a route from $A$ to $C$, going via $B$. For the entire trip from $A$ to $C$, the resistance to motion acting on the lorry is $4800\,\text{N}$.

Energy, work and power

A vehicle begins from rest at point $O$ and travels along a straight line. Its speed $v\,\text{m s}^{-1}$ after $t$ seconds from leaving $O$ is given by the following: when $0 \leq t \leq 60$, $v = k_1 t - 0.005t^2$; and when $t > 60$, $v = \frac{k_2}{\sqrt{t}}$. During the first $60\,\text{s}$, the vehicle covers a distance of $540\,\text{m}$.

Kinematics of motion in a straight line

A particle $P$ with mass $0.3\,\text{kg}$ is fixed to one end of a light inextensible string of length $0.6\,\text{m}$. The other end of the string is fastened to a fixed point $O$ on a smooth horizontal plane. $P$ travels on the plane at a constant speed of $5\,\text{m s}^{-1}$ in a circle centred at $O$.

Probability

A uniform frame is formed by a semicircular arc $ABC$ with radius $0.6\,\text{m}$ and its diameter $AOC$, where $O$ is the centre of the semicircle (see diagram).

Representation of data

A particle $P$ with mass $0.8\,\text{kg}$ travels along the $x$-axis on a horizontal surface. If the displacement of $P$ from the origin $O$ is $x\,\text{m}$, then its velocity is $v\,\text{m s}^{-1}$ in the positive $x$-direction. Two horizontal forces act on $P$. One has magnitude $4e^{-x}\,\text{N}$ and acts in the positive $x$-direction. The other has magnitude $2.4x^2\,\text{N}$ and acts in the negative $x$-direction.

Probability

A small ball $B$ is launched from point $O$ with speed $14\,\text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal.

Representation of data

A particle $P$ has mass $0.2\,\text{kg}$ and is joined to fixed point $A$ by a light inextensible string of length $0.4\,\text{m}$. A second light inextensible string, of length $0.3\,\text{m}$, links $P$ to fixed point $B$, which lies vertically beneath $A$. The particle $P$ travels in a horizontal circle whose centre lies on the line $AB$, and $\angle APB = 90^\circ$ (see diagram).

Probability

$ABCD$ shows the cross-section through the centre of mass of a uniform rectangular block weighing $260\,\text{N}$. The sides $AB$ and $BC$ measure $1.5\,\text{m}$ and $0.8\,\text{m}$ respectively. The block is in equilibrium with point $D$ resting on a rough horizontal floor. A light rope connects point $A$ on the block to point $E$ on the floor, maintaining equilibrium. The points $E$, $A$ and $B$ are on one straight line that is inclined at $30^\circ$ to the horizontal (see diagram).

Probability

A particle $P$ with mass $0.4\,\text{kg}$ is connected to one end of a light elastic string whose natural length is $0.8\,\text{m}$ and modulus of elasticity is $32\,\text{N}$. The opposite end of the string is fixed at point $O$. The particle is released from rest at $O$.

Probability

A particle $P$, with mass $0.3\,\text{kg}$, is fastened to one end of a light inextensible string whose length is $0.6\,\text{m}$. The string’s other end is fixed at a point $O$ on a smooth horizontal plane. $P$ travels at constant speed $5\,\text{m s}^{-1}$ round a circle with centre $O$.

Probability

The uniform frame is formed from a semicircular arc $ABC$ of radius $0.6\,\text{m}$ and its diameter $AOC$, with $O$ at the centre of the semicircle (see diagram).

Representation of data

A particle $P$ with mass $0.8\,\text{kg}$ travels on the $x$-axis over a horizontal plane. If the displacement of $P$ from the origin $O$ is $x\,\text{m}$, its velocity is $v\,\text{m s}^{-1}$ in the positive $x$-direction. Two horizontal forces act on $P$. One has magnitude $4e^{-x}\,\text{N}$ and is directed in the positive $x$-direction. The other has magnitude $2.4x^2\,\text{N}$ and is directed in the negative $x$-direction.

Representation of data

A small ball $B$ is launched from point $O$ at a speed of $14\,\text{m s}^{-1}$ and at an angle of $60^\circ$ above the horizontal.

Probability

Particle P has mass $0.2\,\text{kg}$ and is joined to fixed point A by a light inextensible string of length $0.4\,\text{m}$. A second light inextensible string of length $0.3\,\text{m}$ links P to fixed point B, which lies vertically beneath A. The particle P describes a horizontal circle whose centre lies on the line AB, and angle APB = 90^\circ$ (see diagram).

Probability

$ABCD$ shows a cross-section through the centre of mass of a uniform rectangular block weighing $260\,\text{N}$. The lengths $AB$ and $BC$ are $1.5\,\text{m}$ and $0.8\,\text{m}$ respectively. The block is in equilibrium with point $D$ resting on a rough horizontal floor. A light rope holds the block in equilibrium, being attached to point $A$ on the block and point $E$ on the floor. The points $E$, $A$ and $B$ lie on one straight line that is inclined at $30^\circ$ to the horizontal (see diagram).

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is connected to one end of a light elastic string whose natural length is $0.8\,\text{m}$ and whose modulus of elasticity is $32\,\text{N}$. The free end of the string is fixed at point $O$. The particle is released from rest at $O$.

Probability

A particle $P$ with mass $0.1\,\text{kg}$ is fixed to one end of a light elastic string whose natural length is $0.4\,\text{m}$ and modulus of elasticity is $12\,\text{N}$. The opposite end of the string is fastened to a fixed point $O$ on a smooth horizontal plane. $P$ travels across the surface in a horizontal circle with centre $O$ and radius $0.6\,\text{m}$.

Probability

A particle $P$ with mass $0.5\,\text{kg}$ is let go from rest at point $O$ and moves straight down. After it has descended a distance of $x\,\text{m}$ from $O$, its speed is $v\,\text{m s}^{-1}$. A resisting force of magnitude $0.015x^2\,\text{N}$ acts on $P$.

Representation of data

A particle $P$ with mass $0.5\,\text{kg}$ travels in a horizontal circle on the smooth inner face of a hollow cone, fixed with its axis vertical and vertex downwards. $P$ moves with angular speed $5\,\text{rad s}^{-1}$ in a circle of radius $0.4\,\text{m}$ (see diagram).

Probability