Oct/Nov 2015

For the expansion of $(1 - \frac{2x}{a})(a + x)^5$, where $a$ is a non-zero constant, show that the coefficient of $x^2$ is zero.

Series

A cuboid $OABCPQRS$ is shown with flat base $OABC$, where $AB = 6\,\text{cm}$ and $OA = a\,\text{cm}$, with $a$ constant. The cuboid has height $OP = 10\,\text{cm}$. Point $T$ lies on $BR$ so that $BT = 8\,\text{cm}$, and $M$ is the mid-point of $AT$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OP$ respectively.

Coordinate geometry

The diagram shows a section of the curve $y = \sqrt{1 + 4x}$, together with the point $P(6,5)$ on the curve. The line $PQ$ cuts the $x$-axis at $Q(8,0)$.

Integration

The function $f$ satisfies $f'(x) = 3x^2 - 7$ and $f(3) = 5$.

Functions

Solve the equation $\sin^{-1}(4x^4 + x^2) = \frac{1}{6}\pi$.

Trigonometry

Show that the equation $\frac{4\cos\theta}{\tan\theta} + 15 = 0$ may be rewritten as $4\sin^2\theta - 15\sin\theta - 4 = 0$.

Trigonometry

The equation of a curve is $y = \frac{8}{x} + 2x$.

Differentiation

The curve is defined by $y = x^2 - x + 3$ and the line is defined by $y = 3x + a$, where $a$ is a constant.

Coordinate geometry

A circle of radius $r$ is centred at $A$. The diameters $CAD$ and $BAE$ cross at right angles. A second, larger circle is centred at $B$ and passes through $C$ and $D$.

Coordinate geometry

A progression has first term $4x$ and second term $x^2$.

Series

The function $f: x \mapsto -x^2 + 6x - 5$ has domain $x \geq m$, with $m$ as a constant.

Functions

The functions $f$ and $g$ are given by $f: x \mapsto 3x + 2,\ x \in \mathbb{R}$ and $g: x \mapsto 4x - 12,\ x \in \mathbb{R}$.

Functions

The diagram illustrates part of the curve $y = \sqrt{9 - 2x^2}$. Point $P(2, 1)$ is on the curve, and the normal at $P$ cuts the $x$-axis at $A$ and the $y$-axis at $B$. The shaded area is enclosed by the curve, the $y$-axis and the line $y = 1$.

Integration

In the expansion of $(x + 2k)^7$, where $k$ is a non-zero constant, the coefficients of $x^4$ and $x^5$ are the same. Find the value of $k$.

Series

Fig. 1 illustrates an open tank formed as a triangular prism. The two vertical ends $ABE$ and $DCF$ are congruent isosceles triangles. $ angle ABE = \angle BAE = 30^\circ$. $AD$ has length $40\,\text{cm}$. The tank is held so that the top $ABCD$ lies horizontally. Water is added at a steady rate of $200\,\text{cm}^3\,\text{s}^{-1}$. If the water depth is $h\,\text{cm}$ after $t$ seconds from the start of filling, this is shown in Fig. 2.

Differentiation

Prove the identity $\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right)^2 = \frac{1 - \cos x}{1 + \cos x}$.

Trigonometry

The diagram depicts a metal plate $OABC$, made up of a right-angled triangle $OAB$ and a sector $OBC$ from a circle with centre $O$. Angle $AOB = 0.6$ radians, $OA = 6\,\text{cm}$ and $OA$ is perpendicular to $OC$.

Circular measure

The coordinates of the points are A(-3, 7), B(5, 1) and C(-1, k), with k as a constant.

Coordinate geometry

With O as the origin, the position vectors of points A, B and C are given by $\overrightarrow{OA} = \begin{pmatrix}0\\2\\-3\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}2\\5\\-2\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}3\\p\\q\end{pmatrix}$.

Coordinate geometry

For $x \in \mathbb{R}$, the function $f$ is given by $f: x \mapsto x^2 + ax + b$, with $a$ and $b$ as constants.

Quadratics

The curve $y = f(x)$ has a stationary point at $(2, 10)$, and $f''(x) = \frac{12}{x^3}$ is given.

Differentiation

The line is given by $y = 2x - 7$ and the curve is given by $y = x^2 - 4x + c$, with $c$ a constant. Find the set of values of $c$ for which the line does not intersect the curve.

Quadratics

For $x > -1$, the function $f$ is given by $f(x) = 2x + (x + 1)^{-2}$.

Differentiation

Find the coefficient attached to $x$ in the expansion of $\left(\frac{x}{3} + \frac{9}{x^2}\right)^7$.

Series

Express $3x^2 - 6x + 2$ as $(x + b)^2 + c$, where $b$ and $c$ are constants.

Differentiation

The diagram depicts a metal plate $OABCDEF$ made up of $3$ sectors, all with centre $O$. Sector $COD$ has radius $2r$ and angle $COD$ is $\theta$ radians. Each of the sectors $BOA$ and $FOE$ has radius $r$, and $AOED$ and $CBOF$ are straight lines.

Circular measure

With respect to an origin $O$, the position vectors of points $A$ and $B$ are $\vec{OA} = \begin{pmatrix} p - 6 \\ 2p - 6 \\ 1 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 4 - 2p \\ p \\ 2 \end{pmatrix}$, where $p$ is a constant.

Coordinate geometry

A ball is released from a height of $1$ metre, and on hitting the ground it rebounds straight up to $0.96$ metres. It keeps bouncing on the ground, with the maximum height decreasing each time. Two distinct models, $A$ and $B$, are used to describe this. Model $A$: Each bounce reduces the height reached by $0.04$ metres. Model $B$: Each bounce reduces the height reached by $4\%$.

Series

Show that the equation $\frac{1}{\cos \theta} + 3 \sin \theta \tan \theta + 4 = 0$ can be rewritten as $3 \cos^2 \theta - 4 \cos \theta - 4 = 0$, and hence solve the equation $\frac{1}{\cos \theta} + 3 \sin \theta \tan \theta + 4 = 0$ for $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The rule for $f$ is $f(x) = 3x + 1$ when $x \leq a$, where $a$ is constant. The rule for $g$ is $g(x) = -1 - x^2$ when $x \leq -1$.

Functions

A curve goes through the point $A(4,6)$ and satisfies $\frac{dy}{dx} = 1 + 2x^{-\frac{1}{2}}$. A point $P$ moves along the curve so that the $x$-coordinate of $P$ is rising at a constant rate of $3$ units per minute.

Differentiation

Solve $5^{x+3} = 7^{x-1}$ by using logarithms, and give the result correct to 3 significant figures.

Logarithmic and exponential functions

The curve is given by the equation $y = \frac{3x + 1}{x - 5}$.

Differentiation

Express $8\sin\theta + 15\cos\theta$ in the form $R\sin(\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the value of $\alpha$ correct to 2 decimal places.

Trigonometry

By drawing an appropriate pair of graphs, show that the equation $\ln x = 4 - \frac{1}{2}x$ has exactly one real root, $\alpha$.

Numerical solution of equations

Determine $\int (\tan^2 x + \sin 2x)\,dx$.

Integration

Determine the quotient and the remainder when $x^4 + x^3 + 3x^2 + 12x + 6$ is divided by $(x^2 - x + 4)$.

Algebra

For a curve, the parametric equations are $x = 6\sin^2 t$, $y = 2\sin 2t + 3\cos 2t$, for $0 \leq t < \pi$. The curve meets the $x$-axis at $B$ and $D$, and the stationary points are $A$ and $C$, as the diagram shows.

Differentiation

Solve $|3x - 2| = 5$.

Logarithmic and exponential functions

Starting from $x_1 = 2$, the values generated by the iterative formula $x_{n+1} = 2 + \frac{4}{x_n^2 + 2x_n + 4}$ approach $\alpha$.

Numerical solution of equations

The variables $x$ and $y$ obey the relation $y = Kx^m$, where $K$ and $m$ are constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line running through the points $(0.22, 3.96)$ and $(1.32, 2.43)$, as shown in the diagram.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = 6x^3 + 11x^2 + ax + a$, where $a$ is a constant. It is also stated that $(x + 2)$ is a factor of $p(x)$.

Trigonometry

Find the $x$-coordinates of the stationary points for the curves below:

Logarithmic and exponential functions

The diagram depicts the curve whose parametric equations are $x = 3\cos t$, $y = 2\cos\left(t - \frac{1}{6}\pi\right)$, for $0 \leq t < 2\pi$.

Differentiation

Show that the exact value of $\int_0^{\frac{1}{3}\pi} \left(\cos^2 x + \frac{1}{\cos^2 x}\right) dx$ is $\frac{1}{6}\pi + \frac{9}{8}\sqrt{3}$.

Integration

Find the exact value of $\int_{-1}^{35} \frac{3}{2x+5}\,dx$, and present the result in the form $\ln k$.

Integration

Solve for $x$ in the equation $|2x+3| = |x+8|$.

Logarithmic and exponential functions

The parametric form of a curve is $x = (t+1)e^t$, $y = 6\sqrt{t+4}$.

Differentiation

Find the quotient obtained when $3x^3 + 5x^2 - 2x - 1$ is divided by $(x-2)$, and show that the remainder comes to $39$.

Algebra

You are told that $\int_0^a (3e^{3x} + 5e^x)\,dx = 100$, with $a$ a positive constant.

Numerical solution of equations

Express $(\sqrt{5})\cos\theta - 2\sin\theta$ in the form $R\cos(\theta + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the value of $\alpha$ correct to $2$ decimal places.

Trigonometry

The curve is defined by $y = \frac{\sin 2x}{\cos x + 1}$.

Differentiation

Find the values of $x$ that satisfy $|2x - 5| > 3|2x + 1|$.

Algebra

The diagram depicts the curve $y = \frac{x^2}{1 + x^3}$ for $x \ge 0$, together with its maximum point $M$. The shaded region $R$ lies between the curve, the $x$-axis and the lines $x = 1$ and $x = p$.

Integration

By making the substitution $u = 3^x$.

Logarithmic and exponential functions

The angles $\theta$ and $\phi$ each lie in the interval from $0^\circ$ to $180^\circ$, and they satisfy $\tan(\theta - \phi) = 3$ together with $\tan \theta + \tan \phi = 1$.

Trigonometry

The equation $x^3 - x^2 - 6 = 0$ has a single real root, written as $\alpha$.

Numerical solution of equations

A curve is defined by $y = e^{-2x} \tan x$, with $0 \le x < \frac{\pi}{2}$.

Differentiation

The polynomial $8x^3 + ax^2 + bx - 1$, with $a$ and $b$ taken as constants, is written as $p(x)$. You are told that $(x + 1)$ is a factor of $p(x)$, and that dividing $p(x)$ by $(2x + 1)$ gives remainder $1$.

Algebra

The position vectors of points $A$, $B$ and $C$, measured from the origin $O$, are given by $\vec{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. Plane $m$ is perpendicular to $AB$ and passes through point $C$.

Vectors

The variables $x$ and $\theta$ are related by the differential equation $\frac{dx}{d\theta} = (x + 2) \sin^2 2\theta$, and the condition $x = 0$ applies when $\theta = 0$.

Differential equations

Let the complex number $3 - i$ be written as $u$. Write its complex conjugate as $u^*$.

Complex numbers

Solve the inequality $|2x - 5| > 3|2x + 1|$.

Algebra

The diagram depicts the curve $y = \frac{x^2}{1 + x^3}$ for $x \ge 0$, together with its maximum point $M$. The shaded area $R$ is bounded by the curve, the $x$-axis, and the straight lines $x = 1$ and $x = p$.

Integration

Apply the substitution $u = 3^x$.

Logarithmic and exponential functions

The angles $\theta$ and $\phi$ each lie in the interval from $0^\circ$ to $180^\circ$, and they satisfy $\tan(\theta - \phi) = 3$ and $\tan \theta + \tan \phi = 1$.

Trigonometry

The equation $x^3 - x^2 - 6 = 0$ has a single real root, written as $\alpha$.

Numerical solution of equations

A curve has equation $y = e^{-2x} \tan x$, where $0 \le x < \frac{1}{2}\pi$.

Differentiation

Let $p(x)$ denote the polynomial $8x^3 + ax^2 + bx - 1$, where $a$ and $b$ are constants. It is stated that $(x + 1)$ is a factor of $p(x)$ and that dividing $p(x)$ by $(2x + 1)$ leaves a remainder of $1$.

Algebra

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. The plane $m$ is perpendicular to $AB$ and passes through the point $C$.

Vectors

The variables $x$ and $\theta$ are linked by the differential equation $\frac{dx}{d\theta} = (x + 2) \sin^2 2\theta$, and $x = 0$ when $\theta = 0$ is given.

Differential equations

The complex number $3 - i$ is represented by $u$. Its complex conjugate is represented by $u^*$.

Complex numbers

Draw the graph of $y = e^{ax} - 1$ for a positive constant $a$.

Logarithmic and exponential functions

Naturalists are running a wildlife reserve to raise the population of a rare plant species. The number of plants after $t$ years is represented by $N$, with $N$ taken as a continuous variable.

Differential equations

If $\sqrt[3]{(1 + 9x)} \approx 1 + 3x + ax^2 + bx^3$ for small values of $x$, determine the values of the coefficients $a$ and $b$.

Algebra

The equation of the curve is $y = \frac{2 - \tan x}{1 + \tan x}$.

Differentiation

A curve is defined by the parametric equations $x = t^2 + 3t + 1$, $y = t^4 + 1$. The point $P$ on the curve corresponds to parameter $p$. It is stated that the gradient of the curve at $P$ is $4$.

Numerical solution of equations

Use the substitution $u = 4 - 3\cos x$ to determine the exact value of $\int_0^{\frac{1}{2}\pi} \frac{9 \sin 2x}{\sqrt{(4 - 3 \cos x)}} \, dx$.

Integration

Angles $A$ and $B$ satisfy $\sin(A + 45^\circ) = (2\sqrt{2}) \cos A$ and $4\sec^2 B + 5 = 12 \tan B$.

Trigonometry

Demonstrate that $(x + 1)$ is a factor of $4x^3 - x^2 - 11x - 6$.

Integration

A plane is described by the equation $4x - y + 5z = 39$. A straight line is parallel to the vector $\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$ and goes through the point $A(0, 2, -8)$. This line intersects the plane at point $B$.

Vectors

It is given that $(1 + 3i)w = 2 + 4i$. Show all the working needed to prove that the exact value of $|w^2|$ equals $2$ and determine $\arg(w^2)$ correct to $3$ significant figures.

Complex numbers

A weightlifter carries out an exercise in which he lifts a mass of $200\,\text{kg}$ from rest vertically by a distance of $0.7\,\text{m}$ and then keeps it at that level.

Energy, work and power

A particle of mass $0.5\,\text{kg}$ is released from rest and moves down the line of greatest slope on a smooth plane. The plane is set at an angle of $30^\circ$ to the horizontal.

Kinematics of motion in a straight line

A lorry with mass $24\,000\,\text{kg}$ is moving uphill on a slope that makes an angle of $3^\circ$ with the horizontal. The engine produces constant power, and the resistive force opposing the motion is constant at $3200\,\text{N}$.

Energy, work and power

Blocks $P$ and $Q$, with masses $m\,\text{kg}$ and $5\,\text{kg}$ respectively, are joined by the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a rough plane inclined at $35^\circ$ to the horizontal. Block $P$ rests on the plane, while block $Q$ hangs vertically below the pulley (see diagram). The coefficient of friction between block $P$ and the plane is $0.2$.

Forces and equilibrium

A small bead $Q$ is able to slide without friction along the smooth horizontal straight wire $AB$, which has length $3\,\text{m}$. As shown in the diagram, three horizontal forces with magnitudes $F\,\text{N}$, $10\,\text{N}$ and $20\,\text{N}$ act on the bead. The three forces have a resultant of magnitude $R\,\text{N}$, acting in the direction indicated in the diagram.

Energy, work and power

A particle $P$ travels in a straight line from a starting point $O$. At time $t\,\text{s}$ after leaving $O$, its velocity, in $\text{m s}^{-1}$, is given by $v = 0.6t - 0.03t^2$.

Kinematics of motion in a straight line

A cyclist leaves point A from rest and travels in a straight line with acceleration $0.5\,\text{m s}^{-2}$ over a distance of $36\,\text{m}$. The cyclist then continues at constant speed for $25\,\text{s}$ before decelerating uniformly until coming to rest at point B. The distance $AB$ is $210\,\text{m}$. $24\,\text{s}$ after the cyclist leaves point A, a car sets off from rest from point A, with constant acceleration $4\,\text{m s}^{-2}$, towards B. It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.

Kinematics of motion in a straight line

At point $O$, four horizontal forces are balanced, so the forces are in equilibrium. Their magnitudes are $F$ N, $G$ N, $15$ N and $F$ N, and they act in the directions shown in the diagram.

Forces and equilibrium

A particle is let go from rest at a point $H$ m above the horizontal ground and travels straight downwards. It goes past a point $35$ m above the ground at a speed of $(V - 10)$ m s$^{-1}$ and then hits the ground at speed $V$ m s$^{-1}$. Find

Kinematics of motion in a straight line

A particle $P$ travels in a straight line over $100$ s. It leaves a point $O$, and when $t$ seconds have passed since leaving $O$ its velocity is $v$ m s$^{-1}$, where $v = 0.00004t^3 - 0.006t^2 + 0.288t$.

Kinematics of motion in a straight line

The figure shows a vertical cross-section $ABC$ of a surface. The section of the surface along $AB$ is smooth, and $A$ lies $2.5$ m above the level of $B$. The section of the surface along $BC$ is rough and makes an angle of $45^\circ$ with the horizontal. The length of $BC$ is $4$ m (see diagram). A particle $P$ of mass $0.2$ kg is released from rest at $A$ and moves while remaining in contact with the curve $AB$ and then the straight line $BC$. The coefficient of friction between $P$ and the section of the surface along $BC$ is $0.4$.

Energy, work and power

A smooth inclined plane of length $2.5$ m has one end on the horizontal floor, while the other end is raised to a height of $0.7$ m above the floor. Particles $P$ and $Q$, with masses $0.5$ kg and $0.1$ kg respectively, are connected to the ends of a light inextensible string that passes over a small smooth pulley fixed at the top of the plane. Particle $Q$ is kept at rest on the floor directly below the pulley. The string is taut and $P$ is at rest on the plane (see diagram). $Q$ is released and starts to move vertically upwards towards the pulley, while $P$ moves down the plane.

Newton's laws of motion

A small ring of mass $0.024$ kg is threaded onto a rough horizontal rod that is fixed in place. A light inextensible string is attached to the ring, and it is pulled by a force of magnitude $0.195$ N making an angle $\theta$ with the horizontal, where $\sin\theta = \frac{5}{13}$. When $\theta$ is below the horizontal (see Fig. 1), the ring is in limiting equilibrium.

Newton's laws of motion

A car with mass $1600$ kg is travelling on a straight horizontal road at constant power $14$ kW. It takes $25$ s to go from point $A$ to point $B$ on the road.

Energy, work and power

A ball $B$ of mass $4\,\text{kg}$ is fastened to one end of a light inextensible string. At the opposite end of the string is a particle $P$ of mass $3\,\text{kg}$. The string runs over a fixed smooth pulley. The system is in equilibrium, with the string taut and both straight sections vertical. $B$ rests on a rough plane that is inclined to the horizontal at an angle of $\alpha$, where $\cos\alpha = 0.8$ (see diagram).

Forces and equilibrium

A ring with mass $0.2\,\text{kg}$ is placed on a fixed rough horizontal rod, and a light inextensible string is fixed to the ring at an angle $\alpha$ above the horizontal, where $\cos\alpha = 0.96$. The ring is in limiting equilibrium, and the tension in the string is $T\,\text{N}$ (see diagram). The coefficient of friction between the ring and the rod is $0.25$.

Forces and equilibrium

The diagram shows three horizontal forces with magnitudes $150\,\text{N}$, $100\,\text{N}$ and $P\,\text{N}$ acting in the directions indicated. Their resultant is represented by the broken line in the diagram. This resultant has magnitude $120\,\text{N}$ and forms an angle $75^\circ$ with the $150\,\text{N}$ force.

Forces and equilibrium

Particles $A$ and $B$, with masses $0.35\,\text{kg}$ and $0.15\,\text{kg}$ respectively, are fixed to the ends of a light inextensible string that runs over a fixed smooth pulley. The system is initially at rest, with $B$ resting on the horizontal floor, the string taut and its straight sections vertical. $A$ is $1.6\,\text{m}$ above the floor (see diagram). $B$ is then released and the system starts to move; $B$ does not reach the pulley.

Newton's laws of motion

A cyclist together with his bicycle has a combined mass of $90\,\text{kg}$. He begins moving at a speed of $3\,\text{m s}^{-1}$ from the summit of a straight hill, with length $500\,\text{m}$, that is inclined at an angle of $\sin^{-1} 0.05$ to the horizontal. He travels with constant acceleration and arrives at the bottom of the hill at a speed of $5\,\text{m s}^{-1}$. While going downhill, the cyclist produces $420\,\text{W}$ of power. The resistive force on the cyclist and his bicycle, $R\,\text{N}$, and the cyclist’s speed, $v\,\text{m s}^{-1}$, both change.

Energy, work and power

A particle $P$ leaves rest from a point $O$ on a straight line and travels along that line. Its displacement after $t\,\text{s}$ from leaving $O$ is $x = 0.08t^2 - 0.0002t^3$.

Kinematics of motion in a straight line

A straight hill $AB$ is $400\,\text{m}$ long, with $A$ at the summit and $B$ at the foot, and it makes an angle of $4^\circ$ to the horizontal. A straight level road $BC$ is $750\,\text{m}$ long. A car of mass $1250\,\text{kg}$ has speed $5\,\text{m s}^{-1}$ at $A$ as it begins to travel downhill. During the descent, the resistive force on the car is $2000\,\text{N}$ and the driving force remains constant. The car’s speed on arriving at $B$ is $8\,\text{m s}^{-1}$. At $B$ the car continues along road $BC$. The driving force stays constant and is twice its value on the hill. The resistive force on the car remains $2000\,\text{N$}.

Energy, work and power

A particle $P$ travels along a straight line, crossing point $O$ on the line with velocity $2\,\text{m s}^{-1}$. After $t$ s has elapsed since passing through $O$, the velocity of $P$ is $v\,\text{m s}^{-1}$ and the acceleration of $P$ is $e^{-0.5v}\,\text{m s}^{-2}$. Calculate the velocity of $P$ when $t = 1.2$.

Probability

A uniform rigid rod $AB$ with length $1.2\,\text{m}$ and weight $8\,\text{N}$ carries a particle of weight $2\,\text{N}$ fixed at endpoint $B$. Endpoint $A$ is attached by a free hinge to a fixed point. One end of a light elastic string, whose natural length is $0.8\,\text{m}$ and modulus of elasticity is $20\,\text{N}$, is fastened to the hinge. The string passes over a small smooth pulley $P$ fixed $0.8\,\text{m}$ vertically above the hinge. The other end is connected to a small light smooth ring $R$ that can move along the rod. The system is in equilibrium, with the rod making an angle $\theta^{\circ}$ to the vertical (see diagram).

Probability

A particle $P$ with mass $0.3\,\text{kg}$ travels along a straight line on a smooth horizontal surface. $P$ passes through a fixed point $O$ on the line with velocity $8\,\text{m s}^{-1}$. A force of magnitude $2x\,\text{N}$ acts on $P$ in the direction $PO$, where $x\,\text{m}$ is the displacement of $P$ from $O$.

Probability

A light inextensible string has one end fixed at point $A$. It runs through a smooth bead $B$ of mass $0.3\,\text{kg}$, and the other end is fixed at point $C$, directly beneath $A$. The bead $B$ moves at constant speed in a horizontal circle of radius $0.6\,\text{m}$ whose centre lies between $A$ and $C$. The string is at an angle of $30^{\circ}$ to the vertical at $A$ and at an angle of $45^{\circ}$ to the vertical at $C$ (see diagram).

Probability

A particle $P$ with mass $0.2\,\text{kg}$ is connected to one end of a light elastic string whose natural length is $0.75\,\text{m}$ and modulus of elasticity is $21\,\text{N}$. The opposite end of the string is fixed at a point $A$, which lies $0.8\,\text{m}$ vertically above a smooth horizontal surface. $P$ is at rest in equilibrium on the surface.

Representation of data

A uniform circular disc has centre $O$ and radius $1.2\,\text{m}$. Its centre is positioned at the origin of the $x$- and $y$-axes. Two circular holes, with centres at $A$ and $B$, are cut out of the disc (see diagram). Point $A$ lies on the negative $x$-axis, where $OA = 0.5\,\text{m}$. Point $B$ lies on the negative $y$-axis, where $OB = 0.7\,\text{m}$. The hole centred at $A$ has radius $0.3\,\text{m}$ and the hole centred at $B$ has radius $0.4\,\text{m}$. Find the distance of the centre of mass of the object from

Representation of data

Particle $P$ is fired with speed $V\,\text{m s}^{-1}$ at $60^{\circ}$ above the horizontal from point $O$. Exactly $1\,\text{s}$ later, particle $Q$ is launched from $O$ with the same initial speed at $45^{\circ}$ above the horizontal. The particles meet when $Q$ has been moving for $t\,\text{s}$.

Probability

A particle is projected at a speed of $25\,\text{m s}^{-1}$ at an angle of $50^\circ$ above the horizontal. Calculate the time after projection when the particle has speed $18\,\text{m s}^{-1}$ and is rising.

Probability

One end of a light inextensible string of length $0.5\,\text{m}$ is fastened to a fixed point $A$. A particle $P$ with mass $0.2\,\text{kg}$ is joined to the other end of the string. $P$ travels at constant speed in a horizontal circle with centre $O$, and $O$ is $0.4\,\text{m}$ vertically below $A$.

Representation of data

A particle $P$ is projected from a point $O$ on horizontal ground with speed $V\,\text{m s}^{-1}$ at an angle of $\theta^\circ$ above the horizontal. $4\,\text{s}$ after projection, the particle is at point $A$, where $OA = 40\,\text{m}$ and the line $OA$ is inclined at $30^\circ$ to the horizontal. Calculate $V$ and $\theta$.

Representation of data

A particle $P$, with mass $0.4\,\text{kg}$, travels at constant speed round a horizontal circle on the smooth inside of a fixed hollow hemisphere of radius $0.5\,\text{m}$ and centre $O$ (see diagram).

Probability

A particle $P$ with mass $0.5\,\text{kg}$ is launched vertically upwards from a point on a horizontal surface. A resisting force of magnitude $0.02v^2\,\text{N}$ acts on $P$, where $v\,\text{m s}^{-1}$ is the upward speed of $P$ when it is at a height of $x\,\text{m}$ above the surface. Its initial speed is $8\,\text{m s}^{-1}$.

Probability

An object is made by joining a hemispherical shell of radius $0.2\,\text{m}$ to a solid cone with base radius $0.2\,\text{m}$ and height $h\,\text{m}$, with the two parts meeting along their circumferences. The centre of mass, $G$, of the object lies $d\,\text{m}$ from the vertex of the cone on the object’s axis of symmetry. The object is in equilibrium on a horizontal plane, with the curved surface of the cone touching the plane (see diagram). The object is about to topple.

Representation of data

A particle $P$ with mass $M\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $0.8\,\text{m}$ and whose modulus of elasticity is $12.5\,\text{N}$. The other end of the string is fixed at point $A$. The particle is released from rest at $A$ and moves vertically downward until it is momentarily at rest at point $B$. The largest speed of $P$ during the descent is $4.4\,\text{m s}^{-1}$, and this occurs when the extension of the string is $e\,\text{m}$.

Probability

For a town of this kind, $76\%$ of cars have satellite navigation equipment installed. A random sample of $11$ cars is taken from the town.

The Poisson distribution

The random variable $X$ follows the distribution $N(\mu, \sigma^2)$. It is known that $P(X < 54.1) = 0.5$ and $P(X > 50.9) = 0.8665$.

Continuous random variables

Robert works part-time delivering newspapers. On several days, he recorded the time taken to complete his job, correct to the nearest minute. He used his results to complete the table below; the two missing entries are shown as $a$ and $b$.

Continuous random variables