Oct/Nov 2012

An arithmetic progression has first term 61 and second term 57. The sum of the first $n$ terms is $n$.

Series

The function $f$ is defined by $f(x) = 4x^2 - 24x + 11$, where $x \in \mathbb{R}$.

Functions

The diagram represents the curve $y = (6x + 2)^{\frac{1}{3}}$ and the point $A(1, 2)$, which is on the curve. At $A$, the tangent meets the $y$-axis at $B$, while the normal meets the $x$-axis at $C$.

Differentiation

The curve satisfies $\frac{dy}{dx} = -\frac{8}{x^3} - 1$, and the point $(2, 4)$ is on it.

Integration

Oil is escaping from a pipeline beneath the sea, forming a circular patch of oil on the sea surface. By midday, the patch has radius 50 m and the radius is growing at 3 metres per hour.

Differentiation

Find the first 3 terms when $(2x - x^2)^6$ is expanded in ascending powers of $x$.

Series

The equation of a curve is $y = 2x + \frac{1}{(x - 1)^2}$.

Differentiation

The diagram depicts the sector $OAB$ of a circle whose centre is $O$ and whose radius is $r$. The angle $AOB$ is $\theta$ radians. Point $C$ lies on $OA$ so that $BC$ is perpendicular to $OA$. Point $D$ is located on $BC$, and the circular arc $AD$ has centre $C$.

Circular measure

Solve $2\cos^2\theta = 3\sin\theta$ for $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The diagram depicts the curve $y^2 = 2x - 1$ together with the straight line $3y = 2x - 1$. These two graphs meet when $x = \frac{1}{2}$ and again when $x = a$, with $a$ being a constant.

Integration

Relative to an origin $O$, the position vectors of points $A$ and $B$ are $?

Coordinate geometry

In the expansion of $(x^2 - \frac{a}{x})^7$, the coefficient of $x^5$ comes out as $-280$.

Series

The curve is specified for $x > 0$ and satisfies $\frac{dy}{dx} = x + \frac{4}{x^2}$. The point $P(4, 8)$ lies on it.

Differentiation

The diagram represents a sector of a circle whose centre is $O$ and whose radius is $20\,\text{cm}$. Inside the sector is a circle with centre $C$ and radius $x\,\text{cm}$, touching the sector at $P$, $Q$ and $R$. Angle $POR = 1.2\,\text{radians}$.

Circular measure

For $x \geq -3$, the function $f$ satisfies $f(x) = \sqrt{\left(\frac{x+3}{2}\right)} + 1$.

Functions

A diagram gives a layout for a rectangular park $ABCD$, where $AB = 40\,\text{m}$ and $AD = 60\,\text{m}$. The points $X$ and $Y$ are located on $BC$ and $CD$ respectively, and the paths $AX$, $XY$ and $YA$ enclose a triangular playground. The distance $DY$ is $x\,\text{m}$, while $XC$ is $2x\,\text{m}$.

Quadratics

At the point $P$, the curve $4y = x^2$ has the line $y = \frac{x}{k} + k$ as a tangent, where $k$ is constant.

Coordinate geometry

The diagram depicts triangle $ABC$, with $A$ at $(1, 3)$, $B$ at $(5, 11)$, and angle $ABC$ equal to $90^\circ$. Point $X(4, 4)$ lies on $AC$.

Coordinate geometry

Show that the equation $2\cos x = 3\tan x$ can be rearranged into a quadratic equation in $\sin x$.

Trigonometry

The position vectors of points $A$ and $B$, measured from the origin $O$, are given by $\overrightarrow{OA} = \begin{pmatrix}1\\0\\2\end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix}k\\-k\\2k\end{pmatrix}$, where $k$ is a constant.

Coordinate geometry

In a geometric progression, every term is positive, the second term is $24$ and the fourth term is $13\frac{1}{2}$. Determine the first term.

Series

The diagram depicts part of the curve $y = \frac{9}{2x+3}$, which cuts the $y$-axis at $B(0, 3)$. On the curve, the point $A$ has coordinates $(3, 1)$, and the tangent drawn at $A$ meets the $y$-axis at $C$.

Integration

Determine the coefficient of $x^3$ in the expansion of $(2 - \tfrac{1}{2}x)^7$.

Series

The equation of a straight line is $y = -2x + k$, where $k$ is a constant, while the curve is given by $y = \frac{2}{x - 3}$.

Coordinate geometry

The curve in the diagram is given by $y = x(x - 2)^2$. Its minimum point is $(a, 0)$, and the maximum point has $x$-coordinate $b$, where $a$ and $b$ are constants.

Differentiation

It is stated that $f(x) = \frac{1}{x^3} - x^3$, for $x > 0$.

Differentiation

Solve $7\cos x + 5 = 2\sin^2 x$, for $0^\circ \leq x \leq 360^\circ$.

Trigonometry

In the diagram, D is on side AB of triangle ABC, and CD is an arc from a circle centred at A with radius 2\,\text{cm}. BC has length 2\sqrt{3}\,\text{cm} and meets AC at right angles.

Circular measure

The initial term of a geometric progression is $5\tfrac{1}{3}$, while its fourth term is $2\tfrac{1}{4}$.

Series

For $-\tfrac{1}{2}\pi \leq x \leq \tfrac{1}{2}\pi$, the functions $f$ and $g$ are given by $f(x) = \tfrac{1}{2}x + \tfrac{1}{6}\pi$ and $g(x) = \cos x$.

Functions

The diagram displays a section of the curve $y = 11 - x^2$ together with a segment of the straight line $y = 5 - x$, and they intersect at the point $A(p, q)$, where $p$ and $q$ are positive constants.

Functions

The curve is defined by $\frac{dy}{dx} = 2(3x + 4)^{\tfrac{3}{2}} - 6x - 8$.

Differentiation

Relative to origin $O$, the position vectors of points $A$ and $B$ are given by $\vec{OA} = \begin{pmatrix} p \\ 1 \\ 1 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 4 \\ 2 \\ p \end{pmatrix}$, with $p$ taken as a constant.

Coordinate geometry

Solve the inequality $|x - 2| \ge |x + 5|$.

Algebra

Use logarithms to solve the equation $5^x = 3^{2x-1}$, and give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

Solve the equation $2\cos 2\theta = 4\cos \theta - 3$, taking $0^\circ \le \theta \le 180^\circ$.

Trigonometry

The curve is defined parametrically by $x = \ln(1 - 2t)$ and $y = \frac{2}{t}$, where $t < 0$.

Differentiation

The diagram depicts the curve $y = \cos x$, for $0 \le x \le \frac{\pi}{2}$. A rectangle $OABC$ is shown, with $B$ lying on the curve and having $x$-coordinate $\theta$, while $A$ and $C$ lie on the axes, as illustrated. The shaded region $R$ is enclosed by the curve and by the lines $x = \theta$ and $y = 0$.

Numerical solution of equations

Estimate the value of $\int_0^1 \frac{1}{6 + 2e^x}\,dx$ by applying the trapezium rule with two intervals, and give your answer correct to $2$ decimal places.

Integration

Let $p(x)$ be the polynomial $2x^3 - 4x^2 + ax + b$, with $a$ and $b$ as constants. It is known that the remainder is $4$ when $p(x)$ is divided by $(x + 1)$, and that the remainder is $12$ when $p(x)$ is divided by $(x - 3)$.

Algebra

Differentiate $\frac{1}{\cos \theta}$ to show that, when $y = \sec \theta$, then $\frac{dy}{d\theta} = \tan \theta \sec \theta$.

Integration

Solve the inequality $|2x + 1| < |2x - 5|$ for $x$.

Algebra

The curve with equation $y = \frac{\sin 2x}{e^{2x}}$ has a single stationary point on the interval $0 \le x \le \frac{1}{2}\pi$.

Differentiation

$p(x)$ is the notation used for the polynomial $x^4 - 4x^3 + 3x^2 + 4x - 4$.

Algebra

The diagram illustrates the section of the curve $y = \sqrt{(2 - \sin x)}$ for $0 \le x \le \frac{1}{2}\pi$.

Numerical solution of equations

The variables $x$ and $y$ are linked by $y = A b^{-x}$, with $A$ and $b$ as constants. As illustrated, the graph of $\ln y$ against $x$ is a straight line that goes through the points $(1, 2.9)$ and $(3.5, 1.4)$.

Logarithmic and exponential functions

Calculate $\int 4e^{-\frac{1}{2}x} \, dx$.

Integration

The curve is defined by the equation $3x^2 - 4xy + 2y^2 - 6 = 0$.

Differentiation

If $\tan A = t$ and $\tan(A + B) = 4$, determine $\tan B$ in terms of $t$.

Trigonometry

Solve the inequality $|x - 2| \ge |x + 5|$.

Algebra

Use logarithms to find the solution of the equation $5^x = 3^{2x-1}$, giving your answer correct to $3$ significant figures.

Logarithmic and exponential functions

Solve the equation $2\cos 2\theta = 4\cos \theta - 3$ within $0^\circ \le \theta \le 180^\circ$.

Trigonometry

A curve is given by the parametric equations $x = \ln(1 - 2t)$ and $y = \frac{2}{t}$, with $t < 0$.

Differentiation

The sketch displays the curve $y = \cos x$, for $0 \le x \le \frac{1}{2}\pi$. A rectangle $OABC$ has been drawn, with $B$ located on the curve at the point whose $x$-coordinate is $\theta$, and $A$ and $C$ lying on the axes, as shown. The shaded region $R$ is enclosed by the curve and by the lines $x = \theta$ and $y = 0$.

Numerical solution of equations

Estimate $\int_0^1 \frac{1}{6 + 2e^x}\,dx$ by using the trapezium rule with two intervals, and state your answer correct to $2$ decimal places.

Integration

Let $p(x)$ represent the polynomial $2x^3 - 4x^2 + ax + b$, with $a$ and $b$ as constants. It is stated that the remainder is $4$ when $p(x)$ is divided by $(x + 1)$, and that the remainder is $12$ when $p(x)$ is divided by $(x - 3)$.

Algebra

Differentiating $\frac{1}{\cos \theta}$, demonstrate that when $y = \sec \theta$ then $\frac{dy}{d\theta} = \tan \theta \sec \theta$.

Differentiation

Determine the set of $x$ values that satisfy the inequality $3|x - 1| < |2x + 1|$.

Algebra

With origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\overrightarrow{OA} = \begin{pmatrix}3\\-2\\4\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}2\\-1\\7\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1\\-5\\-3\end{pmatrix}$. The plane $m$ is parallel to $OC$ and passes through $A$ and $B$.

Vectors

Solve $5^{x-1} = 5^x - 5$, and give $x$ correct to $3$ significant figures.

Logarithmic and exponential functions

Solve the equation $\sin(\theta + 45^\circ) = 2\cos(\theta - 30^\circ)$, and give every solution within $0^\circ < \theta < 180^\circ$.

Trigonometry

For $(1 + ax)^{-2}$, where $a$ is a positive constant, the expansion in ascending powers of $x$ has matching coefficients for $x$ and $x^3$.

Differentiation

Differentiate $\frac{1}{\cos x}$ to show that, when $y = \sec x$, $\frac{dy}{dx} = \sec x \tan x$.

Integration

The variables $x$ and $y$ satisfy the differential equation $x \frac{dy}{dx} = 1 - y^2$. Also, when $x = 2$, $y = 0$.

Differential equations

The curve is described by the equation $\ln(xy) - y^3 = 1$.

Differentiation

The graph depicts the curve $y = e^{-\frac{1}{2}x^2}\sqrt{(1 + 2x^2)}$ for $x \geq 0$, together with its highest point $M$.

Numerical solution of equations

Let the complex number $1 + (\sqrt{2})i$ be represented by $u$. Let the polynomial $x^4 + x^2 + 2x + 6$ be represented by $p(x)$.

Complex numbers

Determine the set of $x$ values that satisfy the inequality $3|x - 1| < |2x + 1|$.

Algebra

Taking the origin $O$ as the reference point, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}3 \\ -2 \\ 4\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}2 \\ -1 \\ 7\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1 \\ -5 \\ -3\end{pmatrix}$. The plane $m$ is parallel to $OC$ and passes through $A$ and $B$.

Vectors

Solve the equation $5^{x-1} = 5^x - 5$, and give the result correct to $3$ significant figures.

Logarithmic and exponential functions

Determine all solutions of the equation $\sin(\theta + 45^{\circ}) = 2\cos(\theta - 30^{\circ})$ in the interval $0^{\circ} < \theta < 180^{\circ}$.

Trigonometry

In the expansion of $(1 + ax)^{-2}$, where $a$ is a positive constant, written as a power series in increasing powers of $x$, the coefficients of $x$ and $x^3$ are the same.

Differentiation

By differentiating $\frac{1}{\cos x}$, demonstrate that when $y = \sec x$ then $\frac{dy}{dx} = \sec x \tan x$.

Integration

The variables $x$ and $y$ satisfy the differential equation $x\frac{dy}{dx} = 1 - y^2$. When $x = 2$, $y = 0$.

Differential equations

The curve is given by the equation $\ln(xy) - y^3 = 1$.

Differentiation

The diagram depicts the curve $y = e^{-\frac{1}{2}x^2}\sqrt{(1 + 2x^2)}$ for $x \ge 0$, together with its maximum point $M$.

Numerical solution of equations

Use $u$ to denote the complex number $1 + (\sqrt{2})i$. Write $x^4 + x^2 + 2x + 6$ as $p(x)$.

Complex numbers

Solve this equation

Logarithmic and exponential functions

Solve the equation $iw^2=(2-2i)^2$ without using a calculator.

Complex numbers

Write $24\sin\theta-7\cos\theta$ in the form $R\sin(\theta-\alpha)$, with $R>0$ and $0^\circ<\alpha<90^\circ$. State the value of $\alpha$ correct to 2 decimal places.

Trigonometry

The parametric equations describing a curve are $x=\frac{4t}{2t+3}$, $y=2\ln(2t+3)$.

Differentiation

The variables $x$ and $y$ are linked through the differential equation $(x^2+4)\frac{dy}{dx}=6xy$. It is given that $y=32$ when $x=0$.

Algebra

The function is given by $f(x)=3xe^{-2x}$.

Differentiation

The diagram depicts the curve $y=x^4+2x^3+2x^2-4x-16$, which meets the $x$-axis at the points $(\alpha,0)$ and $(\beta,0)$ with $\alpha<\beta$. It is stated that $\alpha$ is an integer.

Numerical solution of equations

The diagram presents a section of the curve $y=\sin^3 2x\cos^3 2x$. The shaded part shown is enclosed by the curve and the $x$-axis, and its exact area is written as $A$.

Integration

The equations of the two lines are $\mathbf{r}=\begin{pmatrix}5\\1\\-4\end{pmatrix}+s\begin{pmatrix}1\\-1\\3\end{pmatrix}$ and $\mathbf{r}=\begin{pmatrix}p\\4\\-2\end{pmatrix}+t\begin{pmatrix}2\\5\\-4\end{pmatrix}$, with $p$ being a constant. It is stated that the lines intersect.

Vectors

Express $\frac{9-7x+8x^2}{(3-x)(1+x^2)}$ as partial fractions.

Algebra

An object is let go from rest from a height of $125\text{ m}$ above level ground and descends freely under gravity, where it strikes a moving target $P$. The target $P$ travels along the ground in a straight line with constant acceleration $0.8\text{ m s}^{-2}$. At the moment the object is released, $P$ goes through a point $O$ with speed $5\text{ m s}^{-1}$.

Kinematics of motion in a straight line

Two particles, $A$ and $B$, with masses $0.3\text{ kg}$ and $0.2\text{ kg}$ respectively, are fastened to the two ends of a light inextensible string. $A$ is kept at rest on a rough horizontal table, and the string passes over a small smooth pulley at the table edge. $B$ hangs vertically beneath the pulley. The system is released and $B$ begins to move downwards with acceleration $1.6\text{ m s}^{-2}$.

Newton's laws of motion

A particle $P$ with mass $0.5\text{ kg}$ is at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = 0.28$. A force of magnitude $0.6\text{ N}$, applied upwards to $P$ at angle $\alpha$ to a line of greatest slope of the plane, is just enough to stop $P$ from sliding down the plane.

Forces and equilibrium

At one point, three coplanar forces with magnitudes $8\text{ N}$, $12\text{ N}$ and $2\text{ N}$ act. Their resultant has magnitude $R\text{ N}$. The diagram shows the directions of the three forces and the resultant.

Forces and equilibrium

Particle $P$ moves on a straight line from $A$ to $B$ with constant acceleration $0.05\text{ m s}^{-2}$. Its speed at $A$ is $2\text{ m s}^{-1}$, and its speed at $B$ is $5\text{ m s}^{-1}$. Particle $Q$ also moves on the same straight line from $A$ to $B$, beginning from rest at $A$. After $t\text{ s}$ from leaving $A$, the speed of $Q$ is $kt^3\text{ m s}^{-1}$, where $k$ is a constant. $Q$ takes the same amount of time to go from $A$ to $B$ as $P$.

Kinematics of motion in a straight line

The diagram represents the vertical cross-section $ABCD$ of a surface. $BC$ is a circular arc, and $AB$ and $CD$ are tangents to $BC$ at $B$ and $C$ respectively. $A$ and $D$ are at the same horizontal level, while $B$ and $C$ lie $2.7\text{ m}$ and $3.0\text{ m}$ above the level of $A$ and $D$ respectively. A particle $P$ of mass $0.2\text{ kg}$ is projected from $A$ with speed $8\text{ m s}^{-1}$ in the direction of $AB$. The parts of the surface containing $AB$ and $BC$ are smooth. The section containing $CD$ applies a constant frictional force to $P$ as it travels from $C$ to $D$, and $P$ is at rest when it reaches $D$.

Energy, work and power

A car with mass $1200\text{ kg}$ travels in a straight line over level ground. The resistive force on the car is constant, with magnitude $960\text{ N}$. The engine of the car operates at a rate of $17\,280\text{ W}$. The car goes through points $A$ and $B$. During the journey from $A$ to $B$, its speed is constant at $V\text{ m s}^{-1}$. The moment the car arrives at $B$, the engine is turned off and from then on supplies no energy. The car keeps moving along the same straight line until it stops at $C$. The time for the car to go from $A$ to $C$ is $52.5\text{ s}$.

Kinematics of motion in a straight line

A block is moved across a horizontal floor by a force of magnitude $45\,\text{N}$ applied at an angle of $14^\circ$ to the horizontal (as shown in the diagram).

Energy, work and power

Masses $A$ and $B$, with masses $m\,\text{kg}$ and $(1 - m)\,\text{kg}$ respectively, are joined to the two ends of a light inextensible string that runs over a fixed smooth pulley. The system is let go from rest, with the straight sections of the string initially vertical. $A$ moves vertically downwards, and $0.3\,\text{s}$ afterwards its speed is $0.6\,\text{m s}^{-1}$.

Newton's laws of motion

A car moves on a straight road with constant acceleration $a\,\text{m s}^{-2}$. It goes past points $A$, $B$ and $C$; the journey time from $A$ to $B$ is $5\,\text{s}$, and the journey time from $B$ to $C$ is also $5\,\text{s}$. The speed of the car at $A$ is $u\,\text{m s}^{-1}$, while the distances $AB$ and $BC$ are $55\,\text{m}$ and $65\,\text{m}$ respectively.

Kinematics of motion in a straight line

At origin $O$, three coplanar forces with magnitudes $68\,\text{N}$, $75\,\text{N}$ and $100\,\text{N}$ act, as the diagram shows. Their components along the positive $x$-direction are $-60\,\text{N}$, $0\,\text{N}$ and $96\,\text{N}$, respectively.

Forces and equilibrium

$A$, $B$ and $C$ are three points along the line of greatest slope on a plane inclined at $\theta^\circ$ to the horizontal, with $A$ above $B$ and $B$ above $C$. The section between $A$ and $B$ is smooth, while the section between $B$ and $C$ is rough. A particle $P$ is released from rest at $A$ and slides down the line $ABC$. $0.8\,\text{s}$ after leaving $A$, the particle passes through $B$ with speed $4\,\text{m s}^{-1}$.

Newton's laws of motion

A car with mass $1250\,\text{kg}$ travels from the foot to the summit of a straight hill that is $500\,\text{m}$ long. The top is $30\,\text{m}$ above the level of the bottom. The car’s engine has a constant power of $30\,000\,\text{W}$. At the bottom of the hill, the car’s acceleration is $4\,\text{m s}^{-2}$, and at the top it is $0.2\,\text{m s}^{-2}$. The resistance opposing the car’s motion is $1000\,\text{N}$.

Energy, work and power

A particle $P$ leaves point $O$ and then moves along a straight line. At time $t\,\text{s}$ after leaving $O$, the velocity of $P$ is $k(60t^2 - t^3)\,\text{m s}^{-1}$, where $k$ is a constant. The maximum velocity of $P$ is $6.4\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

ABCD is a semi-circular cross-section, in a vertical plane, of the inner surface of half a hollow cylinder of radius $2.5\,\text{m}$, fixed so that its axis is horizontal. AD lies horizontally, B is the lowest point of the cross-section, and C is $1.8\,\text{m}$ above the level of B (see diagram). A particle $P$ of mass $0.8\,\text{kg}$ is released from rest at A and reaches instantaneous rest at C.

Energy, work and power

A particle travels along a straight line. Its velocity $t$ seconds after it leaves a fixed point $O$ on the line is $v\,\text{m s}^{-1}$, where $v = 0.2t + 0.006t^{2}$.

Kinematics of motion in a straight line

Particle $P$ is launched vertically upwards from point $O$ with speed $8\,\text{m s}^{-1}$. Point A is the highest point attained by $P$.

Kinematics of motion in a straight line

A particle $P$ with weight $21\,\text{N}$ is connected to one end of each of two light inextensible strings, $S_1$ and $S_2$, whose lengths are $0.52\,\text{m}$ and $0.25\,\text{m}$ respectively. The opposite end of $S_1$ is fixed at point A, and the opposite end of $S_2$ is fixed at point B, which lies at the same horizontal level as A. Particle P is in equilibrium, hanging $0.2\,\text{m}$ below the level of $AB$, with both strings taut (see diagram).

Forces and equilibrium

A body of mass $12\,\text{kg}$ moves down the line of greatest slope on a smooth plane that is inclined at $10^{\circ}$ to the horizontal. It goes through points A and B with speeds $3\,\text{m s}^{-1}$ and $7\,\text{m s}^{-1}$ respectively.

Energy, work and power

The diagram depicts a particle with mass $0.6\,\text{kg}$ resting on a plane that is inclined at $25^{\circ}$ to the horizontal. A force of magnitude $P\,\text{N}$ acts on the particle up the plane, parallel to a line of greatest slope. The coefficient of friction between the particle and the plane is $0.36$. Assuming that the particle is in equilibrium,

Forces and equilibrium

Particles A and B have masses $0.32\,\text{kg}$ and $0.48\,\text{kg}$ respectively. They are joined by a light inextensible string that passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle B is initially at rest on the table, $1.4\,\text{m}$ from the pulley. A is hanging vertically beneath the pulley, with its height $0.98\,\text{m}$ above the floor (see diagram). A, B, the string and the pulley all lie in the same vertical plane. B is released and A moves downwards.

Newton's laws of motion

ABC is a uniform semicircular arc whose diameter is $AC = 0.5\,\text{m}$. It spins about a fixed axis through $A$ and $C$ at an angular speed of $2.4\,\text{rad s}^{-1}$.

Representation of data

A uniform rod $AB$ weighs $6\,\text{N}$ and measures $0.8\,\text{m}$ in length. It is in limiting equilibrium with $B$ touching a rough horizontal plane, and $AB$ is at an angle of $60^\circ$ to the horizontal. A force keeps the rod in equilibrium in the vertical plane containing $AB$; it acts at $A$ and makes an angle of $45^\circ$ with $AB$ (see diagram).

Probability

A particle $P$ with mass $0.2\,\text{kg}$ is released from rest and then moves vertically downwards. After $t\,\text{s}$ from release, $P$ has speed $v\,\text{m s}^{-1}$. A resisting force of magnitude $0.8v\,\text{N}$ acts on $P$.

Probability

Particle $P$ is travelling inside a smooth hollow cone with its vertex pointing downward and its axis vertical, and the semi-vertical angle is $45^\circ$. A light inextensible string, parallel to the cone’s surface, links $P$ to the vertex. $P$ moves at constant angular speed in a horizontal circle of radius $0.67\,\text{m}$ (see diagram). The string tension is equal to the weight of $P$.

Representation of data

A particle $P$ is launched from a point $O$ on horizontal ground with speed $30\,\text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. At the instant when the speed of $P$ is $17\,\text{m s}^{-1}$ and increasing,

Representation of data

A uniform lamina $OABCD$ is made up of a semicircle $BCD$ with centre $O$ and radius $0.6\,\text{m}$, together with an isosceles triangle $OAB$ joined along $OB$ (see diagram). The triangle has area $0.36\,\text{m}^2$ and $AB = AO$.

Representation of data

A light elastic string has a natural length of $3\,\text{m}$ and a modulus of elasticity $45\,\text{N}$. A particle $P$ with weight $6\,\text{N}$ is fixed to the midpoint of the string. The two ends of the string are fastened to fixed points $A$ and $B$ on the same vertical line, with $A$ above $B$ and $AB = 4\,\text{m}$. The particle $P$ is let go from rest at a position $1.5\,\text{m}$ vertically beneath $A$.

Probability

ABC is a uniform semicircular arc whose diameter is $AC = 0.5\,\text{m}$. It turns about a fixed axis passing through A and C at angular speed $2.4\,\text{rad s}^{-1}$.

Representation of data

A uniform rod $AB$ weighs $6\,\text{N}$ and measures $0.8\,\text{m}$ in length. It is in limiting equilibrium with $B$ touching a rough horizontal plane, and $AB$ is set at $60^\circ$ to the horizontal. Equilibrium is held by a force, within the vertical plane containing $AB$, applied at $A$ and making an angle of $45^\circ$ to $AB$ (see diagram).

Probability

A particle $P$ with mass $0.2\,\text{kg}$ is let go from rest and moves vertically downward. At time $t\,\text{s}$ after release, its speed is $v\,\text{m s}^{-1}$. A resisting force of magnitude $0.8v\,\text{N}$ acts on $P$.

Probability

Particle $P$ travels inside a smooth hollow cone with its vertex pointing downward, its axis vertical, and a semi-vertical angle of $45^\circ$. A light inextensible string, parallel to the cone’s surface, links $P$ to the vertex. $P$ moves at constant angular speed in a horizontal circle of radius $0.67\,\text{m}$ (see diagram). The tension in the string is equal to the weight of $P$.

Probability

A particle $P$ is launched with speed $30\,\text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal from a point $O$ on level ground. At the instant when the speed of $P$ is $17\,\text{m s}^{-1}$ and increasing,

Representation of data