Oct/Nov 2010

Find the value of $\int \left(x + \frac{1}{x}\right)^2 \, dx$.

Integration

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

Differentiation

A curve is given by $y = \frac{9}{2 - x}$.

Integration

For the expansion of $(1 + ax)^6$, with $a$ a constant, the coefficient of $x$ is $-30$.

Series

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

Functions

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

Trigonometry

The diagram depicts a pyramid $OABC$ with a flat base $OAB$ such that $OA = 6\,\text{cm}$, $OB = 8\,\text{cm}$ and $\angle AOB = 90^{\circ}$. Point $C$ lies vertically above $O$ and $OC = 10\,\text{cm}$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OC$ respectively, as illustrated.

Coordinate geometry

The fifth term of an arithmetic progression is $18$ and the total of the first $5$ terms is $75$. Work out the first term together with the common difference.

Series

The function $f$ is specified by $f : x \mapsto 3 - 2\tan\left(\frac{1}{2}x\right)$ for $0 \leq x < \pi$.

Functions

The diagram depicts a metal plate made up of a rectangle with side lengths $x\,\text{cm}$ and $y\,\text{cm}$ together with a quarter-circle of radius $x\,\text{cm}$. Its perimeter measures $60\,\text{cm}$. Since $x$ may vary,

Differentiation

The diagram contains two circles, $C_1$ and $C_2$, which are touching at point $T$. $C_1$ has centre $P$ and radius $8\,\text{cm}$, while $C_2$ has centre $Q$ and radius $2\,\text{cm}$. The points $R$ and $S$ lie on $C_1$ and $C_2$ respectively, and $RS$ is tangent to both circles.

Circular measure

Find the first 3 terms of $(1 - 2x^2)^8$, arranged in ascending powers of $x$.

Series

The sketch depicts a rectangular tank of height $h$ metres with a lid fitted on top. Its base measures $x$ metres by $\frac{1}{2}x$ metres, and the lid is a rectangle with side lengths $\frac{5}{4}x$ metres and $\frac{4}{5}x$ metres. When the tank is completely filled, it contains $4\ \text{m}^3$ of water. The tank is made from material of negligible thickness. The total external surface area of the tank together with the area of the top of the lid is $A\ \text{m}^2$.

Differentiation

The diagram illustrates part of the curve $y = \frac{1}{(3x + 1)^{1/4}}$. The curve intersects the $y$-axis at $A$ and the line $x = 5$ at $B$.

Integration

Prove that $\tan^2 x - \sin^2 x = \tan^2 x\sin^2 x$ holds.

Trigonometry

For a Green Anaconda snake aged $t$ years, its length, $x$ metres, is approximated by the formula $x = 0.7\sqrt{(2t - 1)}$, where $1 \leq t \leq 10$. Using this formula, determine

Differentiation

The diagram depicts points $A$, $C$, $B$, $P$ on the circumference of a circle whose centre is $O$ and whose radius is $3\ \text{cm}$. Angle $AOC=2.3$ radians.

Circular measure

The first two terms of an arithmetic progression are $161$ and $154$ respectively. The sum of the first $m$ terms is zero. Determine the value of $m$.

Series

The curve is defined by $y = kx^2 + 1$ and the line is defined by $y = kx$, where $k$ is a non-zero constant.

Quadratics

The function $f$ is given by $f(x) = x^2 - 4x + 7$ when $x > 2$.

Functions

The diagram depicts a section of the curve $y = \frac{2}{1-x}$ together with the line $y = 3x + 4$. These intersect at $A$ and $B$.

Coordinate geometry

The diagram represents a pyramid $OABCP$. Its square horizontal base $OABC$ has side $10\ \text{cm}$, and the vertex $P$ is positioned $10\ \text{cm}$ vertically above $O$. The points $D$, $E$, $F$, $G$ are on $OP$, $AP$, $BP$, $CP$ respectively, and $DEFG$ is a horizontal square with side $6\ \text{cm}$. The perpendicular distance of $DEFG$ above the base is $a\ \text{cm}$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively.

Coordinate geometry

Find the term in the expansion of $(x - \frac{1}{x^2})^9$ that is independent of $x$.

Series

The figure depicts triangle $OAB$, where the position vectors of $A$ and $B$ relative to $O$ are $\vec{OA} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$ and $\vec{OB} = -3\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$. $C$ lies on $OA$ such that $\vec{OC} = p\vec{OA}$, with $p$ a constant.

Coordinate geometry

The diagram displays sections of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ together with their intersection points $P$ and $Q$. The $x$-coordinates of $P$ and $Q$ are $a$ and $b$ respectively.

Integration

The coordinates of points $A$, $B$ and $C$ are $(2,5)$, $(5,-1)$ and $(8,6)$, respectively.

Coordinate geometry

Solve for $x$ in the equation $15\sin^2 x = 13 + \cos x$ over $0^{\circ} \leq x \leq 180^{\circ}$.

Trigonometry

Sketch the graph of $y = 2\sin x$ for $0 \leq x \leq 2\pi$.

Trigonometry

The curve is defined by $y = \frac{1}{x - 3} + x$.

Differentiation

A curve is given by $y = f(x)$. It is stated that $f'(x) = 3x^2 + 2x - 5$.

Differentiation

The diagram represents the function $f$ on $0 \leq x \leq 6$, with $x \mapsto \frac{1}{2}x^2$ for $0 \leq x \leq 2$ and $x \mapsto \frac{1}{2}x + 1$ for $2 < x \leq 6$.

Functions

The diagram represents rhombus $ABCD$. Points $P$ and $Q$ are located on diagonal $AC$ so that $BPD$ is an arc of a circle with centre $C$ and $BQD$ is an arc of a circle with centre $A$. Each side of the rhombus measures $5\text{ cm}$ and angle $BAD = 1.2$ radians.

Integration

A geometric progression starts with term $100$ and has sum to infinity $2000$. Find the second term.

Series

Solve $|x + 1| > |x - 4|$.

Algebra

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

Logarithmic and exponential functions

Show that the value of $\int_0^1 (e^x + 1)^2 \, dx$ is $\frac{1}{2}e^2 + 2e - \frac{3}{2}$.

Integration

The curve is described parametrically by $x = 1 + \ln(t - 2)$ and $y = t + \frac{9}{t}$, with $t > 2$.

Differentiation

Solve the equation $8 + \cot \theta = 2\cosec^2 \theta$, giving every solution in the interval $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The curve with equation $y = \frac{6}{x^2}$ meets the line $y = x + 1$ at the point $P$.

Numerical solution of equations

The polynomial $3x^3 + 2x^2 + ax + b$, in which $a$ and $b$ are constants, is written as $p(x)$. It is stated that $(x - 1)$ divides $p(x)$ exactly, and that the remainder when $p(x)$ is divided by $(x - 2)$ is $10$.

Algebra

The diagram depicts the curve $y = x \sin x$, for $0 \leq x \leq \pi$. The point $Q\left(\frac{1}{2}\pi, \frac{1}{2}\pi\right)$ is on the curve.

Integration

Solve for $x$ in the inequality $|x + 1| > |x - 4|$.

Algebra

Use logarithms to determine the value of $x$ for $5^x = 2^{2x+1}$, with the answer given correct to $3$ significant figures.

Logarithmic and exponential functions

Show that $\displaystyle \int_0^1 (e^x + 1)^2\,dx$ equals $\frac{1}{2}e^2 + 2e - \frac{3}{2}$.

Integration

The curve is specified parametrically by $x = 1 + \ln(t - 2)$ and $y = t + \frac{9}{t}$, with $t > 2$.

Differentiation

Solve the equation $8 + \cot\theta = 2\csc^2\theta$, and give every solution in the interval $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

The curve given by $y = \frac{6}{x^2}$ meets the line $y = x + 1$ at the point $P$.

Numerical solution of equations

Let $p(x)$ represent the polynomial $3x^3 + 2x^2 + ax + b$, with $a$ and $b$ as constants. It is stated that $(x - 1)$ is a factor of $p(x)$, and that the remainder on dividing $p(x)$ by $(x - 2)$ is $10$.

Algebra

The graph depicts the curve $y = x \sin x$, for $0 \leq x \leq \pi$. The point $Q\left(\frac{\pi}{2}, \frac{\pi}{2}\right)$ is on the curve.

Integration

Solve for $x$ in the inequality $|3x + 1| > 8$.

Algebra

The values produced by the iterative relation $x_{n+1} = \frac{7x_n}{8} + \frac{5}{2x_n^4}$, starting from $x_1 = 1.7$, approach $\alpha$.

Numerical solution of equations

The polynomial $x^3 + 4x^2 + ax + 2$, with $a$ a constant, is written as $p(x)$. You are told that the remainder on dividing $p(x)$ by $(x + 1)$ is the same as the remainder on dividing $p(x)$ by $(x - 2)$.

Algebra

Find $\int e^{1-2x}\,dx$.

Integration

Let $x$ and $y$ be linked by $y = A(b^x)$, with $A$ and $b$ as constants. The plot of $\ln y$ against $x$ is a straight line that passes through the points $(1.4, 0.8)$ and $(2.2, 1.2)$, as the diagram shows.

Logarithmic and exponential functions

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

Trigonometry

The sketch shows the curve $y = \frac{\ln x}{x^2}$ together with its highest point $M$.

Integration

The curve is defined by the equation $x^2 + 2xy - y^2 + 8 = 0$.

Differentiation

Solve $2|x - 3| > |3x + 1|.$

Algebra

A chemical reaction produces a certain substance. Let the mass of substance produced be $x$ grams after $t$ seconds from the start of the reaction. At every moment, the formation rate of the substance is proportional to $(20 - x)$. When $t = 0$, $x = 0$ and $\frac{dx}{dt} = 1.$

Differential equations

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

Logarithmic and exponential functions

Solve the equation $\cos(\theta + 60^{\circ}) = 2\sin\theta$, and list every solution in the interval $0^{\circ} \leq \theta \leq 360^{\circ}.$

Trigonometry

By sketching suitable graphs, show that the equation $4x^2 - 1 = \cot x$ has just one root in the interval $0 < x < \frac{1}{2}\pi$.

Numerical solution of equations

Take $I = \int_{0}^{1} \frac{x^2}{\sqrt{(4 - x^2)}} \, dx.$

Integration

The complex number $z$ is defined by $z = (\sqrt{3}) + i.$

Complex numbers

Relative to the origin $O$, the points $A$ and $B$ have position vectors $overrightarrow{OA} = i + 2j + 2k$ and $overrightarrow{OB} = 3i + 4j.$ Point $P$ is on the line $AB$, and $OP$ is at right angles to $AB.$

Vectors

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

Algebra

The graph depicts the curve $y = x^3 \ln x$ together with its minimum point $M.$

Integration

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

Algebra

A certain substance is produced during a chemical reaction. Let the mass of substance produced after $t$ seconds from the start of the reaction be $x$ grams. At every instant, the rate at which the substance is formed is proportional to $(20 - x)$. When $t = 0$, $x = 0$ and $\frac{dx}{dt} = 1$.

Differential equations

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

Logarithmic and exponential functions

Find all solutions of $\cos(\theta + 60^\circ) = 2\sin \theta$ in the interval $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

By sketching appropriate graphs, show that the equation $4x^2 - 1 = \cot x$ has a single root in the interval $0 < x < \tfrac{1}{2}\pi$.

Numerical solution of equations

Define $I = \int_0^1 \frac{x^2}{\sqrt{4 - x^2}}\,dx$.

Integration

The complex number $z$ is defined as $z = \sqrt{3} + i$.

Complex numbers

Relative to the origin $O$, the position vectors of $A$ and $B$ are $ \overrightarrow{OA} = i + 2j + 2k$ and $\overrightarrow{OB} = 3i + 4j$. Point $P$ is located on the line $AB$, and $OP$ is perpendicular to $AB$.

Vectors

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

Algebra

The diagram depicts the curve $y = x^3 \ln x$ together with its minimum point $M$.

Integration

Expand $(1 + 2x)^{-3}$ as a series in ascending powers of $x$, including terms up to and including the one in $x^2$, and simplify the coefficients.

Algebra

The polynomial $p(z)$ is given by $p(z) = z^3 + mz^2 + 24z + 32$, with $m$ a constant. You are told that $(z + 2)$ divides $p(z)$.

Complex numbers

The parametric equations describing a curve are $x = \frac{t}{2t + 3}$ and $y = e^{-2t}$.

Differentiation

The complex number $w$ is given by $w = 2 + i$.

Complex numbers

Given that $f(x) = 4\cos^2 3x$.

Differentiation

Show that $\int_0^7 \frac{2x + 7}{(2x + 1)(x + 2)}\,dx = \ln 50$.

Integration

The line $l$ passes through the points $(-5, 3, 6)$ and $(5, 8, 1)$. The plane $p$ is given by $2x - y + 4z = 9$.

Vectors

With $\int_1^a \frac{\ln x}{x^2}\,dx = \frac{2}{5}$, show that $a = \frac{5}{3}(1 + \ln a)$.

Numerical solution of equations

Rewrite $(\sqrt{6})\cos \theta + (\sqrt{10})\sin \theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. State the value of $\alpha$ correct to 2 decimal places.

Trigonometry

A biologist is studying how a weed spreads across a particular region. At time $t$ weeks after the investigation begins, the area covered by the weed is $A \, \text{m}^2$. The biologist says that the rate at which $A$ increases is proportional to $\sqrt{(2A - 5)}$.

Differential equations

Particles $P$ and $Q$ travel vertically while gravity acts on them. The graphs give the upward velocity $v\,\text{m s}^{-1}$ against time $t$ s for $0 \leq t \leq 4$. $P$ is projected with velocity $V\,\text{m s}^{-1}$, whereas $Q$ begins from rest.

Kinematics of motion in a straight line

A car with mass $600\,\text{kg}$ is moving on a level straight road, and its engine is delivering power at a rate of $40\,\text{kW}$. The resistive force on the car is steady and has magnitude $800\,\text{N}$. At the point $A$ on the road, the car moves with speed $25\,\text{m s}^{-1}$. The acceleration of the car at point $B$ is one-half of its acceleration at $A$.

Kinematics of motion in a straight line

The diagram depicts three particles $A$, $B$ and $C$ suspended in equilibrium, with each one attached to the end of a string. The other ends of the three strings are fastened together at the point $X$. The strings supporting $A$ and $C$ pass over smooth fixed horizontal pegs $P_1$ and $P_2$ respectively. The weights of $A$, $B$ and $C$ are $5.5\,\text{N}$, $7.3\,\text{N}$ and $W\,\text{N}$ respectively, and the angle $P_1XP_2$ is a right angle.

Forces and equilibrium

A particle $P$ leaves a fixed point $O$ at $t = 0$, where $t$ is measured in seconds, and travels in a straight line with constant acceleration. Its initial velocity is $1.5\,\text{m s}^{-1}$ and its velocity at $t = 10$ is $3.5\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A particle with mass $0.8\,\text{kg}$ travels down a rough inclined plane in the direction of steepest descent along $AB$. The length of $AB$ is $8\,\text{m}$. The particle leaves $A$ with speed $3\,\text{m s}^{-1}$ and has constant acceleration $2.5\,\text{m s}^{-2}$.

Energy, work and power

A smooth slide $AB$ is positioned so that its top end $A$ lies $3\,\text{m}$ above level ground. Point $B$ is $h\,\text{m}$ above the ground. A particle $P$ of mass $0.2\,\text{kg}$ is let go from rest at a point on the slide. It travels down the slide and, after going past $B$, keeps moving until it reaches the ground. The speed of $P$ at $B$ is $v_B$ and the speed at which $P$ hits the ground is $v_G$.

Energy, work and power

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are joined by a light inextensible string. This string goes over a smooth pulley at the edge of a rough horizontal table. $P$ hangs freely, while $Q$ rests on the table. A force of magnitude $3.2\,\text{N}$ is applied to $Q$, directed upwards and away from the pulley, making an angle of $30^\circ$ to the horizontal.

Forces and equilibrium

A block with mass $400\,\text{kg}$ is in limiting equilibrium on level ground. A force of magnitude $2000\,\text{N}$ is applied to the block at an angle of $15^\circ$ to the upwards vertical.

Forces and equilibrium

A cyclist is moving at a constant power of $400\,\text{W}$ along a straight road inclined at $2^\circ$ to the horizontal. The combined mass of the cyclist and cycle is $80\,\text{kg}$. Neglecting all resistance to motion, find, correct to $1$ decimal place, the cyclist’s acceleration when he is travelling

Energy, work and power

Particle $P$ is in equilibrium on a smooth horizontal table, acted on by four horizontal forces of magnitudes $6\,\text{N}$, $5\,\text{N}$, $F\,\text{N}$ and $F\,\text{N}$ in the directions indicated.

Forces and equilibrium

A block of mass $20\,\text{kg}$ is drawn from the base to the top of a slope. The slope is $10\,\text{m}$ long and makes an angle of $4.5^\circ$ to the horizontal. The block has speed $2.5\,\text{m}\,\text{s}^{-1}$ at the bottom of the slope and $1.5\,\text{m}\,\text{s}^{-1}$ at the top of the slope.

Energy, work and power

Particles $P$ and $Q$ are projected vertically upwards from two separate points on level ground, with speeds of $20\,\text{m}\,\text{s}^{-1}$ and $25\,\text{m}\,\text{s}^{-1}$ respectively. $Q$ is launched $0.4\,\text{s}$ after $P$.

Kinematics of motion in a straight line

The diagram presents the velocity-time graph of a particle $P$ moving along the straight line $AB$, with $v\,\text{m}\,\text{s}^{-1}$ representing the velocity of $P$ at time $t\,\text{s}$. The graph is made up of five straight-line sections. At $t = 0$, the particle is at rest at a point $X$ on the line segment joining $A$ and $B$, and it then travels towards $A$. It is again at rest at $A$ when $t = 2.5$.

Kinematics of motion in a straight line

A particle $P$ moves along a straight line. At time $t = 0$, where $t$ is measured in seconds, it passes through point $O$ on the line with velocity $5\,\text{m}\,\text{s}^{-1}$. Once $P$ has left $O$, its velocity is given by $(0.002t^{3} - 0.12t^{2} + 1.8t + 5)\,\text{m}\,\text{s}^{-1}$. The velocity of $P$ is increasing for $0 < t < T_1$ and for $t > T_2$, and it is decreasing for $T_1 < t < T_2$.

Kinematics of motion in a straight line

A particle $P$ starts from rest at a point on a smooth plane that is inclined at $30^\circ$ to the horizontal. Find the speed of $P$.

Kinematics of motion in a straight line

The sketch represents the vertical cross-section $ABC$ of a fixed surface. $AB$ is curved, while $BC$ is a horizontal straight segment. The section of the surface containing $AB$ is smooth, whereas the part containing $BC$ is rough. $A$ is $1.8\,\text{m}$ above $BC$. A particle of mass $0.5\,\text{kg}$ is released from rest at $A$ and moves along the surface to $C$.

Energy, work and power

A tiny smooth pulley is fastened at the highest point $A$ of a triangular-prism cross-section $ABC$. The angles are $ABC = 90^\circ$ and $BCA = 30^\circ$. The prism is set up with the face containing $BC$ resting on a horizontal surface. Particles $P$ and $Q$ are joined to the two ends of a light inextensible string, which runs over the pulley. The particles are in equilibrium, with $P$ hanging vertically beneath the pulley and $Q$ touching $AC$ (see diagram). The resultant force exerted on the pulley by the string is $3\sqrt{3}\,\text{N}$.

Forces and equilibrium

A particle begins from rest at point $X$ and travels in a straight line. After $60$ seconds, it arrives at point $Y$. For time $t$ seconds after leaving $X$, the acceleration of the particle is $0.75\,\text{m s}^{-2}$ for $0 \le t \le 4$, $0\,\text{m s}^{-2}$ for $4 \le t \le 54$, and $-0.5\,\text{m s}^{-2}$ for $54 \le t \le 60$.

Kinematics of motion in a straight line

A force with magnitude $F\,\text{N}$ lies in a horizontal plane and has components $27.5\,\text{N}$ and $-24\,\text{N}$ along the $x$-axis and the $y$-axis respectively. The force is directed at an angle of $\alpha^\circ$ below the $x$-axis.

Forces and equilibrium

A particle moves along a straight line. It is initially at rest at a point $A$ on the line and, $10$ seconds later, it is at rest again at a different point $B$ on the same line. The velocity $t$ seconds after leaving $A$ is $0.72t^2 - 0.096t^3$ for $0 \leq t \leq 5$, and $2.4t - 0.24t^2$ for $5 \leq t \leq 10$.

Kinematics of motion in a straight line

A car with mass $1250\,\text{kg}$ is moving on a level straight road. Its engine delivers constant power of $24\,\text{kW}$, and the resistive force on the car is a constant $R\,\text{N}$. As the car goes through point $A$ on the road, its speed is $20\,\text{m s}^{-1}$ and its acceleration is $0.32\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A horizontal circular disc turns at a steady angular speed of $9\ \text{rad s}^{-1}$ around its centre $O$. A particle with mass $0.05\ \text{kg}$ is positioned on the disc $0.4\ \text{m}$ from $O$. The particle travels together with the disc, with no slipping.

Probability

A bow is formed from a uniform curved segment $AB$ of mass $1.4\ \text{kg}$, together with a uniform taut string of mass $m\ \text{kg}$ that connects $A$ and $B$. The curved segment $AB$ is an arc of a circle with centre $O$ and radius $0.8\ \text{m}$. The angle $AOB$ is $\frac{2}{3}\pi$ radians (see diagram). The centre of mass of the bow, including the string, is $0.65\ \text{m}$ from $O$.

Representation of data

One end of a light inextensible string of length $0.2\ \text{m}$ is fastened to a fixed point $A$, which is positioned above a smooth horizontal surface. A particle $P$ with mass $0.6\ \text{kg}$ is attached to the other end of the string. With the string taut and inclined at $30^\circ$ to the horizontal (see diagram), $P$ travels in a circle on the surface at constant speed $v\ \text{m s}^{-1}$.

Representation of data

A uniform beam $AB$ is $2\ \text{m}$ long and has weight $70\ \text{N}$. It is hinged at $A$ to a fixed point on a vertical wall, and equilibrium is maintained by a light inextensible rope. One end of the rope is fixed to the wall at a point $1.7\ \text{m}$ vertically above the hinge, while the other end is fastened to the beam at a point $0.8\ \text{m}$ from $A$. The rope is perpendicular to $AB$. A load of weight $220\ \text{N}$ acts at $B$ (see diagram).

Representation of data

A particle $P$ with mass $0.28\ \text{kg}$ is fixed to the midpoint of a light elastic string whose natural length is $4\ \text{m}$. The string ends are fastened to fixed points $A$ and $B$, which lie at the same horizontal level and are $4.8\ \text{m}$ apart. $P$ is released from rest at the midpoint of $AB$. During the later motion, the acceleration of $P$ is zero when $P$ is $0.7\ \text{m}$ below $AB$.

Probability

A cyclist and his bicycle together have a mass of $81\ \text{kg}$. He begins from rest and moves along a straight line. The cyclist applies a steady force of $135\ \text{N}$, while the motion is opposed by a resistance of magnitude $9v\ \text{N}$, where $v\ \text{m s}^{-1}$ represents the cyclist's speed at time $t$ after starting.

Representation of data

A particle $P$ is launched from point $O$ with initial speed $10\ \text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal. $P$ later passes through point $A$, which is at an angle of elevation of $30^\circ$ from $O$ (see diagram). After time $t$ from projection, the horizontal displacement of $P$ from $O$ is $x$ and the upward vertical displacement is $y$ m respectively.

Representation of data

A flat circular disc turns about its centre $O$ with a constant angular speed of $9\,\text{rad s}^{-1}$. A particle of mass $0.05\,\text{kg}$ is placed on the disc at a point $0.4\,\text{m}$ from $O$. The particle rotates with the disc and does not slip.

Representation of data

The bow is formed from a uniform curved section $AB$ of mass $1.4\,\text{kg}$, together with a uniform taut string of mass $m\,\text{kg}$ joining $A$ to $B$. The curved section $AB$ is an arc of a circle with centre $O$ and radius $0.8\,\text{m}$. The angle $AOB$ is $\frac{2\pi}{3}$ radians (see diagram). The centre of mass of the bow, including the string, is $0.65\,\text{m}$ from $O$.

Representation of data

A light inextensible string of length $0.2\,\text{m}$ has one end fastened at a fixed point $A$ above a smooth horizontal surface. The other end carries a particle $P$ of mass $0.6\,\text{kg}$. With the string taut and inclined at an angle of $30^\circ$ to the horizontal (see diagram), $P$ travels on the surface in a circle at constant speed $v\,\text{m s}^{-1}$.

Probability

A beam $AB$ is uniform, has length $2\,\text{m}$ and weighs $70\,\text{N}$. It is hinged at $A$ to a fixed point on a vertical wall and stays in equilibrium because of a light inextensible rope. One end of the rope is fixed to the wall at a point $1.7\,\text{m}$ vertically above the hinge. The other end is fastened to the beam at a point $0.8\,\text{m}$ from $A$. The rope is perpendicular to $AB$. A load of weight $220\,\text{N}$ acts at $B$ (see diagram).

Representation of data

A particle $P$ with mass $0.28\,\text{kg}$ is fastened to the midpoint of a light elastic string whose natural length is $4\,\text{m}$. The two ends of the string are attached to fixed points $A$ and $B$, which are at the same horizontal level and are $4.8\,\text{m}$ apart. $P$ is released from rest at the midpoint of $AB$. In the motion that follows, the acceleration of $P$ is zero when $P$ is at a distance $0.7\,\text{m}$ below $AB$.

Probability