Oct/Nov 2019

Find the term with no $x$ in the expansion of $(2x + \frac{1}{4x^2})^6$.

Series

Using origin $O$ as the reference point, the position vectors of points $A$, $B$, $C$ and $D$, as displayed in the diagram, are $\vec{OA} = \begin{pmatrix}-1\\3\\-4\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}2\\-3\\5\end{pmatrix}$, $\vec{OC} = \begin{pmatrix}4\\-2\\5\end{pmatrix}$ and $\vec{OD} = \begin{pmatrix}2\\2\\-1\end{pmatrix}$.

Coordinate geometry

The diagram depicts a shaded region enclosed by the $y$-axis, the line $y = -1$, and the section of the curve $y = x^2 + 4x + 3$ for which $x \geq -2$.

Integration

$f$ is an increasing function, defined for values of $x$ with $x > n$, where $n$ is an integer. It is given that $f'(x) = x^2 - 6x + 8$.

Differentiation

For the curve $y = 2x^3 - 5x^2 - 3x + c$, the line $y = ax + b$ is tangent at the point $(2, 6)$. Find the values of the constants $a$, $b$ and $c$.

Differentiation

A runner preparing for a long-distance race intends to cover a greater distance each day for 21 days. On day 1, she will run $x$ km, and on each later day the distance will increase by $10\%$ of the previous day’s distance. On day 21, she will run $20$ km.

Series

From $4\tan x + 3\cos x + \frac{1}{\cos x} = 0$, show, without using a calculator, that $\sin x = -\frac{2}{3}$.

Trigonometry

A line with gradient $m$ passes through $(0, -2)$.

Differentiation

The functions $f$ and $g$ are specified by $f : x \mapsto \dfrac{3}{2x + 1}$ for $x > 0$, $g : x \mapsto \dfrac{1}{x} + 2$ for $x > 0$.

Functions

The figure shows sector $OAC$ in a circle centred at $O$. Tangents $AB$ and $CB$ to the circle intersect at $B$. Arc $AC$ measures $6\text{ cm}$, and angle $AOC=\frac{3}{8}\pi$ radians.

Circular measure

The curve whose gradient is $\frac{dy}{dx} = \sqrt{5x - 1} - 2$ goes through the point $(2, 3)$.

Differentiation

In $(4 + ax)\left(1 + \frac{x}{2}\right)^6$, the coefficient of $x^2$ is $3$. Determine the constant $a$.

Series

The diagram depicts a section of the curve $y = 1 - \frac{4}{(2x + 1)^2}$. The curve meets the $x$-axis at $A$. The normal drawn to the curve at $A$ cuts the $y$-axis at $B$.

Integration

Point $M$ is the midpoint of the segment joining $(3, 7)$ and $(-1, 1)$. Find the equation of the line through $M$ that runs parallel to the line $\frac{x}{3} + \frac{y}{2} = 1$.

Coordinate geometry

For a certain curve, $\frac{dy}{dx} = \frac{k}{\sqrt{x}}$, with $k$ a constant. The points $P(1,-1)$ and $Q(4,4)$ are on the curve.

Differentiation

The diagram depicts a circle with centre $O$ and radius $r$ cm. The points $A$ and $B$ are on the circle, and angle $AOB = 2\theta$ radians. The tangents drawn to the circle at $A$ and $B$ intersect at $T$.

Circular measure

The diagram illustrates a solid cone with slant height of $15\text{ cm}$ and vertical height $h\text{ cm}$.

Differentiation

Given that $x > 0$, Find the two smallest values of $x$, measured in radians, that satisfy $3\tan(2x + 1) = 1$. Show all necessary working.

Trigonometry

The figure represents a three-dimensional solid $OABCDEFG$. The base $OABC$ and the top face $DEFG$ are matching horizontal rectangles. The parallelograms $OAED$ and $CBFG$ lie in vertical planes. Points $P$ and $Q$ are the mid-points of $OD$ and $GF$ respectively. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $\overrightarrow{OA}$ and $\overrightarrow{OC}$ respectively and the unit vector $\mathbf{k}$ is vertically upwards. The position vectors of $A$, $C$ and $D$ are given by $\overrightarrow{OA} = 6\mathbf{i}$, $\overrightarrow{OC} = 8\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 10\mathbf{k}$.

Coordinate geometry

Across a 21-day training spell, an athlete gets ready for a marathon by adding $1.2\text{ km}$ to the distance she covers each day. On day 1, she runs $13\text{ km}$.

Series

The functions $f$ and $g$ are given by $f(x) = 2x^2 + 8x + 1$ for $x \in \mathbb{R}$, and $g(x) = 2x - k$ for $x \in \mathbb{R}$, where $k$ is a constant.

Differentiation

Expand $(1 + y)^6$ into ascending powers of $y$ up to and including the term in $y^2$.

Series

Taking origin $O$ as the reference point, the position vectors of the points $A$, $B$ and $X$ are given by $\overrightarrow{OA} = \begin{pmatrix} -8 \\ -4 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 10 \\ 2 \\ 11 \end{pmatrix}$ and $\overrightarrow{OX} = \begin{pmatrix} -2 \\ -2 \\ 5 \end{pmatrix}$.

Coordinate geometry

The diagram illustrates a portion of the curve $y = (x - 1)^{-2} + 2$ together with the lines $x = 1$ and $x = 3$. Point $A$ lies on the curve and has coordinates $(2, 3)$. The normal to the curve at $A$ meets the line $x = 1$ at $B$.

Integration

The function $g$ is given by $g(x) = x^2 - 6x + 7$ for $x > 4$.

Functions

A curve is given by $y = x^3 + x^2 - 8x + 7$. There are no stationary points in the interval $a < x < b$. Determine the least possible value of $a$ and the greatest possible value of $b$.

Differentiation

In the diagram, $ACB$ is a semicircle with centre $O$ and radius $r$, and arc $OC$ belongs to a circle centred at $A$.

Circular measure

The cuboid’s dimensions are $x$ cm, $2x$ cm and $4x$ cm, as the diagram indicates.

Differentiation

A line is given by $y = 3kx - 2k$, and a curve is given by $y = x^2 - kx + 2$, where $k$ is a constant.

Quadratics

Demonstrate that the equation $3\cos^4\theta + 4\sin^2\theta - 3 = 0$ may be rewritten as $3x^2 - 4x + 1 = 0$, with $x = \cos^2\theta$.

Trigonometry

The function $f$ is specified for $x > \frac{1}{2}$, and its derivative is $f'(x) = 3\sqrt{2x - 1} - 6$.

Differentiation

The first three terms of a geometric progression are $3k$, $5k - 6$ and $6k - 4$, in that order.

Series

Solve for $x$ in $|2x - 7| < |2x - 9|$.

Algebra

Find the exact value of $\int_{1}^{2} (2e^{2x} - 1)^2\,dx$. Show all working needed.

Integration

A curve is defined by the equation $y = \frac{3 + 2\ln x}{1 + \ln x}$.

Differentiation

The polynomial $p(x)$ is given by $p(x)=ax^3+ax^2-15x-18$, where $a$ is a constant. It is stated that $(x-2)$ is a factor of $p(x)$.

Logarithmic and exponential functions

We are told that $\int_0^a (3x^2 + 4\cos 2x - \sin x)\,dx = 2$, where $a$ is a constant.

Numerical solution of equations

With all required working shown, solve $\sec \alpha \cosec \alpha = 7$ for $0^\circ < \alpha < 90^\circ$.

Trigonometry

The curve is described by $x^2 - 4xy - 2y^2 = 1$.

Differentiation

The polynomial $f(x)$ has definition $f(x) = x^4 - 3x^3 + 5x^2 - 6x + 11$.

Algebra

Solve $|4x + 5| = |x - 7|$.

Logarithmic and exponential functions

The variables $x$ and $y$ obey the relationship $y = kx^a$, where $k$ and $a$ are constants. A plot of $\ln y$ against $\ln x$ is a straight line that passes through the points $(0.22,\, 3.96)$ and $(1.32,\, 2.43)$, as shown in the diagram.

Numerical solution of equations

The sequence $x_1, x_2, x_3, \dots$ defined by $x_1 = 1$, $x_{n+1} = \frac{x_n}{\ln(2x_n)}$ tends to the limit $\alpha$.

Numerical solution of equations

Determine the exact coordinates of the stationary point on the curve given by $y = e^{-\frac{1}{2}x}(2x + 5)$.

Differentiation

Show that $\displaystyle \int_{2}^{18} \frac{3}{2x} \, dx$ equals $\ln 27$.

Integration

The curve is given parametrically by $x = 3\sin 2\theta$, $y = 1 + 2\tan 2\theta$, for $0 \leq \theta < \frac{1}{4}\pi$.

Differentiation

Express $0.5\cos\theta - 1.2\sin\theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$, and give the value of $\alpha$ correct to $2$ decimal places.

Trigonometry

Solve $|2x - 7| < |2x - 9|$.

Algebra

Determine the exact value of $\int_{1}^{2} (2e^{2x} - 1)^2\,dx$. Show all necessary working.

Integration

The curve is given by $y = \dfrac{3 + 2\ln x}{1 + \ln x}$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = ax^3 + ax^2 - 15x - 18$, with $a$ a constant. It is stated that $(x - 2)$ is a factor of $p(x)$.

Logarithmic and exponential functions

The condition is $\int_0^a (3x^2 + 4\cos 2x - \sin x)\,dx = 2$, with $a$ taken as a constant.

Numerical solution of equations

With all required working shown, solve the equation $\sec \alpha \cosec \alpha = 7$ for $0^\circ < \alpha < 90^\circ$.

Trigonometry

The equation of a curve is $x^2 - 4xy - 2y^2 = 1$.

Differentiation

If $\ln\left(1 + e^{2y}\right) = x$, express $y$ in terms of $x$.

Logarithmic and exponential functions

The complex number $u$ is defined by $u = -3 - (2\sqrt{10})i$. With all working shown and no calculator used, determine the square roots of $u$. Present your answers in the form $a + ib$, where $a$ and $b$ are exact real numbers.

Complex numbers

Solve the inequality involving absolute values, $|2x - 3| > 4|x + 1|$.

Algebra

A curve is described parametrically by $x = 2t + \sin 2t$ and $y = \ln(1 - \cos 2t)$.

Differentiation

Let $N$ denote the number of insects $t$ weeks after observations begin. The population is falling at a rate proportional to $Ne^{-0.02t}$. Both $N$ and $t$ are continuous variables, and when $t = 0$, $N = 1000$ and $\frac{dN}{dt} = -10$.

Differential equations

For the curve $y = e^{-2x} \ln(x - 1)$, a stationary point occurs when $x = p$.

Numerical solution of equations

Differentiate $\frac{\cos x}{\sin x}$ to demonstrate that, when $y = \cot x$, $\frac{dy}{dx} = -\cosec^2 x$.

Integration

The two lines $l$ and $m$ are given by the equations $\mathbf{r} = a\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 3\mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$ respectively, with $a$ as a constant. The lines are known to intersect.

Vectors

Define $f(x) = \dfrac{x^2 + x + 6}{x^2(x + 2)}$.

Integration

Starting from the expansion of $\cos(2x + x)$, show that $\cos 3x = 4\cos^3 x - 3\cos x$.

Trigonometry

Solve $5\ln(4 - 3^x) = 6$. Include all necessary working and state the answer correct to $3$ decimal places.

Logarithmic and exponential functions

The line $l$ is given by $\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 3\mathbf{k})$. The plane $p$ is described by $2x + y - 3z = 5$.

Vectors

The curve defined by $y = \frac{e^{-2x}}{1 - x^2}$ has a stationary point somewhere in the interval $-1 \le x \le 1$.

Differentiation

Let $p(x)$ represent the polynomial $x^4 + 3x^3 + ax + b$, where $a$ and $b$ are constants. On division of $p(x)$ by $x^2 + x - 1$, the remainder is $2x + 3$.

Algebra

Rewrite $(\sqrt{6})\sin x + \cos x$ in the form $R\sin(x + \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$. Give the exact value of $R$ and state $\alpha$ correct to $3$ decimal places.

Trigonometry

The curve is defined by the equation $2x^2y - xy^2 = a^3$, where $a$ is a positive constant.

Differentiation

The variables $x$ and $\theta$ are linked by the differential equation $\sin \frac{1}{2}\theta\, \frac{dx}{d\theta} = (x + 2)\cos \frac{1}{2}\theta$ for $0 < \theta < \pi$. It is given that $x = 1$ when $\theta = \frac{1}{3}\pi$.

Differential equations

Find the complex number $z$ that satisfies the equation $z + \frac{iz}{z^*} - 2 = 0$, with $z^*$ as the complex conjugate of $z$. Express your answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Let $f(x) = \dfrac{2x^2 + x + 8}{(2x - 1)(x^2 + 2)}$.

Integration

You are told that $\int_0^a x \cos\!\left(\tfrac{1}{3}x\right) dx = 3$, with the constant $a$ chosen so that $0 < a < \tfrac{3}{2}\pi$.

Numerical solution of equations

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

Algebra

The diagram displays the graph of $y = e^{\cos x} \sin^3 x$ for $0 \leq x \leq \pi$, together with its maximum point $M$. The shaded region $R$ lies between the curve and the $x$-axis.

Integration

Let $p(x)$ represent the polynomial $6x^3 + ax^2 + bx - 2$, with $a$ and $b$ as constants. You are told that $(2x + 1)$ is a factor of $p(x)$, and that the remainder when $p(x)$ is divided by $(x + 2)$ is $-24$.

Algebra

Show all the required working to solve the equation $\frac{3^{2x} + 3^{-x}}{3^{2x} - 3^{-x}} = 4$. Give your answer accurate to 3 decimal places.

Logarithmic and exponential functions

First expand $\tan(2x + x)$ to show that the relation $\tan 3x = 3\cot x$ can be transformed into $\tan^4 x - 12\tan^2 x + 3 = 0$.

Trigonometry

Work with the equation $\ln(x + 2) = 4e^{-x}$.

Numerical solution of equations

For the whole of this question, calculator use is not allowed. The complex number with modulus $1$ and argument $\tfrac{1}{3}\pi$ is called $w$. The complex number $1 + 2i$ is called $u$. The complex number $v$ satisfies $|v| = 2|u|$ and $\arg v = \arg u + \tfrac{1}{3}\pi$.

Complex numbers

Plane $m$ is given by $x + 4y - 8z = 2$. Plane $n$ is parallel to $m$ and goes through $P$, whose coordinates are $(5, 2, -2)$.

Vectors

The diagram displays the graph of $y = \sec x$ for $0 \leq x < \frac{1}{2}\pi$.

Numerical solution of equations

The variables $x$ and $t$ are linked by the differential equation $5\frac{dx}{dt} = (20 - x)(40 - x)$. You are told that $x = 10$ when $t = 0$.

Differential equations

A crane is raising a load of $1250\,\text{kg}$ straight upward at a steady speed $V\,\text{m s}^{-1}$. The crane’s power is a constant $20\,\text{kW}$.

Energy, work and power

A cyclist and her bicycle have a combined mass of $75\,\text{kg}$. She goes up a straight hill that is $0.7\,\text{km}$ long and makes an angle of $1.5^\circ$ to the horizontal. Her speed at the foot of the hill is $10\,\text{m s}^{-1}$ and at the summit it is $5\,\text{m s}^{-1}$. Motion is opposed by a resistance, and the work done against this resistance while the cyclist climbs the hill is $2000\,\text{J}$. The cyclist applies a constant force of magnitude $F\,\text{N}$ in the direction of motion.

Forces and equilibrium

A $3\,\text{kg}$ block is stationary on a rough plane that is tilted at $60^\circ$ to the horizontal. A force of $15\,\text{N}$, applied up the line of greatest slope of the plane, is just enough to stop the block from sliding down the plane.

Forces and equilibrium

Blocks $A$ and $B$, with masses $4\,\text{kg}$ and $5\,\text{kg}$ respectively, are connected by a light inextensible string. They are at rest on a smooth plane inclined at angle $\alpha$ to the horizontal, where $\tan \alpha = \frac{7}{24}$. The string lies parallel to a line of greatest slope of the plane, with $B$ positioned above $A$. A force of magnitude $36\,\text{N}$ acts on $B$, parallel to a line of greatest slope of the plane (see diagram).

Newton's laws of motion

A small ring $P$ is fitted on a fixed smooth horizontal rod $AB$. Three horizontal forces of magnitudes $4.5\,\text{N}$, $7.5\,\text{N}$ and $F\,\text{N}$ act on $P$ (see diagram).

Forces and equilibrium

A particle with mass $0.4\,\text{kg}$ is let go from rest from a point $1.8\,\text{m}$ above the water surface in a tank. Its speed does not change instantaneously as it enters the water. While it is in the water, the water exerts an upward force of $5.6\,\text{N}$ on the particle.

Kinematics of motion in a straight line

A particle travels along a straight line, beginning from rest at the point $O$ and arriving momentarily at rest at the point $P$. After $t\,\text{s}$ from leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 0.6t^2 - 0.12t^3$.

Kinematics of motion in a straight line

A particle travels along a straight line. Its displacement at time $t$ is $s$ m, with $s = t^3 - 6t^2 + 4t$.

Kinematics of motion in a straight line

The diagram presents a velocity-time graph that represents the motion of a tractor. It is made up of four straight-line sections. The tractor passes through point $O$ at time $t = 0$ with speed $U$ m s$^{-1}$. It then increases in speed to $V$ m s$^{-1}$ over $5$ s, after which it continues at that speed for another $25$ s. Next, the tractor accelerates to $12$ m s$^{-1}$ over $5$ s. It then slows down to zero in $15$ s.

Kinematics of motion in a straight line

A particle $P$ of mass $0.3\ \text{kg}$ is kept in equilibrium above a horizontal plane by a force of magnitude $5\ \text{N}$ acting vertically upwards. The particle is linked by two strings $PA$ and $PB$ with lengths $0.9\ \text{m}$ and $1.2\ \text{m}$ respectively. The points $A$ and $B$ are on the plane, and angle $APB = 90^\circ$ (see diagram).

Forces and equilibrium

A lorry with a mass of $25\ 000\ \text{kg}$ is moving along a straight horizontal road. A constant force of $3000\ \text{N}$ acts against its motion.

Energy, work and power

Particles $A$ and $B$ travel along the same vertical straight line. $A$ is launched vertically upwards from the ground at $20\,\text{m s}^{-1}$. After 1 second, $B$ is released from rest from a height of $40\,\text{m}$.

Kinematics of motion in a straight line

A block with mass $3\,\text{kg}$ starts from rest on a rough horizontal plane. A force of magnitude $6\,\text{N}$ acts on the block at an angle of $\theta$ above the horizontal, where $\cos\theta = \frac{24}{25}$. The force acts for $5\,\text{s}$, and during this interval the block travels $4.5\,\text{m}$.

Forces and equilibrium

Particles $P$ and $Q$, with masses $0.3\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string. The string runs over a fixed smooth pulley attached to the edge of a smooth plane. The plane is set at an angle $\theta$ to the horizontal, with $\sin\theta = \frac{3}{5}$. $P$ is on the plane, while $Q$ hangs vertically beneath the pulley at a height of $0.8\,\text{m}$ above the floor. The section of string between $P$ and the pulley lies parallel to a line of greatest slope of the plane. $P$ is let go from rest and $Q$ moves vertically downwards.

Newton's laws of motion

A crate with mass $500\,\text{kg}$ is dragged over rough horizontal ground by a horizontal rope connected to a winch. The winch applies a steady pulling force of $2500\,\text{N}$, and the crate travels at a steady speed.

Forces and equilibrium

A train with mass $150\,000\,\text{kg}$ moves up a straight slope at an angle of $\alpha^{\circ}$ to the horizontal, with a constant driving force of $16\,000\,\text{N}$. When the train is at point $A$ on the slope, its speed is $45\,\text{m s}^{-1}$. Point $B$ is $500\,\text{m}$ farther along the slope than $A$. At $B$, the train's speed is $42\,\text{m s}^{-1}$. A resistance force acts on the train, and between $A$ and $B$ the train does $4 \times 10^{6}\,\text{J}$ of work against this resistance force.

Energy, work and power

Three coplanar forces with magnitudes $50\,\text{N}$, $60\,\text{N}$ and $100\,\text{N}$ are applied at one point. Their combined effect is a resultant force of magnitude $R\,\text{N}$. The force directions are indicated in the diagram.

Forces and equilibrium

A car moves on a straight road with constant acceleration. It goes past points $P$, $Q$, $R$ and $S$. The time taken to move from $P$ to $Q$, from $Q$ to $R$ and from $R$ to $S$ is $10\,\text{s}$ for each section. The distance $QR$ is $1.5$ times $PQ$. At $Q$, the car's speed is $20\,\text{m}\,\text{s}^{-1}$.

Kinematics of motion in a straight line

A cyclist is moving along a straight horizontal road. The combined mass of the cyclist and his bicycle is $80\,\text{kg}$. His power output stays constant at $240\,\text{W}$. When his speed is $6\,\text{m s}^{-1}$, his acceleration is $0.3\,\text{m s}^{-2}$.

Energy, work and power

Particle $P$ moves along the straight line from $A$ to $B$. After leaving $A$, its velocity at time $t\,\text{s}$ is given by $v\,\text{m s}^{-1}$, where $v = 0.04t^3 + ct^2 + kt$. The journey from $A$ to $B$ takes $5\,\text{s}$, and when it reaches $B$ its speed is $10\,\text{m s}^{-1}$. The distance $AB$ is $25\,\text{m}$.

Kinematics of motion in a straight line

Two particles $A$ and $B$ have masses $m$ kg and $km$ kg respectively, with $k > 1$. They are fastened to the two ends of a light inextensible string. The string passes over a fixed smooth pulley, and the particles hang vertically beneath it. At the start, both particles are $0.81$ m above horizontal ground (see diagram). The system is released from rest, and particle $B$ hits the ground $0.9$ s later. In the later motion, particle $A$ does not reach the pulley.

Newton's laws of motion

A uniform solid cone has a weight of $5\,\text{N}$ and a base radius measuring $0.1\,\text{m}$. $AB$ is a diameter of the cone’s base. The cone is in equilibrium, with $A$ touching a rough horizontal surface and $AB$ vertical, while a force is applied at $B$. The force is $3\,\text{N}$ in magnitude and acts parallel to the cone’s axis (see diagram).

Representation of data

A particle is launched from a point on horizontal ground with speed $15\,\text{m s}^{-1}$ at an angle of $\theta^\circ$ above the horizontal. It lands on the ground $2\,\text{s}$ after launch.

Representation of data

On a smooth horizontal surface, fixed points $O$ and $A$ are $0.8\,\text{m}$ apart. A particle $P$ of mass $0.25\,\text{kg}$ is projected horizontally from $A$ with velocity $3\,\text{m s}^{-1}$, in the direction away from $O$. The velocity of $P$ is $v\,\text{m s}^{-1}$ when the displacement of $P$ from $O$ is $x\,\text{m}$. A force of magnitude $k v^2 x^{-2}\,\text{N}$ acts opposite to the motion of $P$.

Probability

A small ball $B$ is launched with speed $30\,\text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal from point $O$. After $t\,\text{s}$, the horizontal displacement of $B$ from $O$ is $x\,\text{m}$ and its upward vertical displacement from $O$ is $y\,\text{m}$ respectively.

Representation of data

A particle $P$ with mass $0.3\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $0.6\,\text{m}$ and modulus of elasticity is $9\,\text{N}$. The other end of the string is fixed at a point $O$ on a smooth plane inclined at $30^\circ$ to the horizontal. $OA$ is the line of greatest slope of the plane, with $A$ below the level of $O$ and $OA = 0.8\,\text{m}$. The particle $P$ is released from rest at $A$.

Probability

Points A and $B$ are fixed on a vertical line, with $A$ positioned $0.6\,\text{m}$ above $B$. A particle $P$ with mass $0.3\,\text{kg}$ is connected to $A$ by a light inextensible string of length $0.5\,\text{m}$. The particle $P$ is also connected to $B$ by a light elastic string whose modulus of elasticity is $46\,\text{N}$. The particle $P$ travels at constant angular speed $8\,\text{rad s}^{-1}$ in a horizontal circle centred at the midpoint of $AB$.

Probability

The uniform prism has cross-section $ABC$ through its centre of mass and is resting with $AB$ on a rough horizontal surface. $AB = 0.4\,\text{m}$ and $C$ stands $0.9\,\text{m}$ above the surface (see diagram). The prism is just on the point of toppling about its edge through $B$.

Probability

A particle with mass $0.3\,\text{kg}$ is fixed to one end of a light elastic string whose natural length is $0.6\,\text{m}$ and whose modulus of elasticity is $9\,\text{N}$. The other end of the string is secured at a fixed point $O$ on a smooth horizontal surface. The particle is launched horizontally from $O$ at speed $4\,\text{m s}^{-1}$.

Probability

A small ball is projected from point $O$ on level ground at an angle of $30^\circ$ above the horizontal. After $t$ s, the ball’s horizontal displacement from $O$ is $x$ m and its vertical displacement upwards is $y$ m respectively. It is given that $x = 40t$.

Representation of data

A particle $P$ of mass $0.5\,\text{kg}$ is fastened to one end of a light elastic string with natural length $0.6\,\text{m}$ and modulus of elasticity $12\,\text{N}$. The opposite end of the string is fixed at point $O$. Particle $P$ is fired vertically downwards at speed $2\,\text{m s}^{-1}$ from the position $0.5\,\text{m}$ vertically beneath $O$. At an instant when the acceleration of $P$ is $4\,\text{m s}^{-2}$ downwards,

Probability

A particle is launched from point $O$ on horizontal ground with speed $V\,\text{m s}^{-1}$ and is projected at an angle of $60^\circ$ above the horizontal. At the moment $3\,\text{s}$ after projection, the direction of motion of the particle is $30^\circ$ below the horizontal.

Representation of data

A and B are fixed points on a vertical line, with $A$ above $B$. A particle $P$ of mass $0.4\,\text{kg}$ is connected to $A$ by a light inextensible string of length $0.5\,\text{m}$. The particle $P$ is also connected to $B$ by a second light inextensible string. $P$ moves at constant speed in a horizontal circle with centre $O$ between $A$ and $B$. Angle $BAP = 30^\circ$ and angle $ABP = 70^\circ$ (see diagram).

Probability

Particle $P$ has mass $0.2\,\text{kg}$ and is launched horizontally from a fixed point $O$ on a smooth horizontal surface. If the displacement of $P$ from $O$ is $x\,\text{m}$, then its velocity is $v\,\text{m}\,\text{s}^{-1}$. A horizontal force of variable magnitude $0.09vx\,\text{N}$, acting away from $O$, acts on $P$. A further force of constant magnitude $0.3\,\text{N}$, acting towards $O$, also acts on $P$.

Representation of data

$ABCD$ is a uniform lamina in the form of a trapezium with centre of mass $G$. The lines $AD$ and $BC$ are parallel and separated by $1.8\,\text{m}$, with $AD=2.4\,\text{m}$ and $BC=1.2\,\text{m}$ (see diagram).

Representation of data

A uniform solid cone has weight $5\,\text{N}$ and a base radius of $0.1\,\text{m}$. $AB$ is a diameter of the cone’s base. The cone is maintained in equilibrium, with $A$ touching a rough horizontal surface and $AB$ vertical, by a force applied at $B$. This force has magnitude $3\,\text{N}$ and is parallel to the axis of the cone (see diagram).

Representation of data