May/June 2013

It is stated that $f(x) = (2x - 5)^3 + x$, for $x \in \mathbb{R}$.

Differentiation

The diagram includes a section of the curve $y = (x - 2)^4$ and the point $A(1, 1)$ on that curve. The tangent drawn at $A$ meets the $x$-axis at $B$, while the normal drawn at $A$ meets the $y$-axis at $C$.

Coordinate geometry

In $(1 - px)^6$, $p$ is a non-zero constant. Find the first three terms of the expansion of $(1 - px)^6$ in ascending powers of $x$.

Series

The diagram shows that $OAB$ is a sector of a circle with centre $O$ and radius $8\text{ cm}$. The angle $BOA$ equals $\alpha$ radians. $OAC$ is a semicircle whose diameter is $OA$. The area of the semicircle $OAC$ is twice the area of the sector $OAB$.

Circular measure

The third term in a geometric progression is $-108$, while the sixth term is $32$.

Series

Show that $\dfrac{\sin\theta}{\sin\theta + \cos\theta} + \dfrac{\cos\theta}{\sin\theta - \cos\theta} = \dfrac{1}{\sin^2\theta - \cos^2\theta}$.

Trigonometry

Taking $O$ as the origin, the position vectors of the three points $A$, $B$ and $C$ are $\vec{OA} = \mathbf{i} + 2p\mathbf{j} + q\mathbf{k}$, $\vec{OB} = q\mathbf{i} - 2p\mathbf{k}$ and $\vec{OC} = -(4p^2 + q^2)\mathbf{i} + 2p\mathbf{j} + q\mathbf{k}$, where $p$ and $q$ are constants.

Coordinate geometry

The curve is given by $y = x^2 - 4x + 4$ and the line is given by $y = mx$, with $m$ as a constant.

Differentiation

Rewrite $2x^2 - 12x + 13$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

Functions

The curve is described by the equation $y = f(x)$, where $f'(x) = 3x^{\frac{1}{2}} + 3x^{-\frac{1}{2}} - 10$.

Differentiation

A curve is defined by $\frac{dy}{dx} = \frac{6}{x^2}$, and the point $(2, 9)$ lies on it.

Integration

In an arithmetic progression, the first and last terms are $12$ and $48$ respectively, and the sum of the first four terms is $57$. Find the number of terms in the progression.

Series

The sketch displays the curve $y = \sqrt{1 + 4x}$, which cuts the $x$-axis at $A$ and the $y$-axis at $B$. The normal drawn to the curve at $B$ meets the $x$-axis at $C$.

Coordinate geometry

Find the coefficient of $x^2$ in the expanded form of $(2x - \frac{1}{2x})^6$.

Series

At point $P$, the straight line $y = mx + 14$ touches the curve $y = \frac{12}{x} + 2$.

Coordinate geometry

The diagram depicts a square $ABCD$ with side length $10\text{ cm}$. Point $O$ is the midpoint of $AD$, and $BXC$ is an arc of a circle centred at $O$.

Circular measure

We are told that $a = \sin \theta - 3 \cos \theta$ and $b = 3 \sin \theta + \cos \theta$, where $0^\circ \leq \theta \leq 360^\circ$.

Trigonometry

Taking O as the origin, the position vectors of points A and B are given as \(\overrightarrow{OA} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}\) and \(\overrightarrow{OB} = 3\mathbf{i} + p\mathbf{j} + q\mathbf{k}\), with p and q as constants.

Coordinate geometry

The point $R$ is the image of $(-1, 3)$ after reflection in the line $3y + 2x = 33$.

Coordinate geometry

A solid circular cylinder with radius $r$ cm has volume $250\pi\text{ cm}^3$.

Differentiation

The function $f$ is given by $f(x) = \frac{5}{1 - 3x}$, where $x \geq 1$.

Differentiation

The curve is defined by $\frac{dy}{dx} = \sqrt{2x + 5}$, and it goes through the point $(2, 5)$.

Integration

The function $f$ is given by $f : x \mapsto 2x + k$, $x \in \mathbb{R}$, where $k$ is a fixed constant.

Functions

The diagram presents a segment of the curve $y = \frac{8}{\sqrt{x}} - x$ together with the points $A(1,7)$ and $B(4,0)$, both of which are on the curve. The tangent to the curve at $B$ meets the line $x = 1$ at $C$.

Coordinate geometry

The diagram illustrates a circle $C$ with centre $O$ and radius $3\,\text{cm}$. The radii $OP$ and $OQ$ are produced to $S$ and $R$ respectively, making $ORS$ a sector of a circle centred at $O$. Given that $PS = 6\,\text{cm}$ and that the shaded region has the same area as circle $C$.

Circular measure

Rewrite $2\cos^2\theta = \tan^2\theta$ as a quadratic in $\cos^2\theta$.

Trigonometry

Find the first three terms of the expansion of $(2 + ax)^5$ in ascending powers of $x$.

Series

Sketch the curves $y = \sin 2x$ and $y = \cos x - 1$ together on the same diagram for $0 \le x \le 2\pi$.

Trigonometry

The non-zero variables $x$, $y$, and $u$ satisfy $u = x^2y$. It is given that $y + 3x = 9$.

Differentiation

The diagram illustrates three points $A(2,14)$, $B(14,6)$ and $C(7,2)$. Point $X$ is located on $AB$, and $CX$ is at right angles to $AB$.

Coordinate geometry

The diagram depicts a parallelogram $OABC$ in which $\vec{OA} = \begin{pmatrix}3\\3\\-4\end{pmatrix}$ and $\vec{OB} = \begin{pmatrix}5\\0\\2\end{pmatrix}$.

Coordinate geometry

In an arithmetic progression, the sum $S_n$ of the first $n$ terms is $S_n = 2n^2 + 8n$. Determine the first term and the common difference of the progression.

Series

Solve the equation $|2^x - 7| = 1$, and give answers to $2$ decimal places when needed.

Logarithmic and exponential functions

Solve $\ln(3 - 2x) - 2\ln x = \ln 5$.

Logarithmic and exponential functions

Show that $12\sin^2 x\cos^2 x = \frac{3}{2}(1 - \cos 4x)$.

Integration

The polynomial $ax^3 - 5x^2 + bx + 9$, with $a$ and $b$ as constants, is written as $p(x)$. You are told that $(2x + 3)$ is a factor of $p(x)$, and that the remainder when $p(x)$ is divided by $(x + 1)$ is $8$.

Algebra

A curve is given by the parametric equations $x = e^{2t}$, $y = 4te^t$.

Differentiation

By sketching an appropriate pair of graphs, show that the equation $\cot x = 4x - 2$, where $x$ is in radians, has a single root for $0 \leq x \leq \frac{\pi}{2}$.

Numerical solution of equations

Express $5\sin 2\theta + 2\cos 2\theta$ in the form $R\sin(2\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$, and give the exact value of $R$ as well as the value of $\alpha$ correct to $2$ decimal places.

Trigonometry

A curve is defined by $\frac{dy}{dx} = \frac{4}{7 - 2x}$. The point $(3, 2)$ lies on the curve.

Integration

Determine the solution of the inequality $|x - 8| > |2x - 4|$.

Algebra

The polynomial $2x^3 + ax^2 - ax - 12$, with $a$ as a constant, is represented by $p(x)$.

Algebra

The variables $x$ and $y$ are linked by the equation $5^{y+1} = 2^{3x}$.

Logarithmic and exponential functions

A curve is given by the equation $x^2 - 2x^2y + 3y = 9$.

Differentiation

Using a suitable pair of sketches, demonstrate that the equation $3e^x = 8 - 2x$ has a single root.

Numerical solution of equations

Find the exact area of the region enclosed by the curve $y = 1 + e^{2x-1}$, the $x$-axis and the lines $x = \frac{1}{2}$ and $x = 2$.

Integration

Show that $\frac{1}{\sin(x - 60^\circ) + \cos(x - 30^\circ)} \equiv \cosec x$ is true.

Trigonometry

Solve $|2^x - 7| = 1$, with answers stated correct to $2$ decimal places where needed.

Logarithmic and exponential functions

Solve for $x$ in the equation $\ln(3 - 2x) - 2\ln x = \ln 5$.

Logarithmic and exponential functions

Show that $12\sin^2 x \cos^2 x$ is equal to $\frac{3}{2}(1 - \cos 4x)$.

Integration

The polynomial $ax^3 - 5x^2 + bx + 9$, where $a$ and $b$ are constants, is represented by $p(x)$. It is stated that $(2x + 3)$ is a factor of $p(x)$, and that dividing $p(x)$ by $(x + 1)$ leaves a remainder of $8$.

Algebra

The parametric form of the curve is given by $x = e^{2t}$ and $y = 4te^t$.

Differentiation

By drawing an appropriate pair of graphs, show that the equation $\cot x = 4x - 2$, where $x$ is in radians, has only one root for $0 \le x \le \tfrac{1}{2}\pi$.

Numerical solution of equations

Rewrite $5\sin 2\theta + 2\cos 2\theta$ as $R\sin(2\theta + \alpha)$, 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

Determine the quotient and remainder when $2x^2$ is divided by $x + 2$.

Algebra

Liquid is entering a small tank that has a leak. At the beginning the tank contains no liquid, and after $t$ minutes the liquid volume in the tank is $V$ cm$^3$. The inflow rate is constant at $80$ cm$^3$ per minute. As a result of the leak, liquid is leaving the tank at a rate that, at any instant, is $kV$ cm$^3$ per minute, where $k$ is a positive constant.

Differential equations

Expand $\frac{1 + 3x}{\sqrt{1 + 2x}}$ in ascending powers of $x$ up to and including the term in $x^2$, and simplify the coefficients.

Algebra

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

Algebra

Solve for $x$ in the equation $|4x - 1| = |x - 3|$.

Logarithmic and exponential functions

For each curve shown below, determine the gradient at the point where the curve meets the $y$-axis:

Differentiation

The points $P$ and $Q$ are specified by position vectors, measured from the origin $O$, as $\overrightarrow{OP} = 7\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}$ and $\overrightarrow{OQ} = -5\mathbf{i} + \mathbf{j} + \mathbf{k}$. Point $A$ is the midpoint of $PQ$. Plane $\Pi$ is perpendicular to the line $PQ$ and goes through $A$.

Vectors

Without a calculator, solve $3w + 2i w^* = 17 + 8i$, where $w^*$ is the complex conjugate of $w$. Give your answer in the form $a + bi$.

Complex numbers

Demonstrate that $\int_2^4 4x \ln x \, dx = 56 \ln 2 - 12$.

Integration

Rewrite $4\cos \theta + 3\sin \theta$ as $R\cos(\theta - \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the value of $\alpha$ correct to 4 decimal places.

Trigonometry

Solve for $x$ in $|x - 2| = |\tfrac{1}{3}x|$.

Algebra

Points $A$ and $B$ are represented by the position vectors $2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ respectively, and plane $p$ is given by $x + y = 5$.

Vectors

The value sequence generated by the iterative formula $x_{n+1} = \dfrac{x_n(x_n^3 + 100)}{2(x_n^3 + 25)}$, and starting from $x_1 = 3.5$, converges to $\alpha$.

Numerical solution of equations

The variables $x$ and $y$ obey the relation $y = Ae^{-kx^2}$, in which $A$ and $k$ are constants. The plot of $\ln y$ against $x^2$ is a straight line that goes through the points $(0.64, 0.76)$ and $(1.69, 0.32)$, as illustrated in the diagram.

Logarithmic and exponential functions

The polynomial $ax^3 - 20x^2 + x + 3$, with $a$ as a constant, is called $p(x)$. It is stated that $(3x + 1)$ is a factor of $p(x)$.

Algebra

The diagram illustrates the curve given by $x^3 + xy^2 + ay^2 - 3ax^2 = 0$, where $a$ is a positive constant. The highest point on the curve is $M$.

Vectors

By differentiating $\dfrac{1}{\cos x}$, demonstrate that the derivative of $\sec x$ is $\sec x \tan x$. Hence demonstrate that if $y = \ln(\sec x + \tan x)$ then $\dfrac{dy}{dx} = \sec x$.

Integration

First expand $\cos(x + 45^\circ)$, then write $\cos(x + 45^\circ) - (\sqrt{2})\sin x$ in the form $R\cos(x + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. State $R$ correct to 4 significant figures and $\alpha$ correct to 2 decimal places.

Trigonometry

Write $\dfrac{1}{x^2(2x + 1)}$ as $\dfrac{A}{x^2} + \dfrac{B}{x} + \dfrac{C}{2x + 1}$.

Differential equations

The complex number $w$ has $\Re w > 0$ and satisfies $w + 3w^* = iw^2$, where $w^*$ is the complex conjugate of $w$. Find $w$, giving the answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Solve $|4x + 3| > |x|$ as an inequality.

Algebra

The line $l$ is given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} + \mathbf{k})$, with $a$ constant. The plane $p$ is defined by $x + 2y + 2z = 6$. Determine the value or values of $a$ for each of the situations below.

Vectors

You are told that $\ln(y + 1) - \ln y = 1 + 3\ln x$. Find $y$ in terms of $x$, writing your answer in a form that does not contain logarithms.

Logarithmic and exponential functions

Solve the equation $\tan 2x = 5 \cot x$, for $0^\circ < x < 180^\circ$.

Trigonometry

Write $(\sqrt{3})\cos x + \sin x$ as $R \cos(x - \alpha)$, with $R > 0$ and $0 < \alpha < \tfrac{1}{2}\pi$, and state the exact values of $R$ and $\alpha$.

Trigonometry

Write the polynomial $8x^3 + ax^2 + bx + 3$ as $p(x)$, where $a$ and $b$ are constants. You are told that $(2x + 1)$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $(2x - 1)$ is $1$.

Algebra

The graph displays the curves $y = e^{2x-3}$ and $y = 2 \ln x$. At $x = a$, the tangents drawn to the two curves are parallel.

Numerical solution of equations

The complex number $z$ is given by $z = a + ib$, where $a$ and $b$ are real numbers. Its complex conjugate is written as $z^*$.

Complex numbers

The variables $x$ and $t$ are connected by the differential equation $t \frac{dx}{dt} = \frac{k - x^3}{2x^2}$, for $t > 0$, where $k$ is a constant. When $t = 1$, $x = 1$ and when $t = 4$, $x = 2$.

Differential equations

The diagram illustrates the curve $y = \sin^2 2x \cos x$ for $0 \leq x \leq \tfrac{1}{2}\pi$, together with its highest point $M$.

Integration

A block is resting on a rough horizontal plane, and the coefficient of friction between the block and the plane is $1.25$.

Newton's laws of motion

A car with mass $1250\text{ kg}$ moves from the foot to the summit of a straight hill of length $600\text{ m}$, inclined at $2.5^\circ$ to the horizontal. The constant resistance to the car's motion is $400\text{ N}$. The work done by the driving force is $450\text{ kJ}$. At the bottom of the hill, the car's speed is $30\text{ m s}^{-1}$.

Energy, work and power

A cliff top is $40\text{ metres}$ above sea level. A man in a boat, near the foot of the cliff, is in trouble and launches a distress signal vertically upward from sea level.

Kinematics of motion in a straight line

A train with mass $400\,000\text{ kg}$ is travelling along a straight horizontal track. The engine provides a constant power of $1500\text{ kW}$, and the force resisting the train’s motion is $30\,000\text{ N}$.

Newton's laws of motion

A light inextensible string carries a particle $A$ of mass $0.26\text{ kg}$ at one end and a particle $B$ of mass $0.54\text{ kg}$ at the other. Particle $A$ is kept at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{5}{13}$. The string is taut and runs parallel to a line of greatest slope on the plane. It passes over a small smooth pulley at the top of the plane. Particle $B$ hangs motionless vertically below the pulley. The coefficient of friction between $A$ and the plane is $0.2$. Particle $A$ is released, and the particles begin to move.

Newton's laws of motion

A particle $P$, with mass $0.5\text{ kg}$, is situated on a smooth horizontal plane. Forces with magnitudes $F\text{ N}$, $2.5\text{ N}$ and $2.6\text{ N}$ act horizontally on $P$. Their directions are as shown in the diagram, where $\tan\alpha = \frac{12}{5}$ and $\tan\beta = \frac{7}{24}$.

Newton's laws of motion

A car driver travels in a straight line from $A$ to $B$, beginning from rest. The car’s speed rises to a highest value and then falls until it is again at rest at $B$. $t$ seconds after leaving $A$, the distance covered by the car is $0.0000117(400t^3 - 3t^4)\text{ metres}$.

Kinematics of motion in a straight line

A block of weight $30\,\text{N}$ rests on a rough horizontal plane and is connected to a string. With the string horizontal and the tension equal to $24\,\text{N}$, the block is in limiting equilibrium.

Newton's laws of motion

A and B lie 50\,\text{m} apart along a straight path inclined at an angle $\theta$ to the horizontal, with $\sin\theta = 0.05$, and A is above B. A block of mass $16\,\text{kg}$ is pulled down the path from A to B. It starts from rest at A and arrives at B with speed $10\,\text{m s}^{-1}$. The work done by the pulling force on the block is $1150\,\text{J}$.

Energy, work and power

A particle $P$ with mass $2.1\,\text{kg}$ is connected to one end of each of two light inextensible strings. Their other ends are fixed to points A and B, which lie on the same horizontal line. $P$ is suspended in equilibrium $40\,\text{cm}$ beneath the level of A and B, and the strings $PA$ and $PB$ are $50\,\text{cm}$ and $104\,\text{cm}$ long respectively (see diagram).

Forces and equilibrium

A particle $P$ is let go from rest at the top of a smooth plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{16}{65}$. The distance covered by $P$ from the top to the bottom is $S\,\text{m}$, and the speed of $P$ at the bottom is $8\,\text{m s}^{-1}$. The time taken by $P$ to move from the top to the bottom of the plane is $T\,\text{s}$.

Kinematics of motion in a straight line

A car with mass $1000\,\text{kg}$ is moving along a straight horizontal road. Its engine delivers constant power $P\,\text{kW}$. The resistive force on the car is $600\,\text{N}$. When the car’s speed is $25\,\text{m s}^{-1}$, its acceleration is $0.2\,\text{m s}^{-2}$.

Energy, work and power

A particle $P$ travels along a straight line. It begins from rest at point $O$ and heads towards point A on the line. For the first $8\,\text{s}$, its speed rises to $8\,\text{m s}^{-1}$ with constant acceleration. Over the following $12\,\text{s}$, its speed falls to $2\,\text{m s}^{-1}$ with constant deceleration. $P$ then continues with constant acceleration for $6\,\text{s}$, arriving at A with speed $6.5\,\text{m s}^{-1}$. The displacement of $P$ from $O$, at time $t$ seconds after $P$ leaves $O$, is $s$ metres.

Kinematics of motion in a straight line

Particles A, of mass $0.26\,\text{kg}$, and B, of mass $0.52\,\text{kg}$, are fastened to the two ends of a light inextensible string. That string runs over a small smooth pulley $P$, which is fixed at the top of a smooth plane. The plane is tilted at an angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{16}{65}$ and $\cos\alpha = \frac{63}{65}$. A is kept at rest $2.5\,\text{m}$ from $P$, with the section $AP$ of the string parallel to a line of greatest slope of the plane. B is hanging freely below $P$ at a point $0.6\,\text{m}$ above the floor (see diagram). A is released and the particles begin to move.

Newton's laws of motion

A straight ice track with a length of $50\,\text{m}$ is tilted at $14^\circ$ to the horizontal. A man sets off from the top of the track on a sledge, with speed $8\,\text{m s}^{-1}$. He goes down the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is $0.02$.

Energy, work and power

Particles $A$, with mass $1.6\,\text{kg}$, and $B$, with mass $2\,\text{kg}$, are connected to the two ends of a light inextensible string. The string runs over a small smooth pulley fixed at the top of a smooth plane inclined at angle $\theta$, where $\sin \theta = 0.8$. Particle $A$ is initially held at rest at the lower end of the plane, while $B$ is suspended at a height of $3.24\,\text{m}$ above the level of the bottom of the plane (see diagram). $A$ is released from rest and the particles begin to move.

Energy, work and power

A car has mass $800\,\text{kg}$. Its engine supplies constant power $P\,\text{kW}$ as the car travels along a straight horizontal road. The resistive force opposing the motion is constant and equal to $R\,\text{N}$. At a speed of $14\,\text{m s}^{-1}$, the car’s acceleration is $1.4\,\text{m s}^{-2}$, and at a speed of $25\,\text{m s}^{-1}$, the acceleration is $0.33\,\text{m s}^{-2}$.

Forces and equilibrium

An aeroplane travels in a straight horizontal line along the runway before lifting off. It is at rest at $O$ and has speed $90\,\text{m s}^{-1}$ at the moment it takes off. At time $t$ seconds after leaving $O$, while it is still on the runway, its acceleration is $(1.5 + 0.012t)\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A particle $P$ is launched vertically upwards from a point on the ground at a speed of $17\,\text{m s}^{-1}$. A second particle $Q$ is launched vertically upwards from the same point at a speed of $7\,\text{m s}^{-1}$. Particle $Q$ is launched $T$ seconds after particle $P$.

Kinematics of motion in a straight line

A box of small size with mass $40\,\text{kg}$ is pulled across a rough horizontal floor by three men. Two of them exert horizontal forces of magnitudes $100\,\text{N}$ and $120\,\text{N}$, directed at angles of $30^\circ$ and $60^\circ$ respectively to the positive $x$-direction. The third man exerts a horizontal force of magnitude $F\,\text{N}$ at an angle of $\alpha^\circ$ to the negative $x$-direction (see diagram). The combined effect of the three horizontal forces on the box is a resultant in the positive $x$-direction with magnitude $136\,\text{N}$.

Forces and equilibrium

Particle $A$ has mass $1.26\,\text{kg}$ and particle $B$ has mass $0.9\,\text{kg}$. They are joined to the two ends of a light inextensible string. This string runs over a small smooth pulley $P$, which is fixed at the edge of a rough horizontal table. $A$ is initially held at rest $0.48\,\text{m}$ from $P$, while $B$ hangs vertically below $P$, with its height $0.45\,\text{m}$ above the floor (see diagram). The coefficient of friction between $A$ and the table is $\frac{2}{7}$. $A$ is released and the particles begin to move.

Newton's laws of motion

A small ball is launched from point $O$ on level ground with speed $20\,\text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal. After time $t$ from projection, its horizontal displacement from $O$ is $x$ m and its upward vertical displacement from $O$ is $y$ m.

Representation of data

A particle $P$ with mass $0.4\,\text{kg}$ is connected to one end of a light elastic string whose natural length is $1.2\,\text{m}$ and modulus of elasticity $19.2\,\text{N}$. The other end of the string is fixed at point $A$. Particle $P$ is released from rest at a point $2.7\,\text{m}$ vertically above $A$. Calculate

Probability

A uniform object $ABC$ is made from two rods $AB$ and $BC$ fixed rigidly together at right angles at $B$. Rod $AB$ is $0.3\,\text{m}$ long and rod $BC$ is $0.2\,\text{m}$ long. The object is in contact with a rough horizontal surface at $A$, with rod $AB$ standing vertically. It is kept in equilibrium by a horizontal force of magnitude $4\,\text{N}$ applied at $B$ and directed along $CB$ (see diagram).

Representation of data

A particle with mass $0.2\,\text{kg}$ is projected vertically downwards at an initial speed of $4\,\text{m s}^{-1}$. While it is descending, a resisting force of magnitude $0.09v\,\text{N}$ acts vertically upwards on the particle, where $v\,\text{m s}^{-1}$ represents the particle's downward velocity at time $t$ after it has been set in motion.

Probability

Particle $P$ is launched from point $O$ with speed $50\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal. At $2.5\,\text{s}$ after launch, calculate

Probability

A light inextensible string of length $0.2\,\text{m}$ has one end fastened to a fixed point $A$ positioned above a smooth horizontal table. The other end carries a particle $P$ of mass $0.3\,\text{kg}$. $P$ travels in a horizontal circle on the table, with the string kept taut and inclined at an angle of $60^\circ$ to the downward vertical (see diagram).

Probability

The cross-section $OABC$ passes through the centre of mass of a uniform prism whose weight is $20\,\text{N}$. This cross-section has the form of a sector of a circle with centre $O$, radius $OA = r\,\text{m}$ and angle $AOC = \frac{2}{3}\pi$ radians. The prism rests on a plane inclined at an angle $\theta$ radians to the horizontal, where $\theta < \tfrac{1}{3}\pi$. The line $OC$ is along a line of greatest slope, with $O$ higher than $C$. The prism is freely hinged to the plane at $O$. A force of magnitude $15\,\text{N}$ acts at $A$, directed towards the plane and at right angles to it (see diagram).

Representation of data

A sphere of mass $0.4\,\text{kg}$ travels at constant speed $1.5\,\text{m s}^{-1}$ round a horizontal circle inside a smooth fixed hollow cylinder with diameter $0.6\,\text{m}$. The cylinder’s axis is vertical, and the sphere touches both the horizontal base and the vertical curved surface of the cylinder.

Probability

A uniform semicircular lamina with radius $0.25\,\text{m}$ has diameter $AB$. It is hung freely from a fixed point at $A$ and comes to equilibrium.

Representation of data

A particle $P$ with mass $0.2\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $1.6\,\text{m}$ and modulus of elasticity is $18\,\text{N}$. The opposite end of the string is fixed at point $O$, which lies $1.6\,\text{m}$ vertically above a smooth horizontal surface. $P$ is placed on the surface directly below $O$ and then projected horizontally. $P$ travels in a straight line along the surface with initial speed $1.5\,\text{m s}^{-1}$. Show that, when $OP = 1.8\,\text{m}$,

Probability

A ball $B$ is launched from point $O$ on horizontal ground at an angle of $40^{\circ}$ above the horizontal. $B$ lands on the ground $1.8\,\text{s}$ after projection. Calculate

Representation of data

A block $B$ with mass $3\,\text{kg}$ is fastened to one end of a light elastic string whose modulus of elasticity is $70\,\text{N}$ and whose natural length is $1.4\,\text{m}$. The opposite end of the string is connected to a particle $P$ of mass $0.3\,\text{kg}$. $B$ is stationary $0.9\,\text{m}$ from the edge of a horizontal table, and the string passes over a small frictionless pulley at the table edge. $P$ is released from rest beside the pulley and moves vertically downwards. At the first moment when $P$ is $0.8\,\text{m}$ below the pulley and still descending, $B$ is in limiting equilibrium with the part of the string attached to $B$ horizontal (see diagram).

Probability

A uniform solid cone, with height $0.6\,\text{m}$ and mass $0.5\,\text{kg}$, is arranged so that its axis of symmetry is vertical and its vertex $V$ is at the top. The cone has a semi-vertical angle of $60^{\circ}$ and its surface is smooth. It is fixed on a horizontal surface. A particle $P$ of mass $0.2\,\text{kg}$ is attached to $V$ by a light inextensible string of length $0.4\,\text{m}$ (see diagram).

Representation of data

A particle $P$ with mass $0.5\,\text{kg}$ travels along a straight line on a smooth horizontal plane. When the displacement of $P$ from $O$ is $x\,\text{m}$, its velocity is $v\,\text{m s}^{-1}$. A lone horizontal force of size $0.16e^{x}\,\text{N}$ acts on $P$ in the direction $OP$. The speed of $P$ at $O$ is $0.8\,\text{m s}^{-1}$.

Probability

A particle $P$ is projected at $15\,\text{m s}^{-1}$ at $60^\circ$ above the horizontal. Determine the direction of motion of $P$ at the instant $0.9\,\text{s}$ after projection.

Representation of data