Oct/Nov 2014

In the expansion of $(2 + ax)^7$, the coefficients of $x$ and $x^2$ are equal. Determine the value of the non-zero constant $a$.

Series

For $p \leq x \leq q$, where $p$ and $q$ are positive constants, the function $f$ is defined by $f : x \mapsto x^2 - 2x - 15$. Its range is stated as $c \leq f(x) \leq d$, where $c$ and $d$ are constants.

Quadratics

The diagram presents parts of the curves $y = \sqrt{4x + 1}$ and $y = \frac{1}{2}x^2 + 1$, which intersect at the points $P(0, 1)$ and $Q(2, 3)$. At $Q$, the angle between the tangents to the two curves is $\alpha$.

Integration

Determine the value of $x$ for which $\sin^{-1}(x - 1) = \tan^{-1}(3)$.

Trigonometry

Find the values of $\theta$ that satisfy $\frac{13\sin^2\theta}{2 + \cos\theta} + \cos\theta = 2$ for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

The line $4x + ky = 20$ goes through the points $A(8, -4)$ and $B(b, 2b)$, with $k$ and $b$ as constants.

Coordinate geometry

Find the set of values of $k$ for which the line $y = 2x - k$ meets the curve $y = x^2 + kx - 2$ at two distinct points.

Quadratics

Relative to the origin $O$, the position vector of $A$ is $3\mathbf{i} + 2\mathbf{j} - \mathbf{k}$, whereas the position vector of $B$ is $7\mathbf{i} - 3\mathbf{j} + \mathbf{k}$.

Coordinate geometry

A geometric progression starts with first term $a$ $(a \neq 0)$ and common ratio $r$, and its sum to infinity is $S$. A second geometric progression starts with the same first term $a$ but has common ratio $2r$ and sum to infinity $3S$. Determine the value of $r$.

Series

The diagram shows $AB$ as an arc of a circle centred at $O$ with radius $4\,\text{cm}$. The angle $AOB$ is $\alpha$ radians. Point $D$ lies on $OB$ so that $AD$ is perpendicular to $OB$. The arc $DC$, also centred at $O$, intersects $OA$ at $C$.

Circular measure

For $x > 0$, the function $f$ satisfies $f'(x) = 2x - \frac{2}{x^2}$. Its graph, $y = f(x)$, goes through the point $P(2, 6)$.

Differentiation

The diagram represents a section of the curve $y = x^2 + 1$.

Integration

The curve obeys $\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4$. It has a stationary point at $P$ for which $x = 2$.

Differentiation

The function $f : x \mapsto 6 - 4\cos\left(\frac{1}{2}x\right)$ is defined over the interval $0 \leq x \leq 2\pi$.

Functions

The diagram depicts triangle $AOB$, where $OA$ measures $12\,\text{cm}$, $OB$ measures $5\,\text{cm}$, and angle $AOB$ is a right angle. Point $P$ lies on $AB$, and $OP$ is an arc of a circle with centre $A$. Point $Q$ lies on $AB$, and $OQ$ is an arc of a circle with centre $B$.

Circular measure

Write down the first $3$ terms, arranged in ascending powers of $x$, in the expansion of $(1 + x)^5$.

Series

The curve is given by $y = \frac{12}{3 - 2x}$.

Differentiation

Demonstrate that $1 + \sin x \tan x = 5\cos x$ can be rearranged into $6\cos^2 x - \cos x - 1 = 0$.

Trigonometry

The curve is given by $y = x^3 + ax^2 + bx$, with $a$ and $b$ as constants.

Differentiation

The diagram represents pyramid $OABCX$. Its horizontal square base $OABC$ has side $8$ units, and the centre of the base is $D$. The apex, $X$, lies vertically above $D$ and $XD = 10$ units. $M$ is the mid-point of $OX$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OC$ respectively, while the unit vector $\mathbf{k}$ points vertically upwards.

Coordinate geometry

The sum $S_n$ of the first $n$ terms of an arithmetic progression is $S_n = 32n - n^2$. Find the first term and the common difference.

Series

The figure depicts trapezium $ABCD$ with $AB$ parallel to $DC$, and angle $BAD$ equal to $90^\circ$. Points $A$, $B$ and $C$ have coordinates $(2,6)$, $(5,-3)$ and $(8,3)$ respectively.

Coordinate geometry

Within the expansion of $(2 + ax)^6$, the coefficient of $x^2$ is the same as the coefficient of $x^3$.

Series

For $x \geq 0$, the functions are given by $f: x \mapsto (ax + b)^{\frac{1}{3}}$, where $a$ and $b$ are positive constants, and $g: x \mapsto x^2$. If $fg(1) = 2$ and $fg(9) = 16$, calculate the values of $a$ and $b$.

Functions

In the diagram, $OADC$ is a sector of a circle with centre $O$ and radius $3\,\text{cm}$. $AB$ and $CB$ are tangents to the circle, and angle $ABC = \frac{1}{3}\pi$ radians. Find, giving your answer in terms of $\sqrt{3}$ and $\pi$,

Integration

Express $9x^2 - 12x + 5$ in the form $(ax + b)^2 + c$.

Differentiation

The first three terms of an arithmetic progression are formed by the sums to infinity of three geometric progressions, $P$, $Q$ and $R$, in that order. Progression $P$ has terms $2, 1, \frac{1}{2}, \frac{1}{4}, \ldots$. Progression $Q$ has terms $3, 1, \frac{1}{3}, \frac{1}{9}, \ldots$.

Series

Show that $\sin^4 \theta - \cos^4 \theta \equiv 2\sin^2 \theta - 1$ by simplifying the left-hand side.

Trigonometry

$A$ denotes the point $(a, 2a - 1)$, while $B$ denotes the point $(2a + 4, 3a + 9)$, where $a$ is a constant.

Coordinate geometry

The points $O$, $A$ and $B$ are given by $\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + p\mathbf{k}$ and $\overrightarrow{OB} = -7\mathbf{i} + (1 - p)\mathbf{j} + p\mathbf{k}$, with $p$ as a constant.

Coordinate geometry

The graph of $y = f(x)$ has a stationary point at $(3, 7)$, and it is given that $f''(x) = 36x^{-3}$.

Differentiation

On the diagram, sections of the graphs of $y = x + 2$ and $y = 3\sqrt{x}$ meet at points $A$ and $B$.

Integration

Use the trapezium rule with four intervals to estimate $\int_{1}^{5} |2^{x} - 8|\,dx$.

Integration

The variables $x$ and $y$ are linked by $y = ab^{x}$, with $a$ and $b$ constant. A graph of $\ln y$ against $x$ is a straight line that goes through $(0.75, 1.70)$ and $(1.53, 2.18)$, as the diagram shows.

Numerical solution of equations

Find the value of $\int 4\cos^{2}\!\left(\tfrac{1}{2}\theta\right)\,d\theta$.

Integration

For each curve below, determine the exact gradient at the marked point:

Differentiation

Given that $(x + 2)$ and $(x + 3)$ are factors of $5x^{3} + ax^{2} + b$, determine the values of the constants $a$ and $b$.

Logarithmic and exponential functions

The diagram displays a segment of the curve $y = \frac{x^{2}}{1 + e^{3x}}$ together with its maximum point $M$. Let the $x$-coordinate of $M$ be $m$.

Numerical solution of equations

The angle $\alpha$ is between $0^{\circ}$ and $90^{\circ}$, and it satisfies $2\tan^{2}\alpha + \sec^{2}\alpha = 5 - 4\tan\alpha$.

Trigonometry

Solve $|3x - 1| = |2x + 5|$ for $x$.

Algebra

Evaluate $\int_0^a \left(e^{-x} + 6e^{-3x}\right)\,dx$, with $a$ a positive constant.

Integration

The equation of a curve is $3\ln x + 6xy + y^2 = 16$.

Differentiation

Determine the value of $x$ that satisfies the equation $2\ln(x - 4) - \ln x = \ln 2$.

Logarithmic and exponential functions

The diagram displays part of the curve $y = 2\cos x - \cos 2x$ together with its maximum point $M$. The shaded area is enclosed by the curve, the axes and the line through $M$ that is parallel to the $y$-axis.

Integration

The polynomial $p(x)$ is given by $p(x) = x^4 - 3x^3 + 3x^2 - 25x + 48$. The diagram depicts the curve $y = p(x)$, which meets the $x$-axis at $(\alpha, 0)$ and $(3, 0)$.

Numerical solution of equations

Write $5\cos \theta - 12\sin \theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$, and give $\alpha$ correct to 2 decimal places.

Trigonometry

Use the trapezium rule with four intervals to estimate $\int_{1}^{5} |2^x - 8| \, dx$.

Integration

The variables $x$ and $y$ are related by the equation $y = a(b^x)$, where $a$ and $b$ are constants. A graph of $\ln y$ plotted against $x$ is a straight line that passes through the points $(0.75, 1.70)$ and $(1.53, 2.18)$, as shown in the diagram.

Numerical solution of equations

Evaluate $\int 4\cos^2\left(\tfrac{1}{2}\theta\right) \, d\theta$.

Integration

For each curve shown below, determine the exact gradient at the marked point:

Differentiation

Since $(x + 2)$ and $(x + 3)$ are factors of $5x^3 + ax^2 + b$, determine the constants $a$ and $b$.

Logarithmic and exponential functions

The diagram depicts a section of the curve $y = \frac{x^2}{1 + e^{3x}}$ together with its maximum point $M$. The $x$-coordinate of $M$ is written as $m$.

Numerical solution of equations

For an angle $\alpha$ between $0^\circ$ and $90^\circ$, it is given that $2\tan^2 \alpha + \sec^2 \alpha = 5 - 4\tan \alpha$.

Trigonometry

Apply logarithms to solve the equation $e^x = 3^{x-2}$, with the answer given correct to 3 decimal places.

Logarithmic and exponential functions

The equation of line $l$ is $\mathbf{r} = 4\mathbf{i} - 9\mathbf{j} + 9\mathbf{k} + \lambda(-2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. Point $A$ is given by the position vector $3\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$.

Vectors

With 3 intervals, apply the trapezium rule to estimate the value of $\int_{\pi}^{\frac{3\pi}{2}} \cosec x\,dx$, and state your answer correct to 2 decimal places.

Numerical solution of equations

The polynomial $ax^3 + bx^2 + x + 3$, with $a$ and $b$ as constants, is written as $p(x)$.

Algebra

For the curve, the parametric equations are $x = \frac{1}{\cos^3 t}$ and $y = \tan^3 t$, where $0 \le t < \frac{pi}{2}$.

Differentiation

For this question, calculator use is not allowed. The complex numbers $w$ and $z$ satisfy the relation $w = \frac{z + i}{iz + 2}$.

Complex numbers

We are told that $\int_{1}^{a} \ln(2x)\,dx = 1$, with $a > 1$.

Numerical solution of equations

In one country, the government levies tax on every litre of petrol sold to motorists. The yearly revenue, in $R$ million dollars, depends on the tax rate $x$ dollars per litre. The relationship between $R$ and $x$ is represented by the differential equation $\frac{dR}{dx} = R\left(\frac{1}{x} - 0.57\right)$, with $R$ and $x$ treated as continuous variables. When $x = 0.5$, $R = 16.8$.

Differential equations

Starting from $\sin(2\theta + \theta)$, expand it to show that $\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$.

Trigonometry

Take $f(x) = \frac{x^2 - 8x + 9}{(1 - x)(2 - x)^2}$.

Algebra

Apply logarithms to solve the equation $e^x = 3^{x-2}$, and give your answer correct to $3$ decimal places.

Logarithmic and exponential functions

The line $l$ is given by equation $\mathbf{r} = 4\mathbf{i} - 9\mathbf{j} + 9\mathbf{k} + \lambda(-2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. Point $A$ has position vector $3\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$.

Vectors

Use the trapezium rule with $3$ intervals to estimate $\int_{\frac{\pi}{6}}^{\frac{2\pi}{3}} \cosec x\,dx$, and give your answer correct to $2$ decimal places.

Numerical solution of equations

The polynomial $ax^3 + bx^2 + x + 3$, with $a$ and $b$ as constants, is written as $p(x)$. You are told that $(3x + 1)$ is a factor of $p(x)$, and that the remainder when $p(x)$ is divided by $(x - 2)$ is $21$.

Algebra

The parametric form of a curve is given by $x = \frac{1}{\cos^3 t},\; y = \tan^3 t$, for $0 \le t < \frac{1}{2}\pi$.

Differentiation

For this question, you must not use a calculator. The complex numbers $w$ and $z$ satisfy the relation $w = \frac{z + i}{iz + 2}$.

Complex numbers

You are given that $\int_1^a \ln(2x)\,dx = 1$, with $a > 1$.

Numerical solution of equations

In one country, the government levies tax on every litre of petrol bought by motorists. The annual revenue is $R$ million dollars when the tax rate is $x$ dollars per litre. The relationship between $R$ and $x$ is represented by the differential equation $\frac{dR}{dx} = R\left(\frac{1}{x} - 0.57\right)$, with $R$ and $x$ treated as continuous variables. When $x = 0.5$, $R = 16.8$.

Differential equations

Expand $\sin(2\theta + \theta)$ first, and show that $\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$.

Trigonometry

Define $f(x)$ by $f(x) = \frac{x^2 - 8x + 9}{(1 - x)(2 - x)^2}$.

Algebra

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

Algebra

Begin by substituting $u = e^x$,

Integration

For $0 < \theta < \frac{1}{2}\pi$, the parametric equations defining the curve are $x = \tan \theta$ and $y = 2\cos^2 \theta \sin \theta$.

Differentiation

Let $p(x)$ be the polynomial $4x^3 + ax^2 + bx - 2$, with $a$ and $b$ as constants. It is stated that $(x + 1)$ and $(x + 2)$ are factors of $p(x)$.

Algebra

Demonstrate that $\cos(\theta - 60^\circ) + \cos(\theta + 60^\circ) = \cos \theta$.

Trigonometry

The complex numbers $w$ and $z$ are given by $w = 5 + 3i$ and $z = 4 + i$.

Complex numbers

You are given that $I = \int_{0}^{0.3} (1 + 3x^2)^{-2} \, dx$.

Integration

The two straight lines are given by $\mathbf{r} = \mathbf{i} + 4\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{k})$ and $\mathbf{r} = a\mathbf{i} + 2\mathbf{j} - 2\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} + 3a\mathbf{k})$, where $a$ is a constant.

Algebra

The variables $x$ and $y$ are linked by the differential equation $\dfrac{dy}{dx} = \frac{1}{5} x y^{\frac{1}{2}} \sin\!\left(\frac{1}{3}x\right)$.

Differential equations

Sketch the curve $y = \ln(x + 1)$ and then, by drawing a second curve, show that the equation $x^3 + \ln(x + 1) = 40$ has exactly one real root. State the equation for the second curve.

Numerical solution of equations

A car with mass 800 kg is travelling along a straight horizontal road, and its engine is doing work at a rate of 22.5 kW.

Newton's laws of motion

Two small blocks $A$ and $B$ are kept at rest on a smooth plane that is inclined at $30^\circ$ to the horizontal. Each block is maintained in equilibrium by a force of magnitude $18\,\text{N}$. For $A$, the force acts up the plane, parallel to a line of greatest slope; for $B$, the force acts horizontally in the vertical plane containing a line of greatest slope (see diagram).

Forces and equilibrium

A block weighing $7.5\,\text{N}$ is stationary on a plane inclined to the horizontal at angle $\alpha$, where $\tan \alpha = \frac{7}{24}$. The coefficient of friction between the block and the plane is $\mu$. A force of magnitude $7.2\,\text{N}$, acting parallel to a line of greatest slope, is applied to the block.

Forces and equilibrium

Particles $P$ and $Q$ travel along the straight line $AOB$. They both set off from $O$ at the same moment, with $P$ heading towards $A$ and $Q$ heading towards $B$. Initially, $P$ has speed $1.3\,\text{m s}^{-1}$ and its acceleration in the $OA$ direction is $0.1\,\text{m s}^{-2}$. $Q$ has acceleration towards $OB$ equal to $0.016t\,\text{m s}^{-2}$, where $t$ seconds is the time since $P$ and $Q$ began moving from $O$. When $t = 20$, particle $P$ reaches $A$ and particle $Q$ reaches $B$.

Kinematics of motion in a straight line

A light inextensible string has a small block $B$ of mass $0.25\,\text{kg}$ fixed at its midpoint. Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are attached to the two ends. The string runs over two smooth pulleys mounted at opposite sides of a rough table, and $B$ is in limiting equilibrium on the table between the pulleys, with particles $P$ and $Q$ and block $B$ all lying in the same vertical plane (see diagram).

Newton's laws of motion

A particle with mass $3\,\text{kg}$ is released from rest from a point $5\,\text{m}$ above the top surface of a liquid in a container. Its speed does not change instantaneously as it crosses into the liquid. The liquid in the container has a depth of $4\,\text{m}$. While the particle is travelling through the liquid, its downward acceleration is $5.5\,\text{m s}^{-2}$.

Forces and equilibrium

A block of mass $60\,\text{kg}$ is drawn up a hill along the line of greatest slope by a force of magnitude $50\,\text{N}$ acting at an angle $\alpha^\circ$ above the hill. The block moves through points $A$ and $B$ with speeds $8.5\,\text{m s}^{-1}$ and $3.5\,\text{m s}^{-1}$ respectively (see diagram). The separation $AB$ is $250\,\text{m}$ and $B$ is $17.5\,\text{m}$ above the level of $A$. The resistance to motion of the block is $6\,\text{N}$.

Energy, work and power

Particle $P$ is projected straight upwards with speed $11\,\text{m s}^{-1}$ from a point on level ground. At the same moment, particle $Q$ is let go from rest at a point $h$ m above the ground. $P$ and $Q$ reach the ground together, and when this happens $Q$ has speed $V\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

Three coplanar forces act at a point. Their magnitudes are $20\,\text{N}$, $25\,\text{N}$ and $30\,\text{N}$, and their directions are indicated in the diagram, where $\sin\alpha = 0.28$ and $\cos\alpha = 0.96$, and $\sin\beta = 0.6$ and $\cos\beta = 0.8$.

Forces and equilibrium

A train with mass $200\,000\,\text{kg}$ travels along a level straight track. It goes past point $A$ at $28\,\text{m s}^{-1}$ and then later passes point $B$. At $B$, the power of the train’s engine is $1.2$ times the power of the train’s engine at $A$. Also, the driving force of the train’s engine at $B$ is $0.96$ times the driving force of the train’s engine at $A$.

Energy, work and power

A block B, with mass 15 kg, is held in equilibrium against a horizontal surface between points A and C by forces of magnitudes X N and 40 N. The two forces lie in the same vertical plane, and their directions are shown in the diagram.

Forces and equilibrium

Particles $A$ and $B$, each with mass $0.3\,\text{kg}$, are joined by a light inextensible string. The string runs over a small smooth pulley fixed at the edge of a rough horizontal surface. Particle $A$ hangs freely while particle $B$ is kept at rest in contact with the surface. The coefficient of friction between $B$ and the surface is $0.7$. Particle $B$ is released and moves along the surface without reaching the pulley.

Energy, work and power

$ABC$ lies along the line of greatest slope on a plane inclined at angle $\alpha$ to the horizontal, with $\sin\alpha = 0.28$ and $\cos\alpha = 0.96$. Point $A$ is at the top of the plane, point $C$ is at the bottom of the plane, and $AC$ has length $5\,\text{m}$. The section of the plane above the level of $B$ is smooth, while the section below the level of $B$ is rough. A particle $P$ is released from rest at $A$ and reaches $C$ with speed $2\,\text{m s}^{-1}$. The coefficient of friction between $P$ and the part of the plane below $B$ is $0.5$.

Kinematics of motion in a straight line

The diagram presents the velocity-time graph for a particle $P$ moving along the straight line $BAC$. It leaves $A$ and reaches $B$ in $5\,\text{s}$. It then reverses direction and goes from $B$ to $C$ in $10\,\text{s}$. During the first $3\,\text{s}$ of $P$’s motion, the acceleration is constant. For the next $12\,\text{s}$, the velocity of $P$ at time $t\,\text{s}$ after leaving $A$ is $v\,\text{m s}^{-1}$, where $v = -0.2t^2 + 4t - 15$ for $3 \le t \le 15$.

Kinematics of motion in a straight line

A car with mass $1400\text{ kg}$ travels along a horizontal straight road. The resistive force opposing the car’s motion is constant at $800\text{ N}$, and the engine power stays constant at $P\text{ W}$. At the moment when the car’s speed is $18\text{ m s}^{-1}$, its acceleration is $0.5\text{ m s}^{-2}$. The car then goes on to another point where its speed is $25\text{ m s}^{-1}$.

Energy, work and power

The upper ends of two smooth inclined planes $A$ and $B$ meet at right angles. Plane $A$ makes an angle $\alpha$ with the horizontal, and plane $B$ makes an angle $\beta$ with the horizontal, where $\sin \alpha = \frac{63}{65}$ and $\sin \beta = \frac{16}{65}$. A small smooth pulley is fixed at the top of the planes, and a light inextensible string passes over it. Two particles $P$ and $Q$, each of mass $0.65\text{ kg}$, are connected to the string, one at each end. Particle $Q$ is held stationary at a point on the same line of greatest slope of plane $B$ as the pulley. Particle $P$ is hanging freely below the pulley in contact with plane $A$ (see diagram). Particle $Q$ is released, and the particles begin moving while the string stays taut.

Forces and equilibrium

Three light inextensible strings each have a particle fixed to one end. Their remaining ends are joined at a point $O$. Two of the strings go over fixed smooth pegs, and the particles are suspended freely in equilibrium. The weights of the particles together with the angles between the sloping sections of the strings and the vertical are indicated in the diagram. It is given that $\sin \beta = 0.8$ and $\cos \beta = 0.6$.

Forces and equilibrium

A particle P leaves from rest and travels in a straight line for 18 s. During the first 8 s, P has constant acceleration 0.25 m s^-2. After that, P’s velocity, v m s^-1 at time t seconds after the motion began, is given by v = -0.1t^2 + 2.4t - k, where 8 ≤ t ≤ 18 and k is a constant.

Kinematics of motion in a straight line

A box with mass $8\text{ kg}$ rests on a rough plane that is tilted at $5^\circ$ to the horizontal. A force of size $P\text{ N}$ is applied to the box, directed upwards and parallel to the line of greatest slope of the plane. For $P = 7X$, the box travels up the line of greatest slope with acceleration $0.15\text{ m s}^{-2}$, whereas for $P = 8X$, it travels up the line of greatest slope with acceleration $1.15\text{ m s}^{-2}$.

Forces and equilibrium

Particles $P$ and $Q$ together have a total mass of $1\text{ kg}$. They are connected by a light inextensible string that runs over a smooth fixed pulley. $P$ is kept at rest while $Q$ hangs freely, and both vertical sections of the string are straight. At the start, both particles are at a height of $h\text{ m}$ above the floor (see Fig. 1). $P$ is released from rest and the particles begin to move with the string taut. Fig. 2 gives the velocity-time graphs for the motion of $P$ and for the motion of $Q$, with vertically upwards taken as the positive direction for velocity.

Newton's laws of motion

A block of mass $3\text{ kg}$ is first at rest at the lower end $O$ of a rough plane inclined at angle $\alpha$ to the horizontal, with $\sin \alpha = 0.6$ and $\cos \alpha = 0.8$. A force of magnitude $35\text{ N}$ acts on the block at angle $\beta$ above the plane, where $\sin \beta = 0.28$ and $\cos \beta = 0.96$. The block begins to move up the line of greatest slope of the plane and reaches point $A$ with speed $4\text{ m s}^{-1}$. The distance $OA$ is $12.5\text{ m}$ (see diagram). When the block is at $A$, the force of magnitude $35\text{ N}$ stops acting.

Energy, work and power

Particle $P$ is projected from point $O$ on horizontal ground with speed $V\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. When $2\,\text{s}$ have passed since projection, $OP$ is at an angle of $15^\circ$ above the horizontal.

Representation of data

A uniform solid cone of height $0.8\,\text{m}$ and semi-vertical angle $30^\circ$ weighs $20\,\text{N}$. It is in equilibrium with one point $P$ on its base touching a rough horizontal plane, while the vertex $V$ is directly above $P$. A force of magnitude $F\,\text{N}$, acting along the cone’s axis of symmetry and applied at $V$, keeps it balanced (see diagram).

Representation of data

One end of a light elastic string with natural length $1.6\,\text{m}$ and modulus of elasticity $28\,\text{N}$ is fastened to a fixed point $O$. Its other end is connected to a particle $P$ of mass $0.35\,\text{kg}$, which is hanging at rest vertically beneath $O$. The particle $P$ is launched vertically upwards from the equilibrium position with speed $1.8\,\text{m s}^{-1}$.

Probability

The centre-of-mass cross-section of a uniform solid prism is $ABCDEF$. $ABCF$ is a rectangle with $AB = CF = 1.6\,\text{m}$ and $BC = AF = 0.4\,\text{m}$. $CDE$ is a triangle with $CD = 1.8\,\text{m}$, $CE = 0.4\,\text{m}$, and angle $DCE = 90^\circ$. The prism rests on a rough horizontal plane. A horizontal force of magnitude $T\,\text{N}$ is applied at $B$ in the direction $CB$ (see diagram). The prism is in equilibrium.

Representation of data

The path of a small ball $B$ launched from a fixed point $O$ is given by $y = -0.05x^2$, where $x$ and $y$ represent the horizontal and vertically upwards displacements of $B$ from $O$, in metres, respectively.

Representation of data

$O$, $A$ and $B$ are three points lying on one straight line on a smooth horizontal surface. A particle $P$ of mass $0.6\,\text{kg}$ travels along the line. At time $t\,\text{s}$, the particle’s displacement from $O$ is $x\,\text{m}$ and its speed is $v\,\text{m s}^{-1}$. The only horizontal force on $P$ has magnitude $0.4v^{\frac{1}{2}}\,\text{N}$ and acts in the direction $OA$. At the start, the particle is at $A$, where $x = 1$ and $v = 1$.

Representation of data

One end of a light elastic string with modulus of elasticity $15\,\text{N}$ has its top end fastened at point $A$, which lies $2\,\text{m}$ directly above a fixed small smooth ring $R$. The string’s natural length is $2\,\text{m}$ and it passes through $R$. The other end is connected to a particle $P$ of mass $m\,\text{kg}$, and $P$ travels with constant angular speed $\omega\,\text{rad s}^{-1}$ in a horizontal circle whose centre is $0.4\,\text{m}$ below the ring. $PR$ makes an acute angle $\theta$ with the vertical (see diagram).

Representation of data

A particle $P$ is launched from $O$ on horizontal ground with speed $V\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. After $2\,\text{s}$, the line $OP$ is inclined at $15^\circ$ above the horizontal.

Representation of data

A uniform solid cone of height $0.8\,\text{m}$ and semi-vertical angle $30^\circ$ weighs $20\,\text{N}$. It is in equilibrium, supported by just one point $P$ on its base touching a rough horizontal surface, with the vertex $V$ positioned vertically above $P$. The equilibrium is kept by a force of magnitude $F\,\text{N}$ acting along the cone’s axis of symmetry and applied at $V$ (see diagram).

Representation of data

One end of a light elastic string, with natural length $1.6\,\text{m}$ and modulus of elasticity $28\,\text{N}$, is fixed at a point $O$. The other end is joined to a particle $P$ of mass $0.35\,\text{kg}$, which is hanging in equilibrium vertically beneath $O$. The particle $P$ is projected vertically upwards from the equilibrium position with speed $1.8\,\text{m s}^{-1}$.

Probability

The uniform solid prism has cross-section $ABCDEF$ passing through its centre of mass. $ABCF$ is a rectangle with $AB = CF = 1.6\,\text{m}$ and $BC = AF = 0.4\,\text{m}$. $CDE$ is a triangle with $CD = 1.8\,\text{m}$, $CE = 0.4\,\text{m}$, and angle $DCE = 90^\circ$. The prism rests on a rough horizontal surface. A horizontal force of magnitude $T\,\text{N}$ is applied at $B$ in the direction $CB$ (see diagram). The prism is in equilibrium.

Representation of data

The trajectory equation for a small ball $B$, launched from a fixed point $O$, is $y = -0.05x^2$, where $x$ and $y$ are the horizontal and vertically upward displacements, in metres, of $B$ from $O$, respectively.

Representation of data

Points $O$, $A$ and $B$ lie on one straight line on a smooth horizontal plane. A particle $P$ with mass $0.6\,\text{kg}$ travels along the line. At time $t\,\text{s}$, its displacement from $O$ is $x\,\text{m}$ and its speed is $v\,\text{m s}^{-1}$. The only horizontal force acting on $P$ has magnitude $0.4v^{\frac{1}{2}}\,\text{N}$ and is directed along $OA$. At the beginning, the particle is at $A$, so $x = 1$ and $v = 1$.

Representation of data

One end of a light elastic string, whose modulus of elasticity is $15\,\text{N}$, is fastened to a fixed point $A$, which lies $2\,\text{m}$ directly above a fixed small smooth ring $R$. The string’s natural length is $2\,\text{m}$, and it passes through $R$. Its other end is attached to a particle $P$ of mass $m\,\text{kg}$, and $P$ travels at constant angular speed $\omega\,\text{rad s}^{-1}$ around a horizontal circle with centre $0.4\,\text{m}$ vertically below the ring. $PR$ forms an acute angle $\theta$ with the vertical (see diagram).

Probability

A golf ball $B$ is projected from a point $O$ on horizontal ground. $B$ first lands on the ground at a point $48\,\text{m}$ from $O$, $2.4\,\text{s}$ after being projected.

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 $0.8\,\text{m}$ and whose modulus of elasticity is $64\,\text{N}$. The opposite end of the string is fixed at point $A$ on a smooth horizontal surface. Particle $P$ is positioned on the surface at a point $0.8\,\text{m}$ from $A$. It is then given a speed of $10\,\text{m s}^{-1}$ in a direction directly away from $A$.

Probability