Oct/Nov 2016

Write $x^2 + 6x + 2$ in the form $(x + a)^2 + b$, with $a$ and $b$ as constants.

Quadratics

The curve is defined by $y = f(x)$, and it is stated that $f'(x) = 3x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}$. Point $A$ is the sole point on the curve where the gradient equals $-1$.

Coordinate geometry

Point $P(3, 5)$ belongs to the curve $y = \frac{1}{x - 1} - \frac{9}{x - 5}$.

Differentiation

Find the term that does not contain $x$ in the expansion of $(2x + \frac{1}{2x^3})^8$.

Series

The diagram shows $OCA$ and $ODB$ as radii of a circle whose centre is $O$ and whose radius is $2r$ cm. $ ngle AOB = \alpha$ radians. $CD$ and $AB$ are arcs of circles with centre $O$ and radii $r$ cm and $2r$ cm respectively. The perimeter of the shaded region $ABDC$ equals $4.4r$ cm.

Circular measure

$C$ is the midpoint of the segment from $A(14, -7)$ to $B(-6, 3)$. The line passing through $C$ and perpendicular to $AB$ meets the $y$-axis at $D$.

Coordinate geometry

For a geometric progression, the combined value of the 1st and 2nd terms is $50$, and the combined value of the 2nd and 3rd terms is $30$. Determine the sum to infinity.

Series

Show that, after expansion, $\cos^4 x = 1 - 2\sin^2 x + \sin^4 x$.

Trigonometry

The diagram shows a section of the curves $y = (2x - 1)^2$ and $y^2 = 1 - 2x$, which intersect at the points $A$ and $B$.

Coordinate geometry

The functions $f$ and $g$ are specified by $f(x) = \frac{4}{x} - 2$ for $x > 0$, and $g(x) = \frac{4}{5x + 2}$ for $x \geq 0$.

Functions

The diagram depicts a cuboid $OABCDEFG$ with a horizontal base $OABC$, where $OA = 4$ cm and $AB = 15$ cm. The cuboid has height $2$ cm. Point $X$ lies on $AB$ in such a way that $AX = 5$ cm, and point $P$ lies on $DG$ so that $DP = p$ cm, where $p$ is a constant. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively.

Coordinate geometry

The curve satisfies $\frac{dy}{dx} = \frac{8}{\sqrt{4x + 1}}$, and the point $(2, 5)$ is on it.

Integration

The function $f$ is defined as $f : x \mapsto 5 - 2\sin 2x$ on $0 \leq x \leq \pi$.

Trigonometry

Rewrite the equation $\sin 2x + 3\cos 2x = 3(\sin 2x - \cos 2x)$ so that it takes the form $\tan 2x = k$, where $k$ is a constant.

Trigonometry

The curve is given by $y = 2x^2 - 6x + 5$.

Differentiation

When $(3 - 2x)\left(1 + \frac{x}{2}\right)^n$ is expanded, the coefficient of $x$ is $7$.

Series

For the line $\frac{x}{a} + \frac{y}{b} = 1$, with $a$ and $b$ taken as positive constants, the $x$- and $y$-intercepts are $A$ and $B$ respectively. The midpoint of $AB$ is on the line $2x + y = 10$, and $AB = 10$.

Coordinate geometry

The diagram depicts a metal plate $ABCD$ formed from two separate parts. The part $BCD$ is a semicircle. The part $DAB$ is a segment of a circle with centre $O$ and radius $10\,\text{cm}$. Angle $BOD$ is $1.2$ radians.

Circular measure

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

Differentiation

A cyclist takes part in a long-distance charity ride across Africa. The overall distance is $3050\,\text{km}$. He begins on May $1$st and rides $200\,\text{km}$ that day. After that, the distance he rides each day goes down by $5\,\text{km}$. How far will he travel on May $15$th?

Series

Taking the origin $O$ as the reference, the position vectors of the points $A$, $B$ and $C$ are $overrightarrow{OA} = \begin{pmatrix}2 \\ -2 \\ -1\end{pmatrix}$, $overrightarrow{OB} = \begin{pmatrix}-2 \\ 3 \\ 6\end{pmatrix}$ and $overrightarrow{OC} = \begin{pmatrix}2 \\ 6 \\ 5\end{pmatrix}$.

Coordinate geometry

Determine the values of $k$ for which the curve $y = kx^2 - 3x$ and the line $y = x - k$ have no points in common.

Quadratics

A curve is given by $\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-3/2}$, with $a$ a positive constant. The point $A\,(a^2, 3)$ is on this curve.

Differentiation

A curve is given by the equation $y = (kx - 3)^{-1} + (kx - 3)$, where $k$ is a non-zero constant.

Integration

In the expansion of $(1 - 3x)^6 + (1 + ax)^5$, the coefficient of $x^3$ is $100$. Determine the value of the constant $a$.

Series

By presenting every required step,

Trigonometry

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 9x + 2$ for $x > n$, where $n$ is an integer. It is stated that $f$ is an increasing function.

Series

The diagram depicts the major arc $AB$ of a circle centred at $O$ with radius $6\text{ cm}$. The points $C$ and $D$ lie on $OA$ and $OB$ respectively, and the line $AB$ is tangent at $E$ to the arc $CED$ of a smaller circle that is also centred at $O$. Angle $COD = 1.8$ radians.

Circular measure

Points $A$, $B$ and $C$ are given such that $B$ is the mid-point of $AC$. Point $A$ has coordinates $(2, m)$ and point $B$ has coordinates $(n, -6)$, where $m$ and $n$ are constants. The straight line $y = x + 1$ goes through $C$ and is perpendicular to $AB$.

Coordinate geometry

The diagram depicts a triangular pyramid $ABCD$. It is given that $\vec{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}$, $\vec{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k}$ and $\vec{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}$.

Coordinate geometry

Write $4x^2 + 12x + 10$ in the form $(ax + b)^2 + c$, with $a$, $b$ and $c$ as constants.

Functions

Two convergent geometric progressions, $P$ and $Q$, share the same sum to infinity. The first and second terms of $P$ are $6$ and $6r$ respectively. The first and second terms of $Q$ are $12$ and $-12r$ respectively. Find the common sum to infinity.

Series

The equation $3^{2x} = 5(3^x) + 14$ is satisfied by $x$.

Logarithmic and exponential functions

The variables $x$ and $y$ are linked by the equation $y = Ae^{px}$, where $A$ and $p$ are constants. A plot of $\ln y$ against $x$ gives a straight line that goes through the points $(5, 3.17)$ and $(10, 4.77)$, as the diagram shows.

Logarithmic and exponential functions

The curve is given by $y = 2\sin 2x - 5\cos 2x + 6$ for $0 \le x \le \pi$.

Differentiation

The positive constant $a$ is such that $\int_{-a}^{a} \left(4e^{2x} + 5\right)\,dx = 100$.

Numerical solution of equations

Show that $\frac{\cos 2x + 9\cos x + 5}{\cos x + 4} = 2\cos x + 1$.

Integration

The curve is defined by $3x^2 + 4xy + y^2 = 24$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = ax^3 + 3x^2 + bx + 12$, with $a$ and $b$ as constants. It is stated that $(x + 3)$ is a factor of $p(x)$, and that the remainder when $p(x)$ is divided by $(x + 2)$ is $18$.

Trigonometry

Find the values of $x$ that satisfy $|0.4x - 0.8| = 2$.

Algebra

If $\frac{1 + 4y}{3 + 2^y} = 5$, determine the value of $2^y$.

Logarithmic and exponential functions

The definite integral $I$ is given by $I = \int_0^2 (4e^{\frac{1}{2}x} + 3) \, dx$.

Integration

The polynomial $p(x)$ is defined as $p(x) = 4x^3 + ax^2 + ax + 4$, with $a$ a constant.

Algebra

The graph illustrates the curve $y = \frac{4 \ln x}{x^2 + 1}$ together with its stationary point $M$. Let the $x$-coordinate of $M$ be $m$.

Numerical solution of equations

Show that $\frac{\cos 2\theta}{1 + \cos 2\theta}$ can be written as $1 - \frac{1}{2} \sec^2 \theta$.

Trigonometry

The diagram represents the curve given by the parametric equations $x = 4 \sin \theta$, $y = 1 + 3 \cos(\theta + \frac{1}{6}\pi)$ for $0 \leq \theta < 2\pi$.

Differentiation

The values generated by the iterative relation $x_{n+1} = \frac{4}{x_n} + \frac{2x_n}{3}$, starting from $x_1 = 2$, approach $\alpha$.

Numerical solution of equations

x and y are linked by $y = Kx^p$, where $K$ and $p$ are constants. As shown in the diagram, the graph of $\ln y$ against $\ln x$ is a straight line that passes through $(1.28,\,3.69)$ and $(2.11,\,4.81)$.

Numerical solution of equations

Find the value of $\int \tan^2 4x \, dx$.

Integration

The polynomial $p(x)$ is given by $p(x) = ax^3 + 3x^2 + 4ax - 5$, where $a$ is a constant. It is stated that $(2x - 1)$ is a factor of $p(x)$.

Logarithmic and exponential functions

The graph displays the curve $y = \sqrt{1 + e^{\frac{1}{3}x}}$ for $0 \leq x \leq 6$. The area enclosed by the curve and the lines $x = 0$, $x = 6$ and $y = 0$ is called $R$.

Integration

The curve is given by the parametric equations $x = \ln (t + 1)$ and $y = t^2 \ln t$.

Differentiation

Express $\sin 2\theta (3\sec \theta + 4\cosec \theta)$ as $a \sin \theta + b \cos \theta$, with $a$ and $b$ integers.

Trigonometry

Solve the equation $\frac{3^x + 2}{3^x - 2} = 8$, and give your answer correct to 3 decimal places.

Logarithmic and exponential functions

A diseased soil infection is spreading across a field with total area $4\text{ km}^2$. After $t$ years, the infected region has area $x\text{ km}^2$, and its growth rate is described by $\frac{dx}{dt} = kx(4 - x)$, where $k$ is a positive constant. It is given that $x = 0.4$ when $t = 0$, and that $x = 2$ when $t = 2$.

Differential equations

Expand $(2 - x)(1 + 2x)^{-\frac{3}{2}}$ into ascending powers of $x$, retaining terms up to and including $x^2$, and simplify the coefficients.

Algebra

Express the equation $\sec \theta = 3 \cos \theta + \tan \theta$ as a quadratic in $\sin \theta$. Hence solve it for $-90^\circ < \theta < 90^\circ$.

Trigonometry

The curve is defined by $xy(x - 6y) = 9a^3$, where $a$ is a non-zero constant.

Differentiation

Prove that the identity $\tan 2\theta - \tan \theta \equiv \tan \theta \sec 2\theta$ holds.

Integration

By drawing a suitable pair of graphs, show that the equation $\csc \frac{1}{2}x = \frac{1}{3}x + 1$ has a single root in the interval $0 < x \leq \pi$.

Numerical solution of equations

The diagram depicts a section of the curve $y = (2x - x^2)e^{\frac{1}{2}x}$ together with its maximum point $M$.

Integration

The two planes are given by $3x + y - z = 2$ and $x - y + 2z = 3$.

Vectors

For this question, calculator use is not allowed.

Complex numbers

Solve the equation $\frac{3^x + 2}{3^x - 2} = 8$, and give your answer correct to 3 decimal places.

Logarithmic and exponential functions

A broad field with area $4\,\text{km}^2$ is being affected by a soil disease. At time $t$ years, the diseased area is $x\,\text{km}^2$, and its growth rate is described by the differential equation $\frac{dx}{dt} = kx(4 - x)$, where $k$ is a positive constant. It is given that when $t = 0$, $x = 0.4$ and that when $t = 2$, $x = 2$.

Differential equations

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

Algebra

Rewrite $\sec \theta = 3 \cos \theta + \tan \theta$ as a quadratic equation in $\sin \theta$. Hence solve the equation for $-90^\circ < \theta < 90^\circ$.

Trigonometry

The curve is described by the equation $x y (x - 6y) = 9 a^3$, where $a$ is a non-zero constant.

Differentiation

Prove that $\tan 2\theta - \tan \theta \equiv \tan \theta \sec 2\theta$.

Integration

By sketching an appropriate pair of graphs, show that the equation $\cosec \frac{x}{2} = \frac{1}{3}x + 1$ has one root in the interval $0 < x \leq \pi$.

Numerical solution of equations

The diagram displays a section of the curve $y = (2x - x^2) e^{\frac{1}{2}x}$ and its maximum point $M$.

Integration

The equations for two planes are $3x + y - z = 2$ and $x - y + 2z = 3$.

Vectors

No calculator may be used anywhere in this question.

Complex numbers

You are told that $z = \ln(y + 2) - \ln(y + 1)$. Write $y$ in terms of $z$.

Logarithmic and exponential functions

The line $l$ is represented by the vector equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + \mathbf{k})$.

Vectors

A curve is given by $y = \dfrac{\sin x}{1 + \cos x}$, where $-\pi < x < \pi$. Show that the gradient of the curve is positive for every $x$ in the stated interval.

Differentiation

Rewrite the equation $\cot 2\theta = 1 + \tan \theta$ as a quadratic in $\tan \theta$. Hence solve it for $0^{\circ} < \theta < 180^{\circ}$.

Trigonometry

Let $p(x)$ denote the polynomial $4x^4 + ax^2 + 11x + b$, where $a$ and $b$ are constants. It is stated that $p(x)$ is divisible by $x^2 - x + 2$.

Algebra

The diagram shows a moving point $P$ at $(x, y)$ and the point $N$, which is the foot of the perpendicular from $P$ to the $x$-axis. $P$ lies on a curve for which, whenever $x > 0$, the gradient of the curve is equal to the area of triangle $OPN$, where $O$ is the origin. The point $(0, 2)$ is on the curve.

Differential equations

Define $I = \displaystyle\int_{1}^{4} \dfrac{(\sqrt{x}) - 1}{2(x + \sqrt{x})} \, dx$.

Integration

You must not use a calculator anywhere in this question. The complex number $z$ is given by $z = (\sqrt{2}) - (\sqrt{6})i$. We denote the complex conjugate of $z$ by $z^*$.

Complex numbers

Define $f(x)$ by $f(x) = \dfrac{3x^2 + x + 6}{(x + 2)(x^2 + 4)}$.

Algebra

The sketch displays the curves $y = x \cos x$ and $y = \dfrac{k}{x}$, with $k$ a constant, for $0 < x \le \dfrac{\pi}{2}$. These two curves meet at the point for which $x = a$.

Numerical solution of equations

Particles $P$ and $Q$, with masses $0.6\,\text{kg}$ and $0.4\,\text{kg}$ respectively, are joined by a light inextensible string. This string runs over a small smooth light pulley that is fixed at the end of a smooth horizontal table. At first, $P$ is held stationary on the table while $Q$ hangs vertically (see diagram). $P$ is then released.

Newton's laws of motion

A particle with mass $0.1\,\text{kg}$ is released from rest on a rough plane that is inclined at $20^{\circ}$ to the horizontal. It is stated that, $5$ seconds after release, the particle is moving at $2\,\text{m s}^{-1}$.

Newton's laws of motion

Particle $P$ is launched vertically upwards from point $O$. At a height of $0.5\,\text{m}$, its speed is $6\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

At the point $P$, three coplanar forces of magnitudes $F\,\text{N}$, $2F\,\text{N}$ and $15\,\text{N}$ act, as shown in the diagram. Since the forces are in equilibrium,

Forces and equilibrium

The diagram depicts a velocity-time graph representing the motion of a cyclist. It is made up of five straight-line sections. Starting from rest, the cyclist speeds up to $5\,\text{m s}^{-1}$ in $10\,\text{s}$, then continues at this constant speed for another $20\,\text{s}$. The cyclist then goes down a hill and accelerates to speed $V\,\text{m s}^{-1}$ over $10\,\text{s}$. That speed is then kept for a further $30\,\text{s}$ before the cyclist slows to rest over $20\,\text{s}$.

Energy, work and power

A block with mass $25\,\text{kg}$ is dragged over level ground by a force of size $50\,\text{N}$ that acts at $10^{\circ}$ above the horizontal. It begins at rest and moves $20\,\text{m}$. A constant resistive force of size $30\,\text{N}$ opposes the motion.

Energy, work and power

A racing car travels along a straight path. Its acceleration $a\,\text{m s}^{-2}$ at time $t\,\text{s}$ after it leaves rest is given by $a = 15t - 3t^2$ for $0 \le t \le 5$, and by $a = \dfrac{625}{t^2}$ for $5 < t \le k$, where $k$ is a constant.

Kinematics of motion in a straight line

A particle with mass $2\text{ kg}$ starts from rest on a rough horizontal plane. A force of size $10\text{ N}$ acts on the particle at $15^\circ$ above the horizontal. It is stated that $10\text{ s}$ after the force is applied, the particle has a speed of $3.5\text{ m s}^{-1}$.

Forces and equilibrium

A particle travels along a straight line. Its displacement $t$ s after leaving a fixed point $O$ on the line is $s\text{ m}$, where $s = 2t^2 - \frac{80}{3}t^{\frac{3}{2}}$.

Kinematics of motion in a straight line

Two people are towing a boat along a river. One walks on a path beside one bank, while the other walks on a path on the opposite bank. The first person applies a horizontal force of $60\text{ N}$ at an angle of $25^\circ$ to the river’s direction. The second person applies a horizontal force of $50\text{ N}$ at an angle of $15^\circ$ to the river’s direction (see diagram).

Forces and equilibrium

A girl riding a sledge begins at the top of a slope that is 100 m long and inclined at an angle of 20^\circ to the horizontal, with a speed of $5\text{ m s}^{-1}$. The sledge moves straight down the slope.

Energy, work and power

A particle with mass $m\text{ kg}$ is at rest on a rough plane that is inclined at $30^\circ$ to the horizontal. A force of magnitude $10\text{ N}$ acting on the particle up a line of greatest slope of the plane is just enough to prevent the particle from sliding down the plane. When a force of $75\text{ N}$ acts on the particle up a line of greatest slope of the plane, the particle is just about to slide up the plane.

Forces and equilibrium

A van with mass $3000\text{ kg}$ is towing a trailer of mass $500\text{ kg}$ on a straight, level road at a steady speed of $25\text{ m s}^{-1}$. The van-and-trailer arrangement is represented by two particles joined by a light inextensible cable. The van experiences a constant resistive force of $300\text{ N}$ and the trailer a constant resistive force of $100\text{ N}$.

Energy, work and power

A car sets off from rest and travels in a straight line from point $A$ with constant acceleration $3\text{ m s}^{-2}$ for $10\text{ s}$. It then moves at a constant speed for $30\text{ s}$ before slowing down uniformly and stopping at point $B$. The distance $AB$ is $1.5\text{ km}$.

Kinematics of motion in a straight line

A crane lifts a block with mass $50\,\text{kg}$ straight upward at constant speed through a vertical distance of $3.5\,\text{m}$. A steady resistive force of $25\,\text{N}$ acts against the motion.

Energy, work and power

The diagram depicts a small object $P$ of mass $20\,\text{kg}$, supported in equilibrium by light ropes fixed at points $A$ and $B$. The rope $PA$ is at an angle of $50^\circ$ above the horizontal, while the rope $PB$ is at an angle of $10^\circ$ below the horizontal, and the two ropes lie in the same vertical plane.

Forces and equilibrium

Particles $P$ and $Q$, whose masses are $7\,\text{kg}$ and $3\,\text{kg}$ respectively, are secured to the two ends of a light inextensible string. The string goes over two small smooth pulleys fixed at the two ends of a horizontal table. Both particles hang vertically beneath the pulleys. Initially, both particles are at rest, $0.5\,\text{m}$ below table level and $0.4\,\text{m}$ above the horizontal floor (see diagram).

Newton's laws of motion

Ball $A$ is let go from rest at the top of a tall tower. After $1$ second, ball $B$ is projected vertically upwards from ground level close to the base of the tower at a speed of $20\,\text{m s}^{-1}$. The balls are at the same height $1.5\,\text{s}$ after ball $B$ is projected.

Kinematics of motion in a straight line

Particle P begins at fixed point O and travels along a straight line. For time t\,\text{s} after leaving O, its velocity v\,\text{m s}^{-1} is given by v = 6t - 0.3t^2. The particle then stops momentarily at X.

Kinematics of motion in a straight line

A cyclist travels on a horizontal straight road with a constant power output of $160\,\text{W}$. The resistive force opposing the motion is constant at $20\,\text{N}$. At the moment when the cyclist’s speed is $5\,\text{m s}^{-1}$, the acceleration is $0.15\,\text{m s}^{-2}$.

Energy, work and power

A box of mass $50\,\text{kg}$ is stationary on a plane that is inclined at $10^\circ$ to the horizontal.

Energy, work and power

Particle $P$, with mass $0.3\,\text{kg}$, is moving round a circle centred at $O$ on a smooth horizontal surface. $P$ is linked to $O$ by a light elastic string whose modulus of elasticity is $12\,\text{N}$ and whose natural length is $l\,\text{m}$. The speed of $P$ is $4\,\text{m s}^{-1}$, and the circle it describes has radius $2l\,\text{m}$.

Probability

A wire of uniform density is shaped into a figure made up of a semicircular arc whose diameter $AB$ measures $1.2\,\text{m}$, together with a smaller semicircular arc whose diameter $BC$ measures $0.6\,\text{m}$. The endpoint $C$ of the smaller arc is located at the centre of the larger arc (refer to the diagram). Both semicircular parts of the wire lie in one plane.

Representation of data

A block $B$ of mass $0.25\,\text{kg}$ is released from rest at $O$ on a smooth horizontal surface. Once it has been released, its velocity is $v\,\text{m s}^{-1}$ when its displacement from $O$ is $x\,\text{m}$. The force on $B$ has magnitude $(2 + 0.3x^2)\,\text{N}$ and acts horizontally away from $O$.

Representation of data

The figure presents the cross-section $ABCD$ passing through the centre of mass of a uniform solid prism. $AB = 0.9\,\text{m}$, $BC = 2a\,\text{m}$, $AD = a\,\text{m}$ and angle $ABC =$ angle $BAD = 90^\circ$.

Representation of data

A small ball $B$ of mass $0.4\,\text{kg}$ travels in a horizontal circle of radius $0.6\,\text{m}$ about centre $O$ on a smooth horizontal surface. One end of a light inextensible string is fixed to $B$; the other end is fastened to a point $0.45\,\text{m}$ vertically above $O$.

Probability

The diagram depicts a smooth narrow tube bent into a fixed vertical circle of radius $0.9\,\text{m}$ and centre $O$. One end of a light elastic string, whose modulus of elasticity is $8\,\text{N}$ and natural length is $1.2\,\text{m}$, is fixed to the highest point $A$ on the inside of the tube. The other end is attached to a particle $P$ of mass $0.2\,\text{kg}$. The particle is released from rest at the lowest point on the inside of the tube. Using energy, calculate

Probability

A particle $P$ is fired from a point $O$ on a horizontal plane with speed $35\,\text{m s}^{-1}$. During its motion, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical upward displacement from $O$ is $y\,\text{m}$. The path of $P$ is given by $y = kx - \frac{(1 + k^2)x^2}{245}$, where $k$ is a constant. $P$ goes through the points $A(14, a)$ and $B(42, 2a)$, where $a$ is a constant.

Representation of data

A stone $S$ is projected horizontally from the top $T$ of a tall tower. $1.6\,\text{s}$ after $S$ is released, the line $ST$ is at an angle of $30^\circ$ below the horizontal.

Representation of data

A particle $P$ with mass $0.5\,\text{kg}$ is fixed to one end of a light elastic string whose modulus of elasticity is $24\,\text{N}$ and whose natural length is $0.6\,\text{m}$. The far end of the string is fastened to a stationary point $A$. In equilibrium, the particle $P$ is suspended vertically beneath $A$.

Probability

A non-uniform rod $AB$ with length $0.5\,\text{m}$ is joined by a free hinge at the fixed point $A$. The rod is balanced at an angle of $30^\circ$ to the horizontal, with $B$ lower than $A$. It is kept in equilibrium by a force of size $F\,\text{N}$ applied at $B$, acting at $45^\circ$ above the horizontal in the vertical plane containing $AB$. The force from the hinge on the rod is $10\,\text{N}$ and acts at an angle of $60^\circ$ above the horizontal (see diagram).

Representation of data

A particle $P$ is launched from point $O$ on horizontal ground with speed $20\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. $P$ then rebounds when it first hits the ground at point $A$.

Probability

Particle $P$, with mass $0.4\,\text{kg}$, is let go from rest at point $O$ on a smooth plane that is inclined at $30^\circ$ to the horizontal. A force of magnitude $3e^{-t}\,\text{N}$, acting up the line of greatest slope, is applied to $P$, where $t$ is the time after release.

Probability

The diagram represents the cross-section $ABCDEF$ through the centre of mass of a uniform prism that stands with $AB$ on rough horizontal ground. $ABCD$ is a rectangle for which $AB = CD = 0.4\,\text{m}$ and $BC = AD = 1.8\,\text{m}$. The remaining part of the cross-section is a semicircle whose diameter is $DF$ and whose radius is $r\,\text{m}$.

Representation of data

A small ball $B$ with mass $0.5\,\text{kg}$ travels round a horizontal circle of radius $0.4\,\text{m}$ and centre $O$ on the smooth inner face of a hollow cone, which is held with its vertex pointing downwards. The cone's axis is vertical and the semi-vertical angle is $60^\circ$ (see diagram).

Probability

A particle $P$ with mass $0.3\,\text{kg}$ travels round a circle centred at $O$ on a smooth horizontal surface. $P$ is connected to $O$ by a light elastic string of modulus of elasticity $12\,\text{N}$ and natural length $l\,\text{m}$. The particle moves at $4\,\text{m s}^{-1}$, and the radius of its circular path is $2l\,\text{m}$.

Probability

A uniform wire is bent into a shape consisting of a semicircular arc with diameter $AB$ and length $1.2\,\text{m}$, together with a smaller semicircular arc with diameter $BC$ and length $0.6\,\text{m}$. The point $C$, at the end of the smaller arc, lies at the centre of the larger arc (see diagram). Both semicircular arcs lie in the same plane.

Representation of data