Feb/March 2016

Determine the coefficients of $x^4$ and $x^5$ in the expansion of $(1 - 2x)^5$.

Series

The diagram shows a section of the curve $y = \frac{1}{16}(3x - 1)^2$, which is tangent to the $x$-axis at $P$. The point $Q(3, 4)$ lies on the curve, and the tangent at $Q$ meets the $x$-axis at $R$.

Coordinate geometry

A curve whose gradient is given by $\frac{dy}{dx} = 3x^2 - \frac{2}{x^3}$ passes through $(-1, 3)$.

Integration

In an arithmetic progression, the 12th term is $17$, and the total of the first $31$ terms is $1023$.

Series

Solve the equation $\sin^{-1}(3x) = -\frac{1}{3}\pi$, and give the solution exactly.

Trigonometry

The coordinates of two points are $A(5, 7)$ and $B(9, -1)$.

Coordinate geometry

A vacuum flask, used to keep drinks hot, is represented by a closed cylinder whose internal radius is $r\,\text{cm}$ and internal height is $h\,\text{cm}$. The flask has volume $1000\,\text{cm}^3$. It is most efficient when the total internal surface area, $A\,\text{cm}^2$, is as small as possible.

Differentiation

The diagram depicts a pyramid $OABC$ with a flat triangular base $OAB$ and perpendicular height $OC$. Angles $AOB$, $BOC$ and $AOC$ are each right angles. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ run parallel to $OA$, $OB$ and $OC$ respectively, and $OA = 4$ units, $OB = 2.4$ units and $OC = 3$ units. Point $P$ lies on $CA$ so that $CP = 3$ units.

Coordinate geometry

The function $f$ is defined by $f(x) = a^2x^2 - ax + 3b$ for $x \leq \frac{1}{2a}$, with $a$ and $b$ as constants.

Functions

In Fig. $1$, $OAB$ is a sector of a circle with centre $O$ and radius $r$. $AX$ is tangent to the arc $AB$ at $A$, and $\angle BAX = \alpha$. Show that $\angle AOB = 2\alpha$.

Circular measure

State the quotient and the remainder after dividing $2x^3 + 3x^2 + 10$ by $(x + 2)$.

Algebra

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

Algebra

Assume that $k$ is a positive constant.

Logarithmic and exponential functions

The values generated by the iterative formula $x_{n+1} = \sqrt{\tfrac{1}{2}x_n^2 + 4x_n^{-3}}$, starting from $x_1 = 1.5$, approach $\alpha$.

Numerical solution of equations

If $\int_0^a 6e^{2x+1}\,dx = 65$, determine the value of $a$ correct to 3 decimal places.

Integration

The diagram depicts the segment of the curve $y = 3e^{-x}\sin 2x$ for $0 \leq x \leq \tfrac{1}{2}\pi$, together with the stationary point $M$.

Differentiation

The curve is described by $2x^3 + y^3 = 24$.

Differentiation

Show that $\sin 2x \cot x \equiv 2\cos^2 x$.

Trigonometry

Solve the equation $\ln(x^2 + 4) = 2\ln x + \ln 4$, and give your answer in exact form.

Logarithmic and exponential functions

Determine the complex number $z$ that satisfies the equation $z^* + 1 = 2iz$, where $z^*$ represents the complex conjugate of $z$. Write your answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Express the equation $\tan(\theta + 45^\circ) - 2\tan(\theta - 45^\circ) = 4$ as a quadratic equation in $\tan\theta$. Hence solve it for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

The equation $x^5 - 3x^3 + x^2 - 4 = 0$ has exactly one positive root.

Numerical solution of equations

Let $p(x)$ represent the polynomial $4x^3 + ax + 2$, where $a$ is a constant. It is stated that $(2x + 1)$ is a factor of $p(x)$.

Algebra

Define $I = \int_{0}^{1} \frac{9}{(3 + x^2)^2}\,dx$.

Integration

The curve is described by $\sin y \ln x = x - 2\sin y$, for $-\frac{1}{2}\pi \leq y \leq \frac{1}{2}\pi$.

Differentiation

The variables $x$ and $y$ obey the differential equation $\frac{dy}{dx} = x e^{x+y}$, and it is known that $y = 0$ when $x = 0$.

Differential equations

The vector equation of line $l$ is $\mathbf{r} = \begin{pmatrix}1\\2\\-1\end{pmatrix} + \lambda \begin{pmatrix}2\\1\\3\end{pmatrix}$. The plane $p$ is given by $\mathbf{r} \cdot \begin{pmatrix}2\\-1\\-1\end{pmatrix} = 6$.

Vectors

Define $f(x)$ by $f(x) = \frac{3x^3 + 6x - 8}{x(x^2 + 2)}$.

Integration

A cyclist of mass $85\,\text{kg}$ and a bicycle of mass $20\,\text{kg}$ travel along a horizontal road while opposing a total resistance force of $40\,\text{N}$.

Energy, work and power

A resistive force of magnitude $1350\,\text{N}$ acts continuously on a car with mass $1200\,\text{kg}$.

Energy, work and power

Three coplanar forces, with magnitudes $50\,\text{N}$, $40\,\text{N}$ and $30\,\text{N}$, are applied at point $O$ in the directions shown in the diagram, where $\tan\alpha = \frac{7}{24}$.

Forces and equilibrium

A particle $P$ with mass $0.8\,\text{kg}$ rests on a rough horizontal table. The coefficient of friction between $P$ and the table is $\mu$. A force of magnitude $5\,\text{N}$, directed upwards at an angle $\alpha$ above the horizontal, where $\tan\alpha = \frac{3}{4}$, acts on $P$. The particle is just about to slide across the table.

Newton's laws of motion

A car with mass $1200\,\text{kg}$ is towing a trailer with mass $800\,\text{kg}$ up a slope making an angle $\alpha$ with the horizontal, where $\sin\alpha = 0.1$. The car and trailer system is treated as two particles joined by a light inextensible cable. The engine provides a driving force of $2500\,\text{N}$, while the resistive forces on the car and trailer are $100\,\text{N}$ and $150\,\text{N}$ respectively.

Newton's laws of motion

Particles $A$ and $B$, with masses $0.8\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are joined by a light inextensible string. $A$ rests on a horizontal plane. The string runs over a small smooth pulley $P$ at the edge of the plane, and $B$ is suspended freely. The horizontal part of the string, $AP$, has length $2.5\,\text{m}$. The particles are let go from rest, with both parts of the string taut.

Newton's laws of motion

A particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = 5t(t - 2)$ for $0 \leq t \leq 4$, $v = k$ for $4 \leq t \leq 14$, $v = 68 - 2t$ for $14 \leq t \leq 20$, where $k$ is constant.

Kinematics of motion in a straight line

A particle is launched from a point on level ground.

Representation of data

A uniform solid hemisphere, with weight $60\text{ N}$ and radius $0.8\text{ m}$, is in limiting equilibrium on a rough horizontal plane with its curved face in contact with the plane. The hemisphere’s axis of symmetry makes an angle $\theta$ with the horizontal, where $\cos\theta = 0.28$. A horizontal force of magnitude $P\text{ N}$ is applied at the lowest point of the circular rim to keep the hemisphere in equilibrium (see diagram).

Probability

A stone is projected at a speed of $9\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal from a point on level ground. Find the distance between the two points where the path of the stone makes an angle of $45^\circ$ with the horizontal.

Representation of data

A uniform lamina is formed by combining rectangle $ABCD$, where $AB = CD = 0.56\text{ m}$ and $BC = AD = 2\text{ m}$, with square $EFGA$ of side $1.2\text{ m}$. The square’s vertex $E$ lies on the side $AD$ of the rectangle (see diagram). The lamina’s centre of mass is $h\text{ m}$ from $BC$ and $v\text{ m}$ from $BAG$.

Representation of data

A particle $P$ with mass $0.6\text{ kg}$ is connected to one end of a light elastic string whose natural length is $0.8\text{ m}$ and whose modulus of elasticity is $24\text{ N}$. The opposite end of the string is fixed at point $A$, and $P$ hangs in equilibrium.

Probability

A particle $P$ with mass $0.2\text{ kg}$ is set free from rest at point $O$ on a plane that is inclined at $30^\circ$ to the horizontal. After $t\text{ s}$ from release, $P$ has velocity $v\text{ m s}^{-1}$ and has moved a distance $x\text{ m}$ down the plane from $O$. The coefficient of friction between $P$ and the plane grows as $P$ moves down the plane, and is given by $0.1x^2$.

Representation of data

One end of a light inextensible string is fixed to the top point $A$ of a solid sphere that is held in place, with centre $O$ and radius $0.6\text{ m}$. The string’s other end is joined to a particle $P$ of mass $0.2\text{ kg}$, which is in contact with the sphere’s smooth surface. The angle $AOP$ is $60^\circ$ (see diagram). The sphere applies a contact force of magnitude $R\text{ N}$ on $P$, and the tension in the string is $T\text{ N}$.

Probability

For 10 values of $x$, the mean is $86.2$ and $\sum (x - a) = 362$.

Sampling and estimation

A flower shop contains $5$ yellow roses, $3$ red roses and $2$ white roses. Martin picks $3$ roses at random.

The Poisson distribution

A fair eight-sided die is numbered $1, 2, 3, 4, 5, 6, 7, 8$ on its faces. When the die is rolled, the score is the number showing on the face it lands on. The die is rolled twice. Event $R$ is ‘one score is exactly $3$ greater than the other score’. Event $S$ is ‘the product of the scores is more than $19$’.

Continuous random variables

A survey was carried out on the journey times of $63$ people who cycle to work in one town. The findings are set out in the cumulative frequency table below: Journey time (minutes) $\leq 10, \leq 25, \leq 45, \leq 60, \leq 80$ with cumulative frequencies $0, 18, 50, 59, 63$.

Sampling and estimation

In one town, $35\%$ of residents take a holiday abroad, while $65\%$ spend their holiday in their own country. Among those who go abroad, $80\%$ choose the seaside, $15\%$ go camping and $5\%$ take a city break. Among those who have a holiday in their own country, $20\%$ go to the seaside and the remaining people are split equally between camping and a city break.

The Poisson distribution

Hannah selects $5$ singers from $15$ applicants to take part in a concert. She arranges the $5$ singers in the sequence they will perform. Out of the $15$ applicants, $10$ are female and $5$ are male.

Sampling and estimation

The lengths of time a garage needs to attach a tow bar to a car are normally distributed with mean $m$ hours and standard deviation $0.35$ hours. It is given that $95\%$ of the times exceed $0.9$ hours.

Continuous random variables