Feb/March 2021

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

Series

The diagram depicts sector $ABC$, which is a section of a circle with radius $a$. Points $D$ and $E$ are located on $AB$ and $AC$ respectively, with $AD = AE = ka$, where $k < 1$. Line $DE$ splits the sector into two areas of equal size.

Circular measure

The diagram displays the curve whose equation is $y = 9\left(x^{\frac{1}{2}} - 4x^{-\frac{3}{2}}\right)$. This curve meets the $x$-axis at $A$.

Differentiation

Use an appropriate substitution to solve the equation $(2x - 3)^2 - \dfrac{4}{(2x - 3)^2} - 3 = 0$.

Quadratics

Solve the equation $\dfrac{\tan\theta + 2\sin\theta}{\tan\theta - 2\sin\theta} = 3$, with $0^{\circ} < \theta < 180^{\circ}$.

Trigonometry

A line is given by the equation $y = 3x + k$ and a curve is given by the equation $y = x^2 + kx + 6$, with $k$ a constant.

Coordinate geometry

The diagram presents the graph of $y = f(x)$ using solid lines. The graph drawn with broken lines is a transformed version of $y = f(x)$.

Functions

The curve satisfies $\frac{dy}{dx} = \frac{6}{(3x - 2)^3}$, and $A\,(1,-3)$ is on it. A point moves along the curve, and at $A$ the $y$-coordinate of the point is rising at $3$ units per second.

Differentiation

Functions $f$ and $g$ are given by: $f: x \mapsto x^2 + 2x + 3$ when $x \leq -1$, $g: x \mapsto 2x + 1$ when $x \geq -1$.

Functions

The points $A(7, 1)$, $B(7, 9)$ and $C(1, 9)$ lie on a circle’s circumference.

Coordinate geometry

The first term in the progression is $\cos \theta$, with $0 < \theta < \tfrac{1}{2}\pi$.

Series

Sketch the graphs of $y = |3x - 5|$ and $y = x + 2$ on the same set of axes.

Algebra

Solve for $\theta$ in the range $0 < \theta < \pi$ when $\sec^2 \theta \cot \theta = 8$.

Trigonometry

A curve is given in parametric form by $x = e^{2t} \cos 4t$ and $y = 3 \sin 2t$.

Differentiation

The diagram illustrates a section of the curve whose equation is $y = \frac{5x}{4x^3 + 1}$. The shaded area is enclosed by the curve together with the lines $x = 1$, $x = 3$ and $y = 0$.

Integration

If $2\ln(x+1)+\ln x=\ln(x+9)$, prove that $x=\sqrt{\frac{9}{x+2}}$.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x)=x^3+ax+b$, with $a$ and $b$ as constants. You are told that $(x+2)$ is a factor of $p(x)$ and that, on dividing $p(x)$ by $(x-3)$, the remainder is $5$.

Logarithmic and exponential functions

Express $5\sqrt{3}\cos x + 5\sin x$ as $R\cos(x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$.

Trigonometry

Solve the equation $\ln(x^3 - 3) = 3\ln x - \ln 3$. State your answer correct to 3 significant figures.

Logarithmic and exponential functions

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

Integration

Write the polynomial $ax^3 + 5x^2 - 4x + b$ as $p(x)$, where $a$ and $b$ are constants. You are told that $(x + 2)$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $(x + 1)$ is $2$.

Algebra

Rearrange the equation $\tan(x + 45^\circ) = 2\cot x + 1$ into a quadratic in $\tan x$, then solve it for $0^\circ < x < 180^\circ$.

Trigonometry

The variables $x$ and $y$ are linked by the differential equation $(1 - \cos x)\frac{dy}{dx} = y\sin x$. It is stated that $y = 4$ when $x = \pi$.

Differential equations

Write $\sqrt{7}\sin x + 2\cos x$ as $R\sin(x + \alpha)$, with $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and state $\alpha$ correct to $2$ decimal places.

Trigonometry

Define $f(x) = \dfrac{5a}{(2x - a)(3a - x)}$, with $a$ as a positive constant.

Integration

The two lines are represented by the equations $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$.

Vectors

The complex numbers $u$ and $v$ are given by $u = -4 + 2i$ and $v = 3 + i$.

Complex numbers

For $x > 0$, define $f(x) = \frac{e^{2x} + 1}{e^{2x} - 1}$.

Numerical solution of equations

Particles $P$ and $Q$, whose masses are $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are able to move only along a horizontal straight line on a smooth horizontal plane. $P$ is launched towards $Q$ at speed $0.5\,\text{m s}^{-1}$. Simultaneously, $Q$ is launched towards $P$ at speed $1\,\text{m s}^{-1}$. In the collision that results, $Q$ is brought to rest.

Momentum

A car with mass $1400\,\text{kg}$ moves at constant speed up a straight hill inclined at $\alpha$ to the horizontal, where $\sin \alpha = 0.1$. A constant resistive force of magnitude $600\,\text{N}$ acts on it. The engine power of the car is $22\,500\,\text{W}$.

Energy, work and power

Particle $Q$ has mass $0.2\,\text{kg}$ and is kept in equilibrium by two light inextensible strings $PQ$ and $QR$. $P$ is a fixed point on a vertical wall, while $R$ is a fixed point on a horizontal floor. The angles that strings $PQ$ and $QR$ make with the horizontal are $60^\circ$ and $30^\circ$ respectively (see diagram).

Forces and equilibrium

An elevator travels vertically while being held by a cable. The diagram presents a velocity-time graph that represents the elevator’s motion. The graph is made up of $7$ straight line segments. The elevator first accelerates upwards from rest to a speed of $2\,\text{m s}^{-1}$ in $1.5\,\text{s}$, then continues at this speed for $4.5\,\text{s}$, before slowing to rest in $1\,\text{s}$. It then stays at rest for $6\,\text{s}$, before accelerating downwards to a speed of $V\,\text{m s}^{-1}$ over $2\,\text{s}$. The elevator then moves at this speed for $5\,\text{s}$, before slowing to rest in $1.5\,\text{s}$.

Newton's laws of motion

A block with mass $5\text{ kg}$ is dragged across a rough level floor by a force of magnitude $X\text{ N}$ acting at $30^\circ$ above the horizontal (see diagram). It starts from rest and covers $2\text{ m}$ in the first $5\text{ s}$ of its motion.

Forces and equilibrium

A particle travels along a straight line. It begins from rest at a fixed point $O$ on the line. At time $t\text{ s}$ after leaving $O$, its velocity is $v\text{ m s}^{-1}$, where $v = t^2 - 8t^{\frac{3}{2}} + 10t$.

Kinematics of motion in a straight line

Particles $P$ and $Q$ have masses $0.5\,\text{kg}$ and $m\,\text{kg}$ respectively, and they are connected by a light inextensible string. The string runs over a fixed smooth pulley mounted at the top of two inclined planes. At the start, both particles are at rest, with $P$ on a smooth plane at $30^\circ$ to the horizontal and $Q$ on a plane at $45^\circ$ to the horizontal. The string is taut, and the particles may travel along the lines of greatest slope of the two planes. A force of magnitude $0.8\,\text{N}$ is applied to $P$ down the plane, making $P$ move down the plane (see diagram).

Energy, work and power

A fair spinner with $5$ sides labelled $1, 2, 3, 4, 5$ is spun many times. Each spin gives the score shown by the side where the spinner stops.

Discrete random variables

Georgie owns a red scarf, a blue scarf and a yellow scarf. On each day, she chooses to wear exactly one of these scarves. The probabilities of selecting the three colours are $0.2$, $0.45$ and $0.35$ respectively. If she wears a red scarf, she always also wears a hat. If she wears a blue scarf, the probability that she wears a hat is $0.4$. If she wears a yellow scarf, the probability that she wears a hat is $0.3$.

Probability

The times spent by shoppers in a large shopping centre are normally distributed, with mean $96$ minutes and standard deviation $18$ minutes.

The normal distribution

The random variable $X$ can take only the values 1, 2, 3, 4. The probability that $X$ has the value $x$ is $kx(5-x)$, where $k$ is a constant.

Discrete random variables

The driver notes the distance covered on each of 150 journeys. The distances, each rounded to the nearest km, are shown in the table below.

Representation of data

Find the total number of distinct arrangements of the $11$ letters in the word CATERPILLAR.

Permutations and combinations

A school in one country has $400$ students. Every student was asked if they preferred swimming, cycling or running, and the outcomes are shown in the table below. One student is selected at random.

The normal distribution

A construction company records, in $t$ days, how long it takes to construct each house of one particular design. For a random sample of $60$ such houses, the summary data are given by: $\sum t = 4820$, $\sum t^2 = 392050$.

Sampling and estimation

The diagram displays the graph of the probability density function, $f$, for the random variable $X$.

Continuous random variables

An architect wants to find out whether, on average, the buildings in one city are taller than those in other cities. He draws a large random sample of buildings from the city and calculates the mean height of the sample. He works out the test statistic, $z$, and obtains $z = 2.41$.

Hypothesis testing

At a particular factory, $1$ in $400$ microchips produced are faulty on average. Let $X$ stand for the number of faulty microchips in a random sample of $1000$.

The Poisson distribution

The juice volumes, measured in litres, for large bottles and small bottles have distributions $N(5.10, 0.0102)$ and $N(2.51, 0.0036)$ respectively.

Linear combinations of random variables

It is given that $8\%$ of adults in one town own a Chantor car. Following an advertising campaign, a car dealer wants to find out whether this proportion has gone up. He selects a random sample of $25$ adults from the town and records how many own a Chantor car.

Hypothesis testing