Feb/March 2022

For the curve given by $y = f(x)$, the derivative is $f'(x) = 2x^{-\frac{1}{3}} - x^{\frac{1}{3}}$. Also, $f(8) = 5$ is known.

Integration

The diagram depicts a circle centred at $A$ with radius $5\,\text{cm}$ and another circle centred at $B$ with radius $8\,\text{cm}$. They are tangent at $C$, with $A$, $C$ and $B$ lying on one straight line. The tangent drawn at $D$ on the smaller circle meets the larger circle at $E$ and goes through $B$.

Integration

A curve is defined by the equation $y = k(3x - k)^{-1} + 3x$, with $k$ taken as a constant.

Differentiation

The curve is defined by $y = x^2 + 2cx + 4$ and the straight line is given by $y = 4x + c$, where $c$ is a constant.

Quadratics

For each expansion, determine the term that does not involve $x$.

Series

The first term of the geometric progression and the first term of the arithmetic progression are each $a$. The third term of the geometric progression matches the second term of the arithmetic progression. The fifth term of the geometric progression matches the sixth term of the arithmetic progression.

Series

Rewrite $2x^2 - 8x + 14$ in the form $2[(x - a)^2 + b]$.

Functions

In the diagram, the circle has equation $(x + 1)^2 + (y - 2)^2 = 85$ and the straight line has equation $y = 3x - 20$. The line cuts the circle at $A$ and $B$, and the centre of the circle is $C$.

Coordinate geometry

Show that the expression $\dfrac{\sin \theta + 2\cos \theta}{\cos \theta - 2\sin \theta} - \dfrac{\sin \theta - 2\cos \theta}{\cos \theta + 2\sin \theta}$ is equal to $\dfrac{4}{5\cos^2 \theta - 4}$.

Trigonometry

The diagram displays the circle whose equation is $(x - 2)^2 + y^2 = 8$. The chord $AB$ on the circle meets the positive $y$-axis at $A$ and lies parallel to the $x$-axis.

Integration

The functions $f$, $g$ and $h$ are given by: $f : x \mapsto x - 4x^{\frac{1}{2}} + 1$ for $x \geq 0$, $g : x \mapsto mx^2 + n$ for $x \geq -2$, where $m$ and $n$ are constants, $h : x \mapsto x^{\frac{1}{2}} - 2$ for $x \geq 0$.

Quadratics

Solve for $x$ in $|5x - 2| = |4x + 9|$.

Algebra

The curve is given by $y = 7 + 4\ln(2x + 5)$.

Differentiation

The variables $x$ and $y$ are related by $y = 3^{2a} a^x$, where $a$ is constant. A plot of $\ln y$ against $x$ produces a straight line with gradient $0.239$.

Logarithmic and exponential functions

Show that $\sin 2\theta \cot \theta - \cos 2\theta = 1$.

Trigonometry

For $y = \tan^2 x$, show that $\frac{dy}{dx} = 2 \tan x + 2 \tan^3 x$.

Integration

The polynomial $p(x)$ has the definition $p(x) = 4x^3 + 16x^2 + 9x - 15$.

Algebra

The curve is defined by $e^{2x}y - e^{y} = 100$.

Numerical solution of equations

Solve for $x$ the inequality $|2x + 3| > 3|x + 2|$.

Algebra

The position vectors of $A$ and $B$ are $2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ respectively. The line $l$ is given by the vector equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \mu(\mathbf{i} - 3\mathbf{j} - 2\mathbf{k})$.

Vectors

The graph depicts the curve $y = \sin x \cos 2x$ over $0 \leq x \leq \frac{1}{2}\pi$, together with its maximum point $M$.

Integration

On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy $|z + 2 - 3i| \leq 2$ and $\arg z \leq \frac{3}{4}\pi$.

Complex numbers

The variables $x$ and $y$ obey the equation $x^n y^2 = C$, where $n$ and $C$ are constants. The straight-line graph of $\ln y$ against $\ln x$ goes through the points $(0.31, 1.21)$ and $(1.06, 0.91)$, as illustrated in the diagram.

Logarithmic and exponential functions

The curve has parametric equations $x = 1 - \cos \theta$ and $y = \cos \theta - \frac{1}{4} \cos 2\theta$.

Differentiation

Angles $\alpha$ and $\beta$ are each between $0^\circ$ and $180^\circ$, and they satisfy $\tan(\alpha + \beta) = 2$ as well as $\tan \alpha = 3 \tan \beta$.

Algebra

Find the complex numbers $w$ that satisfy the equation $w^2 + 2i w^* = 1$ while also meeting $\operatorname{Re} w \leq 0$. Write each answer in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

By sketching an appropriate pair of graphs, show that the equation $4 - x^2 = \sec \frac{1}{2} x$ has exactly one solution in $0 \leq x < \pi$.

Numerical solution of equations

Find the quotient and remainder after dividing $8x^3 + 4x^2 + 2x + 7$ by $4x^2 + 1$.

Integration

The variables $x$ and $y$ are linked by the differential equation $(x + 1)(3x + 1)\frac{dy}{dx} = y$, with the condition that $y = 1$ when $x = 1$.

Differential equations

A crane is used to lift a block of mass $600\,\text{kg}$ straight up at constant speed through a height of $15\,\text{m}$. A resistive force acts on the block, and the crane does $10000\,\text{J}$ of work to overcome it.

Energy, work and power

A particle $P$ is launched vertically upwards from level ground with speed $u\\,\\text{m}\\,\\text{s}^{-1}$. $P$ attains a greatest height of $20\\,\\text{m}$ above the ground.

Kinematics of motion in a straight line

A car with mass $m\,\text{kg}$ is pulling a trailer of mass $300\,\text{kg}$ downhill on a straight slope at $3^{\circ}$ to the horizontal at constant speed. Resistive forces act on both the car and the trailer, and the total work done against these resistive forces over a distance of $50\,\text{m}$ is $40\,000\,\text{J}$. The car’s engine is doing no work, and the tow-bar is light and rigid.

Newton's laws of motion

The combined mass of the cyclist and her bicycle is $70\,\text{kg}$. She is climbing a straight hill at constant power of $180\,\text{W}$, with the hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.05$. At the instant when her speed is $6\,\text{m s}^{-1}$, her acceleration is $-0.2\,\text{m s}^{-2}$. A constant resistive force of magnitude $F\,\text{N}$ acts against the motion.

Energy, work and power

Four forces acting at one point lie in the same plane. Their magnitudes are $10\,\text{N}$, $F\,\text{N}$, $G\,\text{N}$ and $2F\,\text{N}$. The directions of the forces are shown in the diagram.

Forces and equilibrium

A cyclist sets off from rest at the fixed point $O$ and travels in a straight line, then comes to rest $k$ seconds later. At time $t$ s after leaving $O$, the cyclist’s acceleration is $a\,\text{m s}^{-2}$, where $a = 2t^{-\frac{1}{2}} - \frac{3}{5}t^{\frac{1}{2}}$ for $0 < t \le k$.

Kinematics of motion in a straight line

A bead, $A$, with mass $0.1\,\text{kg}$ is placed on a long straight rigid wire inclined at $\sin^{-1}\left(\frac{7}{25}\right)$ to the horizontal. $A$ starts from rest and slides down the wire. The coefficient of friction between $A$ and the wire is $\mu$. After $A$ has moved $0.45\,\text{m}$ down the wire, its speed is $0.6\,\text{m s}^{-1}$.

Momentum

A fair red spinner has edges labelled $1, 2, 2, 3$. A fair blue spinner has edges labelled $-3, -2, -1, -1$. Each spinner is spun once, and the number shown on the edge where each spinner stops is recorded. The random variable $X$ represents the total of the two numbers obtained.

Discrete random variables

In a particular country, the chance of getting more than $10\,\text{cm}$ of rain on any one day is $0.18$, independently of the weather on any other day.

Discrete random variables

In a summer camp, $250$ children sit an arithmetic test. The times taken, correct to the nearest minute, to finish the test were noted. The results are shown in the table: Time taken (in minutes): $1\!\text{-}\!30$, $31\!\text{-}\!45$, $46\!\text{-}\!65$, $66\!\text{-}\!75$, $76\!\text{-}\!100$ Frequency: $21$, $30$, $68$, $86$, $45$.

Representation of data

For male leopards in one particular region, the weights are normally distributed, with mean $55\text{ kg}$ and standard deviation $6\text{ kg}$.

The normal distribution

The group of $12$ people is made up of $3$ boys, $4$ girls and $5$ adults.

Permutations and combinations

A factory makes chocolates in three flavours, lemon, orange and strawberry, in the ratio $3:5:7$ respectively. Nell checks the chocolates on the production line by selecting them at random one at a time.

Probability

A random sample of 12 rods from a particular machine has lengths, in millimetres, given by $200,\;201,\;198,\;202,\;200,\;199,\;199,\;201,\;197,\;202,\;200,\;199$.

Sampling and estimation

Harry has a spinner with five sectors, coloured blue, green, red, yellow and black. Harry suspects that the spinner could be biased. He intends to carry out a hypothesis test with the following hypotheses: $H_0:\;P(\text{the spinner lands on blue}) = \frac{1}{5}$ and $H_1:\;P(\text{the spinner lands on blue}) \neq \frac{1}{5}$. Harry spins the spinner 300 times, and blue comes up on 45 of the spins.

Hypothesis testing

From a random sample of 500 households in a particular town, the confidence interval for the proportion, $p$, of all households in that town owning two or more cars was obtained as $0.355 \le p \le 0.445$.

Sampling and estimation

Previously, the time in minutes that students needed to finish a particular challenge had a mean of $25.5$ and a standard deviation of $5.2$. A different challenge is now set, and it is anticipated that, on average, students will need less than $25.5$ minutes to finish it. A random sample of $40$ students is taken, and the sample mean time for the new challenge is $23.7$ minutes.

Hypothesis testing

The building heights in a large city are modeled by a normal distribution with mean $18.3\,\text{m}$ and standard deviation $2.5\,\text{m}$.

Linear combinations of random variables

A ball is released down a slope in a game and then travels along a track until it comes to rest. Let the distance, in metres, covered by the ball be represented by the random variable $X$ with probability density function $f(x) = \begin{cases} -k(x - 1)(x - 3), & 1 \le x \le 3, \\ 0, & \text{otherwise}. \end{cases}$ where $k$ is a constant.

Continuous random variables

Ponds $A$ and $B$ each hold a very large population of fish. It is known that $2.4\%$ of the fish in pond $A$ are carp, while $1.8\%$ of the fish in pond $B$ are carp. Random samples of $50$ fish from pond $A$ and $60$ fish from pond $B$ are taken. Apply suitable Poisson approximations to determine the following probabilities.

The Poisson distribution