Feb/March 2024

Work out the exact value of $\int_{3}^{\infty} \frac{2}{x^2}\,dx$.

Integration

The circle has centre $C(-4, 5)$ and radius $\sqrt{20}$ units, as shown in the diagram. It meets the $y$-axis at $A$ and $B$. The angle $ACB$ has size $\theta$ radians.

Coordinate geometry

The diagram presents the curve with equation $y = 2x^{\frac{2}{3}} - 3x^{-\frac{1}{3}} + 1$ for $x > 0$. This curve crosses the $x$-axis at points $A$ and $B$ and has a minimum point $M$.

Integration

The diagram shows a section of the curve with equation $y = k\sin\left(\frac{1}{2}x\right)$, where $k$ is a positive constant and $x$ is measured in radians. Point $A$ is a minimum point on the curve.

Functions

A curve is defined by $\frac{dy}{dx} = 3\sqrt{4x + 5}$. The points $(1,9)$ and $(5,a)$ are on the curve.

Integration

Show that $\frac{(\sin\theta + \cos\theta)^2 - 1}{\cos^2\theta} = 2\tan\theta$.

Trigonometry

The equation of a curve is $y = \frac{3}{2x^2 - 5}$.

Differentiation

In the expansion of $(2 + ax)^4(5 - ax)$, the coefficient of $x^3$ is $432$.

Series

The line $y = x + 5$ intersects the curve $2x^2 + 3y^2 = k$ at one only point, $P$.

Coordinate geometry

In an arithmetic progression whose first term is $6$ and whose tenth term is $19.5$, determine the sum of the first $100$ terms of this arithmetic progression.

Series

The functions $f$ and $g$ are given for every real value of $x$ by $f(x) = (3x - 2)^2 + k$ and $g(x) = 5x - 1$, with $k$ a constant.

Functions

Solve the equation $3^{4x+3} = 5^{2x+7}$ by using logarithms. Give your answer correct to $3$ significant figures.

Logarithmic and exponential functions

Sketch the graph of $y = \lvert 3x - 7 \rvert$, giving the coordinates of the points at which the graph crosses the axes.

Algebra

The polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 + 3x - 10$, with $a$ as a constant. It is stated that $(2x - 1)$ is a factor of $p(x)$.

Trigonometry

The curve drawn in the diagram is given by the equation $y = \sqrt{1 + e^{0.5x}}$. The shaded area lies between the curve and the lines $x = 0$, $x = 6$ and $y = 0$.

Integration

The diagram depicts a section of the curve with equation $y = \dfrac{x^3}{x + 2}$. At the point $P$, the gradient of the curve equals $6$.

Numerical solution of equations

The diagram represents the curve with parametric equations $x = 1 + \sqrt{t}$, $y = (\ln t + 2)(\ln t - 3)$, for $0 \le t \le 25$. It meets the $x$-axis at the points $A$ and $B$ and has a minimum point $M$.

Differentiation

Prove that $\sin 2\theta (a \cot \theta + b \tan \theta) \equiv a + b + (a - b) \cos 2\theta$, where $a$ and $b$ are constants.

Trigonometry

Determine the quotient and remainder when $x^4 - 3x^3 + 9x^2 - 12x + 27$ is divided by $x^2 + 5$.

Algebra

Take $f(x) = \dfrac{36a^2}{(2a + x)(2a - x)(5a - 2x)}$, where $a$ is a positive constant.

Integration

The variables $y$ and $\theta$ are related by $(1 + y)(1 + \cos 2\theta) \dfrac{dy}{d\theta} = e^{3y}$. It is also given that $y = 0$ when $\theta = \dfrac{1}{4}\pi$.

Differential equations

Find the coefficient of $x^2$ when $(2x - 5)\sqrt{4 - x}$ is expanded.

Algebra

You are told that $z = -\sqrt{3} + i$.

Complex numbers

The positive numbers $p$ and $q$ satisfy $\ln\left(\frac{p}{q}\right) = a$ and $\ln(q^2p) = b$.

Logarithmic and exponential functions

On an Argand diagram sketch, shade the set of points corresponding to complex numbers $z$ that satisfy the inequalities $|z - 4 - 2i| \leq 3$ and $|z| \geq |10 - z|$.

Complex numbers

A curve is described by $2y^2 + 3xy + x = x^2$.

Differentiation

The diagram displays the curve $y = xe^{2x} - 5x$ together with its minimum point $M$, for which $x = \alpha$.

Numerical solution of equations

Express $3\sin x + 2\sqrt{2}\cos\left(x + \frac{1}{4}\pi\right)$ in the form $R\sin(x + \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact value of $R$, and give $\alpha$ correct to $3$ decimal places.

Trigonometry

Taking the origin $O$ as the reference point, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = 5\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\overrightarrow{OB} = 8\mathbf{i} + 2\mathbf{j} - 6\mathbf{k}$ and $\overrightarrow{OC} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k}$.

Vectors

The displacement of a particle after it has left a fixed point $O$ at time $t$ is $s$ m. The diagram presents a displacement-time graph that represents the particle's motion. The graph is made up of $4$ straight line segments. In the first $10$ s, the particle covers $50$ m, then it moves at $2\,\text{m s}^{-1}$ for $10$ s. After that, the particle remains at rest for $20$ s, before it gets back to its starting point when $t = 60$.

Kinematics of motion in a straight line

A particle is launched vertically upwards from horizontal ground. Two seconds after launch, its speed is $5\,\text{m s}^{-1}$ and it is moving downwards.

Kinematics of motion in a straight line

A $600\,\text{kg}$ crate is pulled up the line of greatest slope on a rough plane, using a rope fixed to a winch, at a steady speed of $2\,\text{m s}^{-1}$. The plane is inclined at $30^\circ$ to the horizontal, and the rope is parallel to the plane. The winch operates at a steady rate of $8\,\text{kW}$.

Forces and equilibrium

At one point, four coplanar forces act. Their magnitudes are $F\text{ N}$, $2F\text{ N}$, $3F\text{ N}$ and $30\text{ N}$. Their directions are illustrated in the diagram.

Forces and equilibrium

A particle travels along a straight line from a point $O$. Its velocity $v\text{ m s}^{-1}$, $t\text{ s}$ after it leaves $O$, is $v = t^3 - \frac{9}{2}t^2 + 1$ for $0 \leq t \leq 4$. You may take the velocity to be positive when $t < \frac{1}{2}$, to be zero when $t = \frac{1}{2}$ and to be negative when $t > \frac{1}{2}$.

Kinematics of motion in a straight line

A car with mass $1800\,\text{kg}$ is pulling a trailer with mass $300\,\text{kg}$ along a straight road that slopes at an angle $\alpha$ to the horizontal, with $\sin \alpha = 0.05$. A tow-bar, which is light, rigid and parallel to the road, links the car and trailer. A resistance force of $800\,\text{N}$ acts on the car, while the trailer experiences a resistance force of $F\,\text{N}$. The car's engine provides a driving force of $3000\,\text{N}$.

Energy, work and power

The diagram shows two particles $P$ and $Q$ on the line of greatest slope of plane $ABC$. Each particle has mass $m\,\text{kg}$. The plane is inclined at an angle $\theta$ to the horizontal, with $\sin \theta = 0.6$. $AB$ is $0.75\,\text{m}$ long and $BC$ is $3.25\,\text{m}$ long. Section $AB$ of the plane is smooth, whereas section $BC$ is rough. The coefficient of friction between each particle and section $BC$ is $0.25$. Particle $P$ is released from rest at $A$. At the same moment, particle $Q$ is released from rest at $B$.

Momentum

A bag holds 9 blue marbles and 3 red marbles. One marble is picked at random from the bag. If the marble picked is blue, it is placed back into the bag. If the marble picked is red, it is not placed back into the bag. A second marble is then picked at random from the bag.

Probability

Sam belongs to a soccer club and is practising how to score goals. On each attempt, the probability that Sam scores a goal is $0.7$, independent of every other attempt.

Discrete random variables

The times, measured in minutes, needed by $150$ students to finish a puzzle are shown in the table. The time intervals are $0 \leq t < 20$, $20 \leq t < 30$, $30 \leq t < 35$, $35 \leq t < 40$, $40 \leq t < 50$, $50 \leq t < 70$ with matching frequencies $8$, $23$, $35$, $52$, $20$, and $12$.

Representation of data

A firm sells rice in small and large bags. For the small bags, the masses follow a normal distribution with mean $1.20\text{ kg}$ and standard deviation $0.16\text{ kg}$.

The normal distribution

Anil is competing in a tournament. In each match of this tournament, a player earns $2$ points for a win, $1$ point for a draw and $0$ points for a loss. For any of Anil’s games, the probabilities that he will win, draw or lose are $0.5$, $0.3$ and $0.2$ respectively. The outcomes of the games are mutually independent. The random variable $X$ represents the total number of points that Anil obtains from his first $3$ games in the tournament.

Discrete random variables

A village social club has just been formed and includes 10 members, comprising 6 men and 4 women. The committee of the club is to be made up of 5 members.

Permutations and combinations

The lengths, $X\text{ cm}$, for a sample of 100 insects of one particular type were summarised like this: $n = 100$, $\sum x = 36.8$, $\sum x^2 = 17.34$.

Sampling and estimation

A simple random sample of 250 residents of Barapet was selected. It was discovered that 78 of these residents owned a BETEC phone.

Sampling and estimation

In this lottery, the average is that 1 in every 10,000 tickets is a prize-winning ticket, and an agent sells 6000 tickets.

The Poisson distribution

Each year, a transport company consumes $X$ litres of gasoline and $Y$ litres of diesel fuel, with $X$ and $Y$ independent and distributed as $X \sim N(10\,700,\,950^2)$ and $Y \sim N(13\,400,\,1210^2)$.

Linear combinations of random variables

On any day, the numbers of girls and boys who are late for her class are represented by the independent random variables $G \sim \text{Po}(0.10)$ and $B \sim \text{Po}(0.15)$, respectively.

The Poisson distribution

The probability density function graph $f$ for a random variable $X$ is symmetric about the line $x = 2$. It is stated that $\mathrm{P}(2 < X < 5) = \frac{117}{256}$.

Continuous random variables

The heights, in centimetres, of adult females in Litania have mean $\mu$ and standard deviation $\sigma$. In 2004, the respective values of $\mu$ and $\sigma$ were $163.21$ and $6.95$. The government states that this year’s value of $\mu$ is higher than the 2004 figure. To check this statement, a researcher intends to perform a hypothesis test at the $1\%$ significance level. He measures the heights of a random sample of $300$ adult females in Litania this year and obtains the sample mean.

Hypothesis testing