Feb/March 2023

The equation of a line is $y = 3x - 2k$ and the equation of a curve is $y = x^2 - kx + 2$, with $k$ constant.

Coordinate geometry

At the point $(4, -1)$ on the curve, the gradient is $-\frac{3}{2}$. It is also given that $\frac{dy}{dx} = x^{-\frac{1}{2}} + k$, where $k$ is constant.

Differentiation

The diagram presents the curve given by $x = y^2 + 1$. The points $A\,(5, 2)$ and $B\,(2, -1)$ are on the curve.

Integration

Let $f$ be given by $f(x) = x^2 - 2x + 5$ for $x \in \mathbb{R}$. In the order shown, the graph of $y = f(x)$ is transformed to produce $y = g(x)$: A stretch parallel to the $x$-axis with scale factor $\frac{1}{2}$. A reflection across the $y$-axis. A stretch parallel to the $y$-axis with scale factor $3$.

Quadratics

A curve is defined by $y = \frac{1}{60}(3x + 1)^2$, and a point moves along this curve.

Differentiation

The outside distance around the trunk of a large tree is measured as $5.00\,\text{m}$. One year later, a second measurement gives $5.02\,\text{m}$.

Series

Points $A\,(7,12)$ and $B$ are on a circle centred at $(-2,5)$. The equation of line $AB$ is $y = -2x + 26$.

Coordinate geometry

For the expansion of $\left(\frac{x}{a} + \frac{a}{x^2}\right)^7$, you are told that the coefficient of $x^4$ divided by the coefficient of $x$ is $3$.

Series

First form a quadratic equation in $\cos\theta$, then solve the equation $\tan\theta\,\sin\theta = 1$ for $0^\circ < \theta < 360^\circ$.

Trigonometry

The diagram depicts triangle $ABC$, where angle $B$ is a right angle. $AB$ has length $8\,\text{cm}$ and $BC$ has length $4\,\text{cm}$. Point $D$ lies on $AB$ so that $AD = 5\,\text{cm}$. The sector $DAC$ belongs to a circle with centre $D$.

Integration

The function $f$ is specified by $f(x) = -3x^2 + 2$ for $x \leq -1$.

Functions

Find the exact value for $\int_{0}^{2\pi} 2\tan^2\left(\frac{1}{2}x\right)\,dx$.

Integration

Solve $\tan(\theta - 60^\circ) = 3\cot\theta$ for $-90^\circ < \theta < 90^\circ$.

Trigonometry

The polynomial $p(x)$ is given by $p(x) = ax^3 - ax^2 + ax + b$, with $a$ and $b$ as constants. You are told that $(x + 2)$ is a factor of $p(x)$, and that the remainder is $35$ when $p(x)$ is divided by $(x - 3)$.

Algebra

On one pair of axes, sketch the graphs of $y = |2x - 11|$ and $y = 3x - 3$.

Logarithmic and exponential functions

Given that $\int_1^a \left( \frac{4}{1+2x} + \frac{3}{x} \right)\, dx = \ln 10$, where $a$ is a constant greater than $1$.

Numerical solution of equations

The diagram depicts the curve given by $y = \frac{4e^{2x} + 9}{e^x + 2}$. This curve has a minimum point $M$ and intersects the $y$-axis at $P$.

Differentiation

The diagram represents the curve with parametric equations $x = k \tan t$, $y = 3 \sin 2t - 4 \sin t$, for $0 < t < \frac{\pi}{2}$. It is stated that $k$ is a positive constant. The curve meets the $x$-axis at the point $P$.

Differentiation

It follows that $x = \ln(2y - 3) - \ln(y + 4)$.

Algebra

With origin $O$ as the reference point, the points $A$, $B$, $C$ and $D$ are described by the position vectors $\overrightarrow{OA} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}$, $\overrightarrow{OC} = \begin{pmatrix} 1 \\ -2 \\ 5 \end{pmatrix}$ and $\overrightarrow{OD} = \begin{pmatrix} 5 \\ -6 \\ 11 \end{pmatrix}$.

Vectors

Let $f(x)$ be defined by $f(x) = \dfrac{5x^2 + x + 11}{(4 + x^2)(1 + x)}$.

Integration

On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy the inequalities $-\frac{1}{3}\pi \leq \arg(z - 1 - 2i) \leq \frac{1}{3}\pi$ and $\Re z \leq 3$.

Complex numbers

The polynomial $2x^4 + ax^3 + bx - 1$, with $a$ and $b$ as constants, is called $p(x)$. If $p(x)$ is divided by $x^2 - x + 1$, the remainder is $3x + 2$.

Algebra

Find the solutions of $\frac{5z}{1 + 2i} - zz^* + 30 + 10i = 0$, and give them in the form $x + iy$, with $x$ and $y$ real.

Complex numbers

The curve is specified by the parametric equations $x = te^{2t}$ and $y = t^2 + t + 3$.

Differentiation

Write $5\sin\theta + 12\cos\theta$ as $R\cos(\theta - \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$.

Trigonometry

The diagram represents a circle centred at $O$ with radius $r$. The minor sector $AOB$ has angle $x$ radians. The area of the major sector is 3 times the area of the shaded region.

Numerical solution of equations

The diagram displays the curve $y = x^3 \ln x$, for $x > 0$, together with its minimum point $M$.

Integration

The differential equation connecting $x$ and $y$ is $\frac{dy}{dx} = e^{3y} \sin^2 2x$. Also, $y = 0$ when $x = 0$.

Differential equations

A crate with mass $200\,\text{kg}$ is being dragged at constant speed across horizontal ground by a horizontal rope connected to a winch. The winch operates at a steady rate of $4.5\,\text{kW}$, and the crate experiences a constant resistive force with magnitude $600\,\text{N}$.

Kinematics of motion in a straight line

A particle $P$ is projected straight upwards from horizontal ground with initial speed $15\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A particle travels in a straight line, beginning from rest at point $O$. At time $t\,\text{s}$ after it leaves $O$, its acceleration is $a\,\text{m s}^{-2}$, where $a = 4t^{\frac{1}{2}}$.

Kinematics of motion in a straight line

A toy railway locomotive, with mass $0.8\text{ kg}$, is pulling a truck of mass $0.4\text{ kg}$ along a straight horizontal track at a steady speed of $2\text{ m s}^{-1}$. The locomotive experiences a constant resistive force of magnitude $0.2\text{ N}$, whereas the truck has no resistive force acting on it. The locomotive and the truck are joined by a light rigid horizontal coupling.

Energy, work and power

The diagram depicts a block $D$ of mass $100\text{ kg}$ held up by two inclined struts $AD$ and $BD$, each fixed at an angle of $45^\circ$ to points $A$ and $B$ respectively on a horizontal floor. The block is additionally secured by a vertical rope $CD$ attached to a fixed point $C$ on a horizontal ceiling. The tension in rope $CD$ is $500\text{ N}$, and the block is in equilibrium.

Forces and equilibrium

A block $B$, with mass $2\,\text{kg}$, rests on a rough inclined plane that makes an angle of $30^\circ$ with the horizontal. A light rope, which is at $20^\circ$ above the line of greatest slope, is fixed to $B$. The rope tension is $T\,\text{N}$. A friction force of $F\,\text{N}$ acts on $B$ (see diagram). The coefficient of friction between $B$ and the plane is $\mu$.

Newton's laws of motion

The diagram represents a smooth track lying in a vertical plane. Section $AB$ is a quarter circle with radius $1.8\,\text{m}$ and centre $O$. Section $BC$ is a horizontal straight segment of length $7.0\,\text{m}$, with $OB$ perpendicular to $BC$. Section $CFE$ is a straight line that is inclined at an angle of $\theta^\circ$ above the horizontal. A particle $P$ of mass $0.5\,\text{kg}$ is released from rest at $A$. Particle $P$ then collides with a particle $Q$ of mass $0.1\,\text{kg}$ that is at rest at $B$. Immediately after the collision, the speed of $P$ is $4\,\text{m s}^{-1}$ in the direction of $BC$. Assume that $P$ is moving horizontally at the moment it collides with $Q$.

Momentum

In each year, the total number of sunshine hours, $x$, in Kintoo is measured for the month of June. The findings for the latest 60 years are shown in the table, with the classes $30 \leq x < 60$, $60 \leq x < 90$, $90 \leq x < 110$, $110 \leq x < 140$, $140 \leq x < 180$ and $180 \leq x \leq 240$ together with their frequencies.

Representation of data

Alisha owns four coins. One of them is biased so that the probability of getting a head is $0.6$. The other three coins are fair. Alisha tosses all four coins at the same time. The random variable $X$ represents the number of heads obtained.

Discrete random variables

Eighty percent of the people living in Kinwawa support the idea of a leisure centre being constructed in the town. Twenty Kinwawa residents are picked at random and, one by one, are asked whether they are in favour of the leisure centre.

Discrete random variables

Let the chance that it rains on any particular day be $x$. If it is raining, the probability that Aran wears a hat is $0.8$, whereas if it is not raining, the probability that he wears a hat is $0.3$. No matter whether it is raining, if Aran is wearing a hat, the probability that he wears a scarf is $0.4$. If he is not wearing a hat, the probability that he wears a scarf is $0.1$. On a randomly selected day, the probability that it is not raining and Aran is wearing neither a hat nor a scarf is $0.36$.

Probability

Marco has four boxes, labelled $K$, $L$, $M$ and $N$. He arranges them in a single row as $K$, $L$, $M$, $N$, with $K$ on the far left. Marco also has four coloured marbles: one red, one green, one white and one yellow. He puts one marble into each box, chosen at random. Events $A$ and $B$ are defined as follows: $A$: The white marble is in box $L$ or box $M$. $B$: The red marble lies to the left of both the green marble and the yellow marble.

Probability

In a cycling event, the completion times for the course are represented using a normal distribution with mean $62.3$ minutes and standard deviation $8.4$ minutes.

The normal distribution

Find how many different arrangements of the $9$ letters in DELIVERED have the three Es together and the two Ds not adjacent.

Permutations and combinations

Anita conducted a survey of 140 students chosen at random from her college. She found that 49 of those students watched a TV programme called Bunch.

Sampling and estimation

Over an 8-hour working day, the count of orders reaching a shop is represented by the random variable $X$ with distribution $\text{Po}(25.2)$.

The Poisson distribution

The diagram displays the graph of the probability density function, $f$, for a random variable $X$ that can take only values from $x = 0$ to $x = 3$. The graph is symmetric about the line $x = 1.5$.

Continuous random variables

For a certain road, the number of accidents in each 3-month period follows the distribution $\text{Po}(\lambda)$. Historically, $\lambda$ has had the value $5.7$. After some changes to the road, the council carries out a hypothesis test to decide whether $\lambda$ has fallen. If a randomly chosen 3-month period contains fewer than $3$ accidents, the council will decide that $\lambda$ has decreased.

Hypothesis testing

For large and small packets of Maxwheat cereal, the masses in grams are independently distributed as $N(410.0,\,3.6^2)$ and $N(206.0,\,3.7^2)$, respectively.

Linear combinations of random variables

During the previous year, the average amount of time that students at a school needed to finish a particular test was $25$ minutes. Akash thinks that the average time needed by this year’s students is below $25$ minutes. To check this idea, he selects a large random sample of this year’s students and records the time for each student. He carries out a test, at the $2.5\%$ significance level, for the population mean time, $\mu$ minutes. Akash uses the null hypothesis $H_0: \mu = 25$.

Hypothesis testing