Feb/March 2025

The curve is defined by $y = 5 + 3x - 2x^2$, and the straight line is defined by $y = kx + 13$, where $k$ is a constant.

Quadratics

The diagram presents the curve given by $y = 4\sqrt{3x + 4} - 2x - 6$ for $x$ values satisfying $0 \leq x \leq 7$. The tangent drawn to the curve at $P(7, 0)$ intersects the $y$-axis at $Q$. Region $A$ is enclosed by the curve and the two axes. Region $B$ is enclosed by the curve, the line segment $PQ$ and the $y$-axis.

Integration

For every real value of $x$, the functions f and g are defined by $f(x) = 4x^2 - c$ and $g(x) = 2x + k$, where $c$ and $k$ are positive constants. It is stated that $g^{-1}(3k + 1) = c$.

Functions

The diagram shows the curve whose equation is $y = 2x^2 - \frac{5}{x} + 3$. The curve intersects the $x$-axis at $P(1, 0)$, and $M$ is a minimum point.

Differentiation

Obtain the full expansion of $(2x - \frac{3}{x})^4$.

Series

The diagram depicts triangle $OAB$ with $OA = OB = 10\text{ cm}$ and $\angle AOB = 0.8$ radians. Let $C$ and $D$ lie on $OA$ and $OB$ respectively, so that the arc $CD$ belongs to a circle centred at $O$ with radius $6\text{ cm}$. The shaded part is enclosed by the arc $CD$ together with the line segments $CA$, $AB$ and $BD$.

Coordinate geometry

An arithmetic progression starts with term $5$ and has common difference $6$.

Series

The figure depicts a circle $C$ with radius $r$, and every point on $C$ has $x > 0$ and $y > 0$. The smallest distance from any point on $C$ to the $x$-axis is $8$ units, while the smallest distance from any point on $C$ to the $y$-axis is $5$ units.

Coordinate geometry

Show that $3\tan^2\theta + 5\sin^2\theta$ can be written as $\dfrac{8\sin^2\theta - 5\sin^4\theta}{1 - \sin^2\theta}$.

Trigonometry

A geometric progression has second term $-120$ and sum to infinity $160$.

Series

A curve is defined by $\frac{d^2 y}{dx^2} = \frac{6}{x^4} - \frac{5}{x^3}$. It is stated that this curve has a stationary point at $\left(\frac{1}{2}, 9\right)$.

Differentiation

Solve for $x$ in $\ln(3x + 1) - \ln(x - 5) = \ln 7$.

Logarithmic and exponential functions

The curve passes through the point with coordinates $\left(\frac{1}{2}\pi, 5\right)$, and it satisfies $\frac{dy}{dx} = 4\sec^2\left(\frac{1}{2}x\right)$.

Integration

The diagram depicts the curves $y = e^{2x}$ and $y = 8e^{-x}$, with the shaded area enclosed by both curves and the $y$-axis.

Integration

The curve is given by the equation $y = \dfrac{4\sin x}{3 + \cos 2x}$ for values of $x$ such that $0 \leq x \leq 2\pi$.

Differentiation

Let the x-coordinates of the intersection points be represented by $\alpha$ and $\beta$, with $\alpha < \beta$.

Numerical solution of equations

Find the quotient and remainder after dividing $18x^3 - 6x^2 - 30x + 4$ by $(3x - 1)$.

Integration

Express $6\sin\theta-4\cos\theta$ in the form $R\sin(\theta-\alpha)$, with $R>0$ and $0^\circ<\alpha<90^\circ$. Give the exact value of $R$ and state $\alpha$ correct to 2 decimal places.

Trigonometry

The curve is given by $3e^{2x}y+4e^{3x}+y^3=18$.

Differentiation

Find the solution of the equation $\ln(1 - e^{-2x}) + 3 = 0$. Give your final answer to 4 decimal places.

Logarithmic and exponential functions

Define $f(x)$ by $f(x) = \dfrac{-7x^2 + 2x - 6}{(1 + x)(4 + x)^2}$.

Integration

Work out the exact value of $\int_0^{\pi} x^2 \cos^2\!\left(\frac{1}{3}x\right) \, dx$.

Integration

The curve is defined by $xy^2 + \ln(x + 2y) = 1$.

Differentiation

On the Argand diagram, the shaded area contains the points for complex numbers $z$ that meet two inequalities. A circle and a straight line parallel to the real axis form the boundary, and those boundary lines are included in the shaded area.

Complex numbers

Start by casting the equation $\tan(x - 60^\circ) = 2\cot x$ into a quadratic in $\tan x$, then solve it for $0^\circ \leq x \leq 180^\circ$.

Trigonometry

The square roots of $-4 + 6\sqrt{5}i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ real and exact.

Complex numbers

The variables $x$ and $\theta$ are related by the differential equation $\dfrac{dx}{d\theta} = \left(\dfrac{1}{5}x + 1\right)\sin^2 2\theta$, with $x = 5$ when $\theta = 0$.

Differential equations

The diagram displays the curve $y = x^3 \cos 2x$ for $0 \leq x \leq \tfrac{1}{4}\pi$. This curve reaches a maximum point at $M$, where $x = p$.

Numerical solution of equations

The equations of two lines are $\mathbf{r} = \begin{pmatrix} -1 \\ 3 \\ -4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}$.

Vectors

Let $p(x)$ stand for the polynomial $6x^3 + ax^2 + bx + 9$, where $a$ and $b$ are constants. It is stated that $(x - 3)$ is a factor of $p(x)$, and that dividing the first derivative $p'(x)$ by $(x - 3)$ leaves a remainder of $72$.

Algebra

At a point, three coplanar forces with magnitudes $40\,\text{N}$, $30\,\text{N}$ and $X\,\text{N}$ act in the directions indicated in the diagram.

Forces and equilibrium

A cyclist moves along a straight horizontal road at a speed of $4\,\text{m s}^{-1}$ as she passes point $O$. She then accelerates uniformly over a distance of $42\,\text{m}$ until her speed becomes $V\,\text{m s}^{-1}$. After that, she continues at $V\,\text{m s}^{-1}$ for $50\,\text{m}$ and then slows down at $2\,\text{m s}^{-2}$ before stopping completely. The distance covered during deceleration is $16\,\text{m}$.

Kinematics of motion in a straight line

The aeroplane is travelling at a steady speed.

Energy, work and power

Particles $A$ and $B$ have masses $0.3\,\text{kg}$ and $0.1\,\text{kg}$ respectively. They are fixed to the two ends of a light inextensible string. The string runs over a fixed smooth pulley, and the particles hang vertically below the pulley. At the start, both particles are at a height of $x\,\text{m}$ above the horizontal ground (see diagram). The system is released from rest.

Newton's laws of motion

Particles $P$, $Q$ and $R$ have masses $0.6\,\text{kg}$, $0.4\,\text{kg}$ and $0.8\,\text{kg}$ respectively, and they are initially at rest on a smooth horizontal plane in a straight line. The separation from $P$ to $Q$ is $3\,\text{m}$, and the separation from $Q$ to $R$ is also $3\,\text{m}$. Particle $P$ is then projected directly towards $Q$ with speed $3\,\text{m s}^{-1}$. Once $P$ and $Q$ have collided, $P$ goes on moving in the same direction with speed $1.5\,\text{m s}^{-1}$.

Momentum

A block of mass $12\,\text{kg}$ rests on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\alpha = \tan^{-1}(0.5)$. A force of $X\,\text{N}$ acts on the block, directed straight up the plane. The coefficient of friction between the block and the plane is $\mu$.

Forces and equilibrium

A particle travels along a straight line. Its velocity $v\,\text{m s}^{-1}$, $t\,\text{s}$ after it leaves a fixed point $O$, is defined by $v = k(20 + pt - 6t^2)$, where $k$ and $p$ are constants. At $t = 1$, the particle has acceleration $42\,\text{m s}^{-2}$ and displacement from $O$ equal to $93\,\text{m}$.

Kinematics of motion in a straight line

Jacob flips three coins at the same time. The first coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{3}$. The second coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{4}$. The third coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{5}$. Let the random variable $X$ represent the total number of heads obtained.

Discrete random variables

During the previous year, an online store sold a large number of computers. $55\%$ of these computers were made by company $F$, $30\%$ by company $G$ and $15\%$ by company $H$. A random sample of $3$ customers, each of whom bought a computer from this store, is selected.

Probability

For a particular plant species, the lengths of $250$ leaves are measured to the nearest centimetre. The findings are shown in the table below: Length (cm): $5$-$9$, $10$-$14$, $15$-$19$, $20$-$24$, $25$-$29$, $30$-$39$ with matching frequencies $18$, $28$, $60$, $72$, $48$, $24$.

Representation of data

Eddie owns 16 toy cars: 8 are white, 5 are black and 3 are silver. He puts every car into a bag, then chooses three at random without replacement.

Probability

The daily mass of peaches sold in a supermarket follows a normal distribution with mean $65.8\text{ kg}$ and standard deviation $9.6\text{ kg}$.

The normal distribution

Alissa owns $10$ books from the Squares and Circles series. Apart from their colours, the books all appear alike. The set contains $3$ blue books, $2$ red books, $2$ yellow books, $1$ orange book, $1$ purple book and $1$ green book. She lines the books up on a shelf, and she cares only about the colour sequence.

Permutations and combinations

The random variables $X$ and $Y$ are independent, with distributions $N(44, 16)$ and $N(30, 9)$ respectively.

Linear combinations of random variables

A researcher has recorded the time, $T$ seconds, needed by adults to finish a questionnaire. The findings for a random sample of $60$ adults who completed the questionnaire this year are summarised as follows: $n = 60$, $\sum t = 3678$, $\sum t^2 = 226313.36$.

Hypothesis testing

The distribution of the random variable $X$ is $\operatorname{Po}(1.5)$.

Linear combinations of random variables

The diagram is the graph of the probability density function, $f$, for a random variable $X$. It is a straight line joining $(0, a)$ to $(2, b)$, with $a$ and $b$ both positive constants. For all other values of $x$, $f(x) = 0$.

Continuous random variables

Amir thinks that 20% of the students at his college are left-handed. His friend thinks that the true proportion, $p$, is below 20%. Amir intends to use the binomial distribution to test the null hypothesis, $H_0: p = 0.2$, against the alternative hypothesis, $H_1: p < 0.2$. He plans to select 35 students at random. If 3 or fewer of these students are left-handed, Amir will reject his belief.

Hypothesis testing

Nikki is looking into what students at her school think about the school sports facilities. She intends to hand out a survey to a sample of students.

Sampling and estimation