May/June 2025

Solve $6\sin\theta = 1 + \frac{2}{\sin\theta}$ over the range $-180^{\circ} < \theta < 180^{\circ}$.

Trigonometry

The functions $f$ and $g$ are given by $f(x)=\sqrt{x}$ for $x\geq 0$, and $g(x)=3\sqrt{x+2}-5$ for $x\geq -2$.

Functions

The curve is defined by the equation $\frac{dy}{dx} = 4(2x - 5)^3 - 9x^{\frac{1}{2}}$, and it passes through the point $A\left(4, -\frac{11}{2}\right)$.

Differentiation

In a geometric progression, the third term is 18 and the total of the first three terms is 26. The common ratio is stated to be negative.

Series

The diagram illustrates the curve with equation $y = 5x^2 - 20x$ together with the straight line with equation $y = x - 16$. The $x$-coordinates of the intersection points of the curve and line are 1 and 16.

Integration

Write the first three terms, in ascending powers of $x$, for the expansion of $(2 - px)^5$.

Series

The curve is described by $2x^2 - kxy + 2 = 0$ and the line by $y = px + 3$, with $k$ and $p$ as constants.

Coordinate geometry

A curve is given by $y = 4x^2 + \tfrac{9}{x^2} - 8$.

Differentiation

The circle with equation $x^2 + y^2 - 6x + 10y - 27 = 0$ meets the line $x = -2$ at points $P$ and $Q$.

Coordinate geometry

The diagram depicts a sector $ABC$ of a circle with centre $A$ and radius $r$ cm. The angle $BAC$ is $\alpha$ radians, where $0 < \alpha < \frac{1}{2}\pi$.

Circular measure

The diagram displays the graphs of the equations $y = f(x)$ and $y = g(x)$.

Functions

The first three terms of an arithmetic progression are $4k$, $k^2$ and $8k$ respectively, with $k$ a non-zero constant.

Series

Write $x^2 + 4x + 2$ in the form $(x + a)^2 + b$, where $a$ and $b$ are integers.

Functions

Find the coordinates of the points where the curve and the line with equations $2xy + 5y^2 = 24$ and $2x + y + 4 = 0$ meet.

Coordinate geometry

In the expansion of $(px^2 + \frac{4}{p}x)^5$, the coefficient of $x^7$ equals $1280$.

Series

A point $P$ travels along the curve with equation $y = ax^{\frac{3}{2}} - 12x$ so that the $x$-coordinate of $P$ is increasing at a steady rate of 5 units per second.

Differentiation

The curve is given by $y = 4\cos 2x + 3$ for $0 \leq x \leq 2\pi$.

Trigonometry

The diagram displays the curve given by equation $y = \frac{9}{\sqrt{5x + 4}}$ together with the line $y = 6 - 3x$. The line and the curve meet at point $P$, and $P$ has $y$-coordinate 3.

Integration

Prove that $$\frac{\tan \theta + 7}{\tan^2 \theta - 3} = \frac{\sin \theta \cos \theta + 7 \cos^2 \theta}{1 - 4 \cos^2 \theta}$$ is an identity.

Trigonometry

The diagram represents the circle with equation $x^2 + y^2 - 14x + 8y + 36 = 0$ together with the line $y = -2$. This line cuts the circle at $A$ and $B$. The circle’s centre is $C$.

Circular measure

For the curve, $\frac{d^2 y}{dx^2} = -\frac{24}{x^3}$. It is also stated that the curve has a stationary point at $(-2, 19)$.

Differentiation

The curve is given by $y = 2x + \frac{12}{x^2}$.

Differentiation

A curve $C$ is given by the equation $y = \frac{9}{2x - 5} + 2x - 5$.

Differentiation

The function $f$ is given by $f(x) = x^2 + 4ax + a$ for every $x \in \mathbb{R}$, with $a$ a constant. The function $g$ is defined so that $g^{-1}(x) = \sqrt[3]{2x - 4}$ for every $x \in \mathbb{R}$.

Functions

The progression’s first two terms are $4\sin^2\theta$ and $8\sin^3\theta$, with $\theta$ an angle for which $0 < \theta \leq \frac{1}{6}\pi$.

Series

If $\int_{1}^{3} \left( \frac{a}{(4x-3)^2} + 2 \right) \, dx = 12$, find the value of the constant $a$.

Integration

Find the first three terms of $(2 - \tfrac{3}{2}x)^5$ in ascending powers of $x$.

Series

Solve the equation $4 \sin \theta \tan \theta = 1 + 5 \cos \theta$ for $-180^\circ < \theta < 180^\circ$.

Trigonometry

An arithmetic progression starts with first term $a$ and has common difference $2$. Its $N$th term is $55$, and the total of the first $3N$ terms is $5760$.

Series

The curve is defined by $\frac{dy}{dx} = 3x^2 + 10x - 8$.

Differentiation

The diagram depicts a square $ABCD$ with each side measuring $12\ \text{cm}$. The points $E$ and $F$ are located on the sides $BC$ and $CD$ respectively, and satisfy $BE = \frac{1}{3}BC$ and $DF = \frac{1}{3}DC$. The arc $EF$ is a segment of a circle centred at $A$. The shaded region is enclosed by the arc $EF$ and the line segments $EC$ and $FC$.

Circular measure

Points $P$, $Q$ and $R$ are given by the coordinates $P(-13,\ 5)$, $Q(5,\ 1)$ and $R(2,\ k)$, where $k$ is a constant. The angle $PRQ$ is a right angle.

Coordinate geometry

A curve is defined by $\frac{dy}{dx} = 12(2x - 5)^2 + 8x$. The curve is known to pass through $(2, 4)$.

Integration

A circle is described by $x^2 + y^2 + 4x - 8y - 12 = 0$.

Coordinate geometry

When $(3 + ax)^5 + (6 - x)^4$ is expanded, the coefficient of $x^2$ is six times the coefficient of $x$.

Series

Use completing the square to determine the exact solutions of the equation $4x^2 - 4x - 1 = 0$.

Quadratics

The diagram shows a section of the curve $y = x^2 - \frac{1}{x^2}$. The shaded area is enclosed by the curve, the line $x = 2$ and the $x$-axis.

Integration

A sector $ABD$ of a circle is shown, with centre $A$ and radius $10\text{ cm}$. The perpendicular bisector of $AB$ passes through $D$.

Circular measure

On her birthday each year, Ananya is given some money by each parent. For Ananya’s first birthday, her father gives her $\$10$. In every later year, he increases this by $\$5$ compared with the previous year. For Ananya’s first birthday, her mother also gives her $\$10$. In every later year, she increases this by $20\%$ compared with the previous year.

Series

In parallelogram $ABCD$, $A$ has coordinates $(3, 7)$, $B$ has coordinates $(6, p)$, and $D$ has coordinates $(1, p)$. It is stated that the gradient of $AB$ is $-\frac{2}{3}$.

Coordinate geometry

A curve is given by $y = x^3 + ax^2 + bx + 5$. It has a stationary point at $(1, 9)$.

Differentiation

The functions $f$ and $g$ are given by $f(x) = \cos x$ for $0 \leq x \leq \pi$, and by $g(x) = 3\cos(x - \pi) + 2$ for $\pi \leq x \leq 2\pi$.

Functions

If $y = 6x \cos(x^2 + 1)$, then...

Differentiation

Solve the inequality $4^x < 0.05$ by using logarithms. Present your answer in the form $x < a$, with $a$ correct to 3 significant figures.

Logarithmic and exponential functions

On one diagram, sketch the graphs of $y = 3e^{-2x}$ and $y = \sec x$ for values of $x$ such that $0 \leq x < \frac{1}{2}\pi$.

Numerical solution of equations

The diagram illustrates the curve given by $y = 6e^{2x} - e^{3x}$. The shaded area lies between the axes and the curve.

Integration

The polynomial $p(x)$ is given by $p(x) = ax^3 + bx^2 - ax - 24$, where $a$ and $b$ are constants. It is stated that $(2x - 3)$ is a factor of $p(x)$ and that, when $p(x)$ is divided by $(x + 1)$, the remainder is $-15$.

Trigonometry

For the curve, the parametric equations are $x = \frac{2t + 1}{3t + 4}$ and $y = 2\ln(3t + 4)$, with $t > -\frac{4}{3}$.

Differentiation

Show that $\sin^2 2x + 4\cos^2 x \cos 2x = 4\cos^4 x$.

Trigonometry

Show that $\int_{2}^{11} \frac{8}{4x + 1} \, dx = \ln a$, where $a$ is an integer to be found.

Integration

On one diagram, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

Algebra

Find the coordinates of the stationary points on the curve with equation $y = \frac{8x}{2x + 3} - 6x + 5$.

Differentiation

The diagram displays sections of the curves given by $y = 4e^{-2x}$ and $y = 1 + 0.5\sin 3x$. $P$ marks one point where the curves intersect, and the shaded part is enclosed by the two curves together with the $y$-axis.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = ax^4 + bx^3 + 13x^2 - 35x + 15$, where $a$ and $b$ are constants. It is stated that $(2x - 1)$ and $(x - 3)$ are factors of $p(x)$.

Trigonometry

The curve is defined by $(x^2 - 3)\ln y + 6x = 14$.

Differentiation

Write $4 \cos \theta \sin(\theta + 30^\circ)$ as $R \cos(2\theta - \alpha) + k$, with $R > 0$, $0^\circ < \alpha < 90^\circ$ and $k$ constant.

Trigonometry

Show that $\int_{-2}^{11} \frac{8}{4x + 1} \, dx = \ln a$, where $a$ is the integer to be determined.

Integration

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

Algebra

Identify the coordinates of the stationary points of the curve with equation $y = \frac{8x}{2x + 3} - 6x + 5$.

Differentiation

The diagram displays sections of the curves $y = 4e^{-2x}$ and $y = 1 + 0.5\sin 3x$. Point $P$ is where the curves intersect, and the shaded area is enclosed by the two curves and the $y$-axis.

Numerical solution of equations

The polynomial $p(x)$ has the form $p(x) = ax^4 + bx^3 + 13x^2 - 35x + 15$, where $a$ and $b$ are constants. It is stated that $(2x - 1)$ and $(x - 3)$ are factors of $p(x)$.

Trigonometry

The curve is given by the equation $(x^2 - 3)\ln y + 6x = 14$.

Differentiation

Express $4\cos\theta\sin(\theta + 30^\circ)$ as $R\cos(2\theta - \alpha) + k$, where $R > 0$, $0^\circ < \alpha < 90^\circ$ and $k$ is a constant.

Trigonometry

Show that $\int_{2}^{11} \frac{8}{4x + 1}\,dx = \ln a$, where $a$ is an integer to be found.

Integration

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

Algebra

Determine the coordinates of the stationary points for the curve given by $y = \frac{8x}{2x + 3} - 6x + 5$.

Differentiation

The diagram displays portions of the curves with equations $y = 4e^{-2x}$ and $y = 1 + 0.5\sin 3x$. Point $P$ is where the curves meet, and the shaded part is enclosed by the two curves and the $y$-axis.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = ax^4 + bx^3 + 13x^2 - 35x + 15,$ where $a$ and $b$ are constants. It is stated that $(2x - 1)$ and $(x - 3)$ are factors of $p(x)$.

Trigonometry

The curve is defined by the equation $(x^2 - 3)\ln y + 6x = 14.$

Differentiation

Write $4\cos\theta\sin(\theta + 30^\circ)$ in the form $R\cos(2\theta - \alpha) + k$, where $R > 0$, $0^\circ < \alpha < 90^\circ$ and $k$ is a constant.

Trigonometry

Sketch the graph of $y = |2x - 3|$.

Algebra

Determine the quotient and remainder when $x^3 + 5x^2 - 2x - 15$ is divided by $x^2 - 3$.

Differential equations

The graph depicts the curve $y = \cos x\sqrt{\sin 2x}$ for $0 \leq x \leq \tfrac{1}{2}\pi$. The curve reaches a maximum at $M$, where $x = a$.

Integration

The equation is $2\ln p + \ln(p - 1) - \tfrac{1}{2}\ln(q + 1) = 3.$

Algebra

Determine the complex numbers $z$ for which $\frac{z + 5i}{z - 5}$ is real and $|z| = \sqrt{17}$. Give your answers in the form $z = x + iy$, where $x$ and $y$ are real.

Complex numbers

The parametric form of the curve is $x = e^{\tan t}$, $y = 3\tan^2 t$.

Differentiation

The polynomial $3x^3 + pax^2 + 7a^2x + qa^3$ is represented by $f(x)$, with $p$, $q$ and $a$ as constants and $a \neq 0$. If $f(x)$ is divided by $(x + 2a)$, the remainder is $-22a^3$. If $f(x)$ is divided by $(3x - a)$, the remainder is $-a^3$.

Algebra

Given that $z_1 = 3e^{\frac{\pi i}{4}}$, $z_2 = \tfrac{3}{2}e^{\frac{3\pi i}{4}}$ and $\omega = 2e^{\frac{\pi i}{2}}$.

Complex numbers

Express $5\sin\left(x + \tfrac{\pi}{6}\right) - 4\cos x$ as $R\sin(x - \alpha)$, where $R > 0$ and $0 < \alpha < \tfrac{\pi}{2}$. State the exact value of $R$ and give $\alpha$ correct to 3 decimal places.

Trigonometry

With reference to the origin $O$, the position vectors of $A$ and $B$ are $2\mathbf{i} + 4\mathbf{k}$ and $5\mathbf{i} + \mathbf{j} + 6\mathbf{k}$ respectively. The line $l_1$ passes through $A$ and $B$.

Vectors

The constant $a$ is defined by $\int_1^a 6x\ln x\,dx = 4.$

Numerical solution of equations

Solve the equation $\dfrac{e^x + 2e^{-x}}{e^x - 3} = 4$. State your answer to 3 decimal places.

Logarithmic and exponential functions

Find quotient and remainder when $x^2$ is divided by $1 + 4x^2$.

Integration

The diagram displays the graph of $y = 5\sin 2x\cos^2 x$ for $0 \leq x \leq \tfrac{1}{2}\pi$ and its maximum point $M$.

Integration

Expand $(6 - x)(1 - 2x)^{-\frac{3}{2}}$ in ascending powers of $x$, up to and including the term in $x^2$, with the coefficients simplified.

Algebra

On an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both inequalities.

Complex numbers

Solve the equation $3\cot x - 4\cot 2x = 3$ for angles $x$ with $0^{\circ} \leq x \leq 180^{\circ}$.

Trigonometry

The square roots of $-1 - 4\sqrt{5}i$ may be written in Cartesian form as $x + iy$, with $x$ and $y$ both real and exact.

Complex numbers

Using a pair of suitable sketches, show that the equation $|x - 2| = 2\sin\tfrac{1}{2}x$ has only one root in the interval $0 < x < \pi$.

Numerical solution of equations

Write $7\sin\theta + 24\cos\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \tfrac{1}{2}\pi$. Give $\alpha$ correct to $4$ decimal places.

Trigonometry

The variables $x$ and $\theta$ are linked by the differential equation $\sin 2\theta\,\frac{dx}{d\theta} = (4x + 3)\cos 2\theta$, and $x = 0$ at $\theta = \tfrac{1}{12}\pi$.

Differential equations

Relative to origin O, the position vectors of points A, B and C are given by $\overrightarrow{OA} = \begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} -2 \\ 1 \\ 3 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}$.

Vectors

Given a positive constant $a$, sketch the graph of $y = |3x - 2a|$.

Algebra

The variables $x$ and $y$ are linked by the differential equation $\sin 4y \frac{dy}{dx} = x \sin 2y \sin 3x$. It is also stated that $y = \frac{1}{12}\pi$ when $x = \frac{1}{2}\pi$.

Differential equations

The sketch illustrates the curve $y = \sqrt{x} \sin 2x$ for $0 \leq x \leq \frac{1}{2}\pi$. This curve reaches a highest point at $M$, where $x = a$.

Numerical solution of equations

Solve for $x$ in $2\ln(2x + 3) - \ln(2x + 5) = \ln(3x)$.

Logarithmic and exponential functions

Find the exact value for $\int_{\frac{5\pi}{12}}^{\frac{\pi}{4}} 3\cos^2 5x\,dx$.

Integration

You are given $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$. Show that $(z_1 z_2)^* = z_1^* z_2^*$.

Complex numbers

The curve is defined by $xy + y^2 e^{-x} = 4$.

Differentiation

Find all complex numbers $z$ such that $\displaystyle \frac{z + 4}{z + 4i}$ is real and $|z| = \sqrt{10}$. Give your answers in the form $z = x + iy$, where $x$ and $y$ are real.

Complex numbers

Consider $f(x) = \frac{3a - 5x}{(3a + 2x)(2a - x)}$, where $a$ is a positive constant.

Algebra

Prove that the identity $\cot^2 \theta - \tan^2 \theta = 4 \cot 2\theta \cosec 2\theta$ holds.

Trigonometry

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = i + 2j$, $\overrightarrow{OB} = i + 3j - 2k$ and $\overrightarrow{OC} = 2i - j + 3k$. The line $l$ goes through $B$ and $C$.

Vectors

Solve the equation $3^{4-2x} = 5\left(6^{x-1}\right)$. State your answer correct to $3$ significant figures.

Logarithmic and exponential functions

Relative to the origin $O$, the position vectors for $A$, $B$ and $C$ are given by $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j} - 6\mathbf{k}$, $\overrightarrow{OB} = b\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OC} = -4\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}$.

Vectors

The variables $x$ and $y$ are related by the differential equation $(x^2 + 3)\frac{dy}{dx} = e^{3y}(x - 2)$. It is known that $y = 0$ when $x = 0$.

Differential equations

Solve $3\cot\theta - 4\csc^2\theta + 5 = 0$ for $-\pi \leq \theta \leq \pi$.

Trigonometry

The complex numbers $s$ and $t$ are specified by $s = 5(\cos 0.25 + i\sin 0.25)$ and $t = 6e^{3i}$.

Complex numbers

Determine the exact coordinates of the stationary point on the curve with equation $y = 3x^3 \ln x^4$, for $x > 0$.

Differentiation

The diagram displays the locus of points for the complex numbers $z$ that satisfy $|z + 5 - 4i| = 3$.

Complex numbers

For this curve, the parametric equations are $x = \frac{2}{\cos 3t}$ and $y = \tan 3t$, with $0 \leq t \leq 2\pi$.

Differentiation

A curve is described by $y = \tan^{-1}(4x)$.

Integration

Using sketches of an appropriate pair of graphs, show that the equation $\sec 2x = -2x - \frac{1}{2}$ has precisely one root in the interval $0 \le x \le \frac{\pi}{2}$.

Numerical solution of equations

Express $\frac{12x^2 + 55x - 2}{(3x - 2)(x + 6)}$ as a sum of partial fractions.

Algebra

A block with mass $12\,\text{kg}$ is pulled upward by a rope on a rough plane. The plane makes an angle of $20^\circ$ to the horizontal. The rope acts parallel to the line of greatest slope of the plane. The coefficient of friction between the block and the plane is $0.4$. The block’s acceleration is $2\,\text{m s}^{-2}$.

Forces and equilibrium

Three coplanar forces with magnitudes $P\,\text{N}$, $5\,\text{N}$ and $10\,\text{N}$ act at a point $O$, as indicated in the diagram. Their resultant has magnitude $Q\,\text{N}$ and is directed perpendicular to the force of magnitude $P\,\text{N}$.

Forces and equilibrium

The diagram displays the cyclist’s velocity-time graph. It is made up of three straight-line sections. The cyclist goes through point O at speed $3\,\text{m s}^{-1}$, then accelerates for $10\,\text{s}$ at constant acceleration $0.5\,\text{m s}^{-2}$. After that, he continues at constant speed for $30\,\text{s}$ before slowing down, finally coming to rest at point P, and he covers a distance of $80\,\text{m}$ during the deceleration.

Kinematics of motion in a straight line

A lorry with mass $18000\,\text{kg}$ is moving along a straight road.

Energy, work and power

For a particle $P$ with mass $m\,\text{kg}$ and speed $u\,\text{m s}^{-1}$, the momentum is $4\,\text{N s}$ and the kinetic energy is $16\,\text{J}$.

Momentum

Particles $P$ and $Q$ have masses $0.3\,\text{kg}$ and $0.6\,\text{kg}$ respectively, and are connected to the two ends of a light inextensible string. The string runs over a smooth pulley at the point $B$, where the inclined planes $AB$ and $BC$ meet. $P$ is on the smooth plane $AB$, which is inclined at an angle $\theta$ to the horizontal and has $\sin \theta = 0.4$. $Q$ is on plane $BC$, which is inclined at $30^\circ$ to the horizontal. The string is taut, and the particles can move along the lines of greatest slope of the planes (see diagram). The particles are released from rest.

Energy, work and power