Oct/Nov 2025

Determine the set of values of the constant $k$ for which the quadratic equation $3kx^2 + (k + 8)x + 3 = 0$ has two distinct real roots.

Quadratics

A circle is given by $x^2 + y^2 + 4y - 21 = 0$, while a straight line is given by $2x + y - 8 = 0$. These two graphs meet at two points.

Coordinate geometry

The curve goes through the point $P(4, 3)$ and satisfies $\frac{dy}{dx} = \frac{8}{x^2} - \frac{10}{(2x - 3)^2}$.

Differentiation

A geometric progression starts with first term $a$ and has common ratio $\cos\theta$, where $0 < \theta < \frac{1}{2}\pi$. The second term is given as $8$ and the fifth term as $\frac{1}{8}$.

Series

When $(px + 3)^5 - \left(x^3 + \frac{p}{x}\right)^4$ is expanded, the coefficient of $x^4$ is $216$.

Series

Write $1-6x-x^2$ in the form $a-(x+b)^2$, with $a$ and $b$ as constants.

Quadratics

Show, using identities, that $\tan^4\theta-1\equiv\frac{1-2\cos^2\theta}{\cos^4\theta}$.

Trigonometry

Functions $f$ and $g$ are given by $f(x)=(x+3)^2-12$ for $x\geq 0$, and $g(x)=2x-5$ for $x\in\mathbb{R}$.

Functions

The diagram illustrates a sector of a circle with centre $O$ and radius $r$ cm. The shaded part is enclosed by the chord $AB$ and the arc $AB$. Angle $AOB$ measures $\frac{2}{3}\pi$ radians. The circle's radius is increasing at $0.4\text{ cm s}^{-1}$.

Integration

The sketch depicts the curve given by the equation $y=\frac{1}{2}\sqrt{x}$ and the point $P$ with coordinates $\left(9,\frac{3}{2}\right)$. The shaded area is enclosed by the curve and the lines $x=0$ and $y=\frac{3}{2}$.

Integration

An arithmetic progression starts with term $2$ and has common difference $d$. Write the sum of the first $n$ terms as $S_n$.

Series

Express $9x^2 - 36x + 8$ in the form $p(x + q)^2 + r$, where $p$, $q$ and $r$ are constants.

Quadratics

The diagram contains a circle centred at A with radius $r$ that passes through B, C and D. There is also a larger circle, with centre C and radius $s$, which passes through B and D. Also, the length $BD$ is $s$.

Circular measure

Determine the term without \(x\) in the expansion of $(2x^2 - \frac{3}{x})^6$.

Series

The graph of $y=f(x)$ changes into the graph of $y=f(3x)+2$. Describe fully the two transformations that have been combined to produce the new graph.

Functions

The curve is defined by $\frac{dy}{dx}=kx^3+\frac{2}{x^2}$, where $k$ is a constant. It passes through $S(2,20)$, and the gradient at $S$ is $\frac{65}{2}$.

Differentiation

The curve is defined by $y=4x^{\frac{1}{2}}-x$. It reaches a maximum when $x=a$ and intersects the $x$-axis at $(b,0)$, where $b>0$. The shaded area is enclosed by the curve, the line $x=a$ and the $x$-axis (see diagram).

Differentiation

Draw the graph of $y = 3\sin x + 2$ for $0 \leq x \leq 2\pi$.

Trigonometry

The points $P$ and $Q$ have coordinates $(1, 1)$ and $(7, 11)$, respectively. The line segment $PQ$ is a diameter of a circle.

Coordinate geometry

The first three terms in a geometric progression are $a$, $b$ and $c$ respectively, with $a$, $b$ and $c$ all positive constants. The first three terms in an arithmetic progression are $a$, $b$ and $-3c$ respectively. You are told that $a = 9$ and that $c$ is the smaller of the two possible values.

Series

The function f is given by $f(x)=\frac{4}{(3x-6)^2}+\frac{1}{(3x-6)^3}$ for $x>2$. The function g is given by $g(x)=4x-3$ when $x>a$.

Functions

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

Series

The function $f$ has the rule $f(x) = px^2 + 4x + q$ for $x \in \mathbb{R}$, where $p$ and $q$ are constants.

Quadratics

The curve is defined by $y=\frac{8}{3x-8}-\frac{6}{x-1}$.

Differentiation

Solve the equation $\tan^{-1}(5x-3)=-\frac{1}{4}\pi$ for $x$.

Trigonometry

The curve is described by the equation $y=f(x)$, where $f(x)=\frac{1}{2}x^{\frac{2}{3}}(x-2)^2$. The points below are on the curve. Any non-exact values are correct to 6 decimal places. $A(8,72)$, $B(8.001,k)$, $C(8.01,72.300388)$, $D(8.1,75.038882)$.

Differentiation

The progression has $20$, $k$ and $k-5$ as its first, second and third terms respectively.

Series

The diagram illustrates a section of a circle with centre $O$ and radius $4\text{ cm}$. The chord $PQ$ measures $4\sqrt{3}\text{ cm}$, and $POQ=\theta$ radians. Point $X$ is located on the circle.

Circular measure

The diagram displays the graphs of $y=x^3$ and $y=f(x)$. The graph of $y=x^3$ is changed into the graph of $y=f(x)$ by a chain of transformations.

Functions

For $x>\frac{3}{a}$, the function $g$ is defined by $g(x)=\frac{2}{ax-3}+\frac{1}{2}$, where $a$ is a positive constant.

Functions

The points $(6,1)$ and $(-2,7)$ form the two endpoints of a diameter of a circle.

Coordinate geometry

The sketch depicts a segment of the curve $y=\frac{1}{2}x+\frac{4}{x}$ together with the line $y=4.5$.

Integration

The circle’s centre is $(2, 6)$, and its radius is $10$.

Coordinate geometry

For $x>2$, the function $f$ is given by $f(x)=3+\frac{7}{x-2}$.

Functions

The diagram illustrates the curve $y=4x^2-x^3$ together with the tangent drawn at point $P$. The $x$-coordinate of $P$ is $3$.

Integration

The first term of a geometric progression is $3 + 4\sqrt{2}$, and its second term is $5 - \sqrt{2}$.

Series

Write $4x^2 + 10x + 6$ in the form $a(x+b)^2 + c$, where $a$, $b$ and $c$ are rational constants that must be found.

Quadratics

The diagram illustrates a company’s new logo design. The circle sector, with centre $O$, has radius $r$ cm. The acute angle $AOC$ is $\frac{1}{3}\pi$ radians. The quadrilateral $OABC$ is a rhombus.

Circular measure

Solve the equation $x^3 - 28 + \frac{27}{x^3} = 0$ for $x$.

Quadratics

Show that the equation $6\sin\theta + \frac{1}{\tan\theta} = \frac{4}{\sin\theta}$ may be rewritten in the form $6\cos^2\theta - \cos\theta - 2 = 0$.

Trigonometry

A manufacturer plans to create an open cylindrical tank, as illustrated. The tank has a base but no lid. The tank's external surface area is fixed at $600\pi\text{ cm}^2$. Its radius $r$ cm and height $h$ cm may change.

Differentiation

The curve is defined by $y=\sqrt{6x+5}$. The shaded area is enclosed by the curve, the $x$-axis and the lines $x=a$ and $x=2a$, where $a$ is a positive constant. When this shaded area is rotated through $360^\circ$ about the $x$-axis, a solid is produced. The solid has volume, $V$, with $V>46\pi$.

Integration

In the expansion of $(p+qx)^4$, the coefficient of $x$ matches the coefficient of $x^2$. The constants $p$ and $q$ are both positive.

Series

Find the value of $\int 6\sin^2 x\, dx$.

Integration

Solve for $x$ in the equation $e^{2x}(e^{2x}-8)=48$.

Logarithmic and exponential functions

Solve the equation $|2x-3|=|5x+2|$.

Trigonometry

Solve $\cot\theta\tan(\theta+45^\circ)=7$ for $0^\circ<\theta<90^\circ$.

Trigonometry

The diagram depicts the curve with equation $y=8e^{-\frac{1}{3}x}-1$. It crosses the axes at the points $A$ and $B$. The shaded part is enclosed by the curve and the line segment $AB$.

Logarithmic and exponential functions

The parametric form of a curve is given by $x = \tan \theta$, $y = \sin \theta - 2\sin^3 \theta$, for $0 < \theta < \frac{1}{2}\pi$.

Differentiation

The polynomial $p(x)$ is given by $p(x)=2x^4+kx^3+kx^2+17x+18$, with $k$ a constant. You are told that $(x + 2)$ is a factor of $p(x)$. You are also told that the equation $p(x) = 0$ has exactly two real roots, called $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer.

Numerical solution of equations

The curve is given by $y=4e^{1-2x}\sqrt{3x-1}$.

Differentiation

Solve the equation $\ln(3x + 5) - \ln(x - 2) = 4$. Present your answer exactly.

Logarithmic and exponential functions

Solve the equation $2\tan^2\theta + 3\sec\theta = 18$ for $-180^\circ < \theta < 180^\circ$.

Trigonometry

Solve for $x$ in the inequality $|3x - 4| \leq |2x + 5|$.

Algebra

The polynomial $p(x)$ is given by $p(x) = x^4 - 10x^3 + 20x^2 - 30x + 40$.

Algebra

The diagram depicts the curve given by equation $y = 4\cos 2x + 8\sin x$ for $0 \leq x \leq \pi$. The curve’s maximum points are labelled $A$ and $B$, and the shaded region is enclosed by the line segment $AB$ and the curve.

Trigonometry

If \(\int_{-2a}^{a} \left( \frac{1}{2}e^{2x} + \frac{1}{4}e^{-x} \right) \, dx = 5\), where \(a\) is a positive constant, show that \(a = \frac{1}{2}\ln\left(10 + \frac{1}{2}e^{-a} + \frac{1}{2}e^{-4a}\right)\).

Numerical solution of equations

The curve is given by the equation $5x^2y + 4e^{2y} - 7x + 10 = 0$.

Differentiation

Calculate $\int 6\sin^2 x\, dx$.

Integration

Solve for $x$ in the equation $e^{2x}(e^{2x}-8)=48$.

Logarithmic and exponential functions

Solve the equation $|2x-3|=|5x+2|

Algebra

Solve $\cot\theta\tan(\theta+45^\circ)=7$ for $0^\circ<\theta<90^\circ$.

Trigonometry

The graph is the curve with equation $y=8e^{-\frac{1}{3}x}-1$. It crosses the axes at the points $A$ and $B$. The shaded part lies between the curve and the line segment $AB$.

Integration

The curve is described by the parametric equations $x = \tan \theta$, $y = \sin \theta - 2\sin^3 \theta$, for $0 < \theta < \frac{1}{2}\pi$.

Differentiation

The polynomial $p(x)$ is given by $p(x) = 2x^4 + kx^3 + kx^2 + 17x + 18$, with $k$ constant. It is stated that $(x + 2)$ is a factor of $p(x)$. It is also stated that the equation $p(x) = 0$ has exactly two real roots, named $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer.

Numerical solution of equations

A curve has equation $y=4e^{1-2x}\sqrt{3x-1}$.

Differentiation

Find the value of $\int 6\sin^2 x\, dx$.

Integration

Solve $e^{2x}(e^{2x}-8)=48$.

Logarithmic and exponential functions

Solve $|2x-3|=|5x+2|$.

Algebra

Solve the equation $\cot\theta\tan(\theta+45^\circ)=7$ for $0^\circ<\theta<90^\circ$.

Trigonometry

The sketch depicts the curve whose equation is $y=8e^{-\frac{1}{3}x}-1$. The curve intersects the axes at the points $A$ and $B$. The shaded region is enclosed by the curve and the line segment $AB$.

Integration

The curve is defined by the parametric equations $x = \tan \theta$, $y = \sin \theta - 2\sin^3 \theta$, for $0 < \theta < \frac{1}{2}\pi$.

Differentiation

The polynomial $p(x)$ is specified by $p(x) = 2x^4 + kx^3 + kx^2 + 17x + 18$, with $k$ a constant. You are told that $(x + 2)$ is a factor of $p(x)$. You are also told that the equation $p(x) = 0$ has precisely two real roots, named $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer.

Numerical solution of equations

A curve is given by the equation $y=4e^{1-2x}\sqrt{3x-1}$.

Differentiation

Determine the exact value of $\int_{1}^{2} \ln 3x\, dx$. Express your answer in the form $a + \ln b$, where $a$ and $b$ are integers.

Integration

Define $f(x)$ by $f(x) = \frac{x^3 + 2x - 11}{(3 + x)(2 + x^2)}$.

Algebra

Relative to the origin $O$, the position vectors of the points $A$, $B$, $C$ and $D$ are given by $\overrightarrow{OA}=\begin{pmatrix}1\\5\\3\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\4\\1\end{pmatrix}$, $\overrightarrow{OC}=\begin{pmatrix}1\\-3\\1\end{pmatrix}$ and $\overrightarrow{OD}=\begin{pmatrix}3\\-5\\4\end{pmatrix}$. The line $m$ goes through the points $A$ and $B$.

Vectors

Show that the equation $\log_{4}(2x + 1) = 2\log_{4}(3x - 1) - 2$ can be rewritten as a quadratic in $x$.

Logarithmic and exponential functions

Write \(3\sqrt{2}\sin(x + 45^{\circ}) + \cos x\) in the form \(R\cos(x - \alpha)\), where \(R > 0\) and \(0^{\circ} < \alpha < 90^{\circ}\).

Trigonometry

The diagram displays the graph of $y=e^{\sin 2x}\cos 4x$ for $0\leq x\leq \frac{1}{4}\pi$, together with its maximum point $M$.

Logarithmic and exponential functions

In the Argand diagram, the shaded set consists of complex numbers $z$ satisfying two inequalities. It is enclosed by a circle and by a line parallel to the imaginary axis, and the boundary curves are included in the shaded set.

Complex numbers

Solve the quadratic equation $(2+i)w^2+4w+2-i=0$. Give the solutions in the form $x+iy$, with $x$ and $y$ real.

Complex numbers

The curve is given in parametric form by $x=t^2-\ln(2t+1)$ and $y=\frac{t}{2t+1}$.

Differentiation

The variables $x$ and $y$ are linked by the differential equation $(x^2+1)\frac{dy}{dx}=kxe^{2y}$, where $k$ is a constant. It is also stated that $y=0$ when $x=0$ and that $y=-\frac{1}{2}$ when $x=1$.

Differential equations

Take the equation $\sec 2x = -e^{-x}$.

Numerical solution of equations

with $a$ taken as a positive constant.

Algebra

The diagram depicts a tank used to store water. It has the form of a cube with side $50\text{ cm}$. At time $t$ seconds, the water depth in the tank is $h\text{ cm}$. Water is fed into the tank at $5000\text{ cm}^3\text{ s}^{-1}$. Water leaves the tank through a hole in the base at a rate proportional to $h^2$. When $h=20$, the water depth is rising at $0.4\text{ cm s}^{-1}$.

Differential equations

Taking the origin $O$ as the reference point, the position vectors of points $A$, $B$ and $C$ are $\overrightarrow{OA}=4\mathbf{i}-2\mathbf{j}$, $\overrightarrow{OB}=2\mathbf{i}+8\mathbf{j}+4\mathbf{k}$, and $\overrightarrow{OC}=-2\mathbf{i}+6\mathbf{k}$. Point $M$ is the midpoint of $AB$, as the diagram shows.

Vectors

Solve the equation $3 \times 2^{x+1} = 4 \times 3^{2x-3}$ and provide your answer correct to 3 significant figures.

Logarithmic and exponential functions

The shaded part of the Argand diagram, enclosed by a straight line and a circle, shows the complex numbers $z$ for which $\operatorname{Re} z \leq 2$ and $|z - (3 + i)| \leq 2$ hold. The point $P$ marked on the diagram is one of the intersection points of the line and the circle.

Complex numbers

Determine the exact value of $\int_{0}^{1} x\tan^{-1}x\,dx$.

Integration

It is stated that $f(x)=(x-a)^2g(x)$, with $f(x)$ and $g(x)$ both polynomials. Show that $(x-a)$ is a factor of $f'(x)$.

Algebra

Take the equation $\cot 2x = 2\sin 2x - 1$.

Numerical solution of equations

A curve satisfies the equation $2y^3-3x^2y-x^3=16$.

Differentiation

Prove the identity $\sin 4x = 4\sin x\left(2\cos^3 x-\cos x\right)$.

Trigonometry

Define $f(x)=\dfrac{x^2+4ax+6a^2}{(x+2a)(x+3a)}$, where $a$ is a positive constant.

Integration

Solve for x in the inequality $|3x + 2| < 3|2x - 1|$.

Algebra

Express $\frac{2}{1 - 9y^2}$ using partial fractions.

Differential equations

The graph shown is $y=\sec^2 x\sqrt{3+2\tan x}$ over $-\frac{\pi}{4}\leq x\leq \frac{\pi}{4}$, and the minimum point is $M$.

Integration

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

Algebra

Solve the equation $2^{3x-4} = \frac{3}{5^x}$. Write your answer in the form $\frac{\ln m}{\ln n}$, where $m$ and $n$ are integers.

Logarithmic and exponential functions

On an Argand diagram, shade the set of points for complex numbers $z$ satisfying both $|z + 2i| \leq 3$ and $|z + 2i| \leq |z - 2 + 4i|$.

Complex numbers

Demonstrate that $\cos 4x + 2\sin^2 x - 1 = 8\sin^4 x - 6\sin^2 x$.

Trigonometry

Determine the exact value of $\int_{0}^{\frac{1}{2}\pi} x^2 \sin 2x\, dx$.

Integration

Solve the equation $\frac{5z}{2-i} - zz^{*} + 20 + 8i = 0$. Present your answers in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

The curve with equation $y=e^{-5x}\ln(5x)$ has a stationary point at $x=p$.

Numerical solution of equations

The line $l_1$ goes through the point $(3, 1, -6)$ and is parallel to the vector $2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$. The line $l_2$ goes through the point $(-1, 3, -6)$ and is perpendicular to the vector $3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$. The direction vector for $l_2$ has no component in the $x$-direction.

Vectors

Sketch the graph for $y = |3x - 6|$.

Algebra

The constant $a$ is chosen so that $\int_{0}^{a} x e^{\frac{1}{2}x}\,dx = 6$.

Numerical solution of equations

A fungal disease is now affecting some of the trees in a forest. Let $x$ be the fraction of trees affected after $t$ years. The rate of increase of $x$ is proportional to the product of the fraction already affected and the fraction still unaffected.

Differential equations

On an Argand diagram sketch, shade the set of complex numbers $z$ that satisfy both $|z - 1 - 3i| \leq 2$ and $|z| \geq 4$.

Complex numbers

The variables $x$ and $y$ are linked by the equation $Ay = b^x$, where $A$ and $b$ are constants.

Logarithmic and exponential functions

A curve is described by $x^2\ln(2y)-y\ln(2+x^2)=\ln 6$.

Differentiation

Find the complex numbers, $z$, for which $zz^*+5iz+2-10i=0$ holds. Give your answers in the form $x+iy$, where $x$ and $y$ are real.

Complex numbers

The equation of a curve is $y=\frac{\tan x}{5+2\sin x}$.

Differentiation

Use the substitution $u = x^2 - 3$ to show that $\int_{\sqrt{7}}^{\sqrt{12}} \frac{4x^3}{\sqrt{x^2 - 3}}\,dx = \int_{a}^{b} \frac{2u(u+3)}{\sqrt{u}}\,du$, with $a$ and $b$ as the values to determine.

Integration

Express $3\sqrt{3}\sin\left(\theta + \frac{\pi}{6}\right) - 2\sin\theta$ in the form $R\sin(\theta + \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. State the exact value of $R$ and give the value of $\alpha$ correct to 3 decimal places.

Trigonometry

The two lines are described by $l_1:\ \mathbf{r}=(2\mathbf{i}+\mathbf{j}+4\mathbf{k})+\lambda(\mathbf{i}+2\mathbf{j}-3\mathbf{k})$, $l_2:\ \mathbf{r}=(3\mathbf{i}-\mathbf{j}+5\mathbf{k})+\mu(2\mathbf{i}+3\mathbf{j}+a\mathbf{k})$.

Vectors

A car with mass 900 kg is travelling along a straight horizontal road, and it experiences a constant resistive force of 350 N. At the moment when its speed is $15\text{ m s}^{-1}$, its acceleration is $0.25\text{ m s}^{-2}$.

Newton's laws of motion

Particles $P$ and $Q$, with masses 3 kg and 5 kg respectively, start from rest on a smooth horizontal plane. $P$ is then projected directly towards $Q$ at a speed of $4\text{ m s}^{-1}$. After the collision, $P$ moves with speed $1\text{ m s}^{-1}$.

Momentum