Cambridge AS & A Level. 2025. Mathematics (9709). May/June - Paper 1 (13) · Pure Mathematics 1

A curve has equation $y = 2x + \dfrac{12}{x^2}$.

Find the equation of the tangent to the curve at the point $(-2, -1)$. Give your answer in the form $y = mx + c$.

The first two terms of a geometric progression are

$4\sin^2\theta,\quad 8\sin^3\theta,$

where $\theta$ is an angle such that $0 \lt \theta \lt \frac{1}{6}\pi$.

Given that the sum to infinity of the progression is $\frac{1}{2}$, find the value of $\theta$. Give your answer in the form $\sin^{-1}k$, where $k$ is a rational number.

Given that $\displaystyle\int_1^3 \left(\frac{a}{(4x-3)^2} + 2\right)dx = 12$, find the value of the constant $a$.

**(a)** Find the first three terms in the expansion of $\left(2 - \frac{3}{2}x\right)^5$ in ascending powers of $x$.

**(b)** (In part (a) you found that the first three terms in the expansion of $\left(2 - \frac{3}{2}x\right)^5$ are $32 - 120x + 180x^2$.)

Use your answer to part (a), with a suitable value of $x$, to find an approximation to $1.985^5$.

Solve the equation

$4\sin\theta\tan\theta = 1 + 5\cos\theta$

for $-180^\circ \lt \theta \lt 180^\circ$.

An arithmetic progression has first term $a$ and common difference 2. The $N$th term is 55 and the sum of the first $3N$ terms is 5760.

Find the values of $N$ and $a$.

A curve is such that $\dfrac{dy}{dx} = 3x^2 + 10x - 8$.

**(a)** Find the set of values of $x$ for which $y$ decreases as $x$ increases.

A curve is such that $\dfrac{dy}{dx} = 3x^2 + 10x - 8$.

**(b)** (In part (a) you found that $y$ decreases for $-4 \lt x \lt \frac{2}{3}$.)

It is given that the maximum point of the curve has $y$-coordinate 27.

Find the equation of the curve.

The diagram shows a square $ABCD$ where each side has length 12 cm. Points $E$ and $F$ lie on the sides $BC$ and $CD$ respectively and are such that $BE = \frac{1}{3}BC$ and $DF = \frac{1}{3}DC$. The arc $EF$ is part of a circle with centre $A$. The shaded region is bounded by the arc $EF$ and the line segments $EC$ and $FC$.

**(a)** Show that the size of angle $EAF$ is 0.9273 radians, correct to 4 significant figures.

The diagram shows a square $ABCD$ where each side has length 12 cm. Points $E$ and $F$ lie on the sides $BC$ and $CD$ respectively and are such that $BE = \frac{1}{3}BC$ and $DF = \frac{1}{3}DC$. The arc $EF$ is part of a circle with centre $A$. The shaded region is bounded by the arc $EF$ and the line segments $EC$ and $FC$.

**(b)** (In part (a) you showed that angle $EAF = 0.9273$ radians.)

Find the perimeter of the shaded region.

The diagram shows a square $ABCD$ where each side has length 12 cm. Points $E$ and $F$ lie on the sides $BC$ and $CD$ respectively and are such that $BE = \frac{1}{3}BC$ and $DF = \frac{1}{3}DC$. The arc $EF$ is part of a circle with centre $A$. The shaded region is bounded by the arc $EF$ and the line segments $EC$ and $FC$.

**(c)** (In part (a) you showed that angle $EAF = 0.9273$ radians.)

Find the area of the shaded region.

Three points $P$, $Q$ and $R$ have coordinates $P(-13, 5)$, $Q(5, 1)$ and $R(2, k)$, where $k$ is a constant. It is given that the angle $PRQ$ is a right angle.

**(a)** Show that one of the possible values of $k$ is 10, and find the other possible value.

Three points $P$, $Q$ and $R$ have coordinates $P(-13, 5)$, $Q(5, 1)$ and $R(2, k)$, where $k$ is a constant. It is given that the angle $PRQ$ is a right angle.

**(b)** It is now given that $k = 10$. A circle passes through the points $P$, $Q$ and $R$.

Find the equation of the tangent to the circle at $R$. Give your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

A curve $C$ has equation $y = \dfrac{9}{2x - 5} + 2x - 5$.

**(a)** Find the coordinates of the two stationary points.

A curve $C$ has equation $y = \dfrac{9}{2x - 5} + 2x - 5$.

**(b)** (In part (a) you found that the stationary points are $(1, -6)$ and $(4, 6)$.)

Find $\dfrac{d^2y}{dx^2}$ and hence determine the nature of each stationary point.

A curve $C$ has equation $y = \dfrac{9}{2x - 5} + 2x - 5$.

**(c)** The curve $C$ is transformed to the curve $C_1$ using a translation of $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$ followed by reflection in the $x$-axis.

(The stationary points of $C$ are the maximum $(1, -6)$ and the minimum $(4, 6)$.)

**(i)** State the coordinates of the maximum point of $C_1$.

Write your answer as $(x, y)$.

A curve $C$ has equation $y = \dfrac{9}{2x - 5} + 2x - 5$.

**(c)** The curve $C$ is transformed to the curve $C_1$ using a translation of $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$ followed by reflection in the $x$-axis.

**(ii)** Find the equation of $C_1$ in the form $y = \dfrac{a}{bx + c} + dx + e$, where $a$, $b$, $c$, $d$ and $e$ are integers.

The function f is defined by $\text{f}(x) = x^2 + 4ax + a$ for $x \in \mathbb{R}$, where $a$ is a constant.

The function g is such that $\text{g}^{-1}(x) = \sqrt[3]{2x - 4}$ for $x \in \mathbb{R}$.

**(a)** Given that the range of f is $\text{f}(x) \ge -33$, find the possible values of $a$.

The function f is defined by $\text{f}(x) = x^2 + 4ax + a$ for $x \in \mathbb{R}$, where $a$ is a constant.

The function g is such that $\text{g}^{-1}(x) = \sqrt[3]{2x - 4}$ for $x \in \mathbb{R}$.

**(b)** Given instead that $\text{fgg}(0) = 96$, find the value of $a$.