2025 Paper 2

A function is defined by $\displaystyle f(x) = \cos^{-1} 4x.$
Find $\displaystyle f'(x).$

A curve is defined by the equation $\displaystyle 2y^2 + 4xe^{2y} = 3x.$
Find an expression for $\displaystyle \frac{dy}{dx}$ in terms of $\displaystyle x$ and $\displaystyle y.$

Express $\displaystyle \frac{2x^2 - 18x + 4}{(x - 1)(x - 3)(x + 5)}$ in partial fractions.

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 1118 and 416.

(b)Hence find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 1118a + 416b = d.$

A curve is defined by $\displaystyle y = x^{\cot x}.$
Use logarithmic differentiation to find $\displaystyle \frac{dy}{dx}.$
Write your answer in terms of $\displaystyle x.$

(a)Find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^4$, for $\displaystyle \cos 3x.$

(b)Hence find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^4$, for $\displaystyle \cos^2 3x.$

A curve is defined on a suitable domain by the equations $\displaystyle x = t^2$ and $\displaystyle y = \tan t.$
Find in terms of $\displaystyle t$:

(a)$\displaystyle \frac{dy}{dx}$

(b)$\displaystyle \frac{d^2y}{dx^2}$

The matrix $\displaystyle A$ has the following property:

$\displaystyle A^2 = 6A - I$, where $\displaystyle I$ is the identity matrix.

(a)Express $\displaystyle A^3$ in the form $\displaystyle pA + qI$, where $\displaystyle p, q \in \mathbb{R}.$

Matrix $\displaystyle A$ is non-singular.

(b)Find a similar expression for $\displaystyle A^{-1}$ in terms of $\displaystyle A$ and $\displaystyle I.$

Relative to a fixed origin, the velocity, $\displaystyle v$ metres per second, of an object at time $\displaystyle t$ seconds is given by $\displaystyle v = 2t + e^{5t}.$

(a)Find an expression for the displacement of the object, $\displaystyle s$ metres, in terms of $\displaystyle t$, given that when $\displaystyle t = 0, s = 0.$

(b)Show that the acceleration of the object is always positive.

Find and fully factorise an expression for $\displaystyle \sum_{r=1}^{n} (r^3 - 3r).$

(a)Using the substitution $\displaystyle u = 2x^2$, or otherwise, find $\displaystyle \int xe^{-2x^2} dx.$

The diagram shows part of the curve with equation $\displaystyle y = \frac{4\sqrt{x}}{e^{x^2}}.$

A solid is generated by rotating the curve through $\displaystyle 2\pi$ radians about the $\displaystyle x$-axis from $\displaystyle x = 0$ to $\displaystyle x = 1.$

(b)Calculate the exact value of the volume generated.

Solve the differential equation

$\displaystyle \frac{d^2y}{dx^2} - 8\frac{dy}{dx} + 15y = 15x^2 - 31x + 40$

given that $\displaystyle y = 4$ and $\displaystyle \frac{dy}{dx} = 13$ when $\displaystyle x = 0.$

An infinite geometric sequence of positive numbers has second term 100 and fourth term 16.

(a)Determine:

(i) the common ratio
(ii) the first term of this sequence.

(b)Explain why the associated geometric series has a sum to infinity.

(c)Determine this sum to infinity.

A new geometric sequence is formed by multiplying each term in the sequence above by the real number $\displaystyle k$, where $\displaystyle k \neq 0.$

(d)State the effect that this will have on:

(i) the common ratio
(ii) the sum to infinity of the associated series.

Find the general solution of the differential equation

$\displaystyle \frac{dy}{dx} - \frac{2}{x}y = x^2 \sec^2 3x.$

Give your answer in the form $\displaystyle y = f(x).$

Prove by induction that, for all positive integers $\displaystyle n$,

$\displaystyle \sum_{r=1}^{n} \frac{1}{(2r + 1)(2r - 1)} = \frac{n}{2n + 1}.$

Use integration by parts to find

$\displaystyle \int e^{2x} \sin 5x \,dx.$

The volume, $\displaystyle V$ cm$\displaystyle ^3$, of water in a tank is given by

$\displaystyle V = \frac{1}{5}h^3$, where $\displaystyle h$ cm is the depth of water in the tank.

Water is being piped into the tank at a rate of 6 cm$\displaystyle ^3$/second.
Water is leaking from the bottom of the tank at a rate of $\displaystyle \frac{1}{10}\sqrt{h}$ cm$\displaystyle ^3$/second.

Calculate the rate of change of the depth of water when $\displaystyle h = 400.$

Let $\displaystyle z = x + iy$ be a complex number, where $\displaystyle x, y \in \mathbb{R}.$

(a)(i) Express $\displaystyle \bar{z} + iz$ in Cartesian form, where $\displaystyle \bar{z}$ is the complex conjugate of $\displaystyle z.$
(ii) Given $\displaystyle x > y$, find the argument of $\displaystyle \bar{z} + iz.$

When $\displaystyle x < y$, $\displaystyle \bar{z} + iz = r\left(\cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right)\right)$ where $\displaystyle r$ is the modulus of $\displaystyle \bar{z} + iz.$

(b)Use de Moivre's theorem to find, in polar form, both square roots of $\displaystyle \bar{z} + iz.$