2026 Paper 2

Differentiate $\displaystyle f(x)=3\sin^{-1}7x.$

Write down the binomial expansion of

$\displaystyle \left(x^{2}-\frac{5}{x}\right)^{4}$

and simplify your answer.

(a)Find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^{3}$, for:

(i) $\displaystyle e^{3x}$

(ii) $\displaystyle \ln(1+x).$

(b)Hence find and simplify the Maclaurin expansion, up to and including the term in $\displaystyle x^{3}$, for $\displaystyle e^{3x}\ln\left(\frac{1}{1+x}\right).$

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 1428 and 567.

(b)Find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 1428a+567b=d.$

Given $\displaystyle y=x^{4x}$, use logarithmic differentiation to find $\displaystyle \frac{dy}{dx}.$
Write your answer in terms of $\displaystyle x.$

(a)An arithmetic sequence has terms $\displaystyle u_{3}=6$ and $\displaystyle u_{11}=10.$
For this sequence, find the:

(i) common difference

(ii) first term

(iii) sum of the first 109 terms.

(b)The terms $\displaystyle v_{3}=18$ and $\displaystyle v_{4}=27$ form part of a geometric sequence.
For this sequence, find:

(i) the common ratio

(ii) the first term

(iii) an expression, in terms of $\displaystyle n$, for the sum of the first $\displaystyle n$ terms.

(c)Find algebraically the least value of $\displaystyle n$ such that the sum of the geometric series exceeds the sum of the first 109 terms in the arithmetic sequence.

A curve is defined parametrically by

$\displaystyle x=3\ln(2t+1),\quad y=t-\frac{1}{2}t^{2},$ where $\displaystyle t>0.$

(a)Find an expression for $\displaystyle \frac{dy}{dx}.$ Simplify your answer.

(b)Find the coordinates of the stationary point on the curve.

The volume, $\displaystyle V$ cubic metres, of water held in a reservoir is given by

$\displaystyle V=18(2+\sqrt{h})^{6}-1152$

where $\displaystyle h$ metres is the depth of water.
Water is pumped out of the reservoir at a constant rate of $\displaystyle 0.6$ m³s⁻¹.
Find the rate of change of the depth of water in the reservoir when the depth is 9 metres.

Express $\displaystyle 3442_{5}$ in base 9.

A curve is defined by the equation $\displaystyle x^{2}e^{6y}+x^{2}+y^{5}=50.$

(a)Find $\displaystyle \frac{dy}{dx}$ in terms of $\displaystyle x$ and $\displaystyle y.$

(b)Given $\displaystyle x>0$, explain why the derivative is never zero.

Find

$\displaystyle \int\frac{2x+4}{x^{2}+4}\,dx.$

Prove by induction that $\displaystyle 7^{n}+2$ is divisible by 3 for all $\displaystyle n\in\mathbb{N}.$

(a)Express using partial fractions $\displaystyle \frac{x+3}{(x+7)(x+5)}.$

(b)Hence find the particular solution of the differential equation

$\displaystyle \frac{dy}{dx}+\frac{2}{x+3}y=\frac{1}{(x+7)(x+5)(x+3)},$ where $\displaystyle x\ge0,$

given that $\displaystyle y=0$ when $\displaystyle x=3.$

The plane $\displaystyle \pi$ contains the points $\displaystyle P(2,3,-4)$, $\displaystyle Q(3,5,1)$ and $\displaystyle R(6,0,-6).$

(a)Determine the Cartesian equation of $\displaystyle \pi.$

The line $\displaystyle L$ has symmetric equations

$\displaystyle \frac{x-8}{2}=\frac{y+2}{-1}=\frac{z+2}{3}.$

$\displaystyle L$ intersects $\displaystyle \pi$ at the point $\displaystyle S.$

(b)Find the coordinates of $\displaystyle S.$

(c)Calculate the size of the acute angle between $\displaystyle L$ and $\displaystyle \pi.$

Let $\displaystyle r$ be a positive real number and consider the following statement:

If $\displaystyle r$ is irrational then $\displaystyle \sqrt{r}$ is irrational.

(a)Write down the contrapositive of the statement.

(b)Hence prove that the statement is true.

(a)Given $\displaystyle y=\ln(\cos x),\ 0\le x<\frac{\pi}{2}$, show that $\displaystyle \frac{dy}{dx}=-\tan x.$

For a function $\displaystyle g(x)$, it is known that

$\displaystyle \int xg(x)\,dx=2x\tan 2x-\int2\tan 2x\,dx.$

(b)

(i) Determine the exact value of $\displaystyle \int_{0}^{\frac{\pi}{6}}xg(x)\,dx.$

(ii) Find an expression for $\displaystyle g(x)$ in terms of $\displaystyle x.$