2023 Paper 2

The function $\displaystyle f$ is defined by $\displaystyle f(x)=2 \sin^{-1} 3x.$
Find $\displaystyle f'(x).$

Find $\displaystyle \int \frac{x^2}{x^3+10}dx.$

Matrix $\displaystyle A$ is defined by

$\displaystyle A = \begin{pmatrix} 2 & 2x & 4 \\ x & -1 & 0 \\ 1 & 0 & -2 \end{pmatrix}$

where $\displaystyle x \in \mathbb{R}.$

(a)Find a simplified expression for the determinant of $\displaystyle A.$

(b)Hence, determine whether $\displaystyle A^{-1}$ exists for all values of $\displaystyle x.$

Calculate the gradient of the tangent to the curve with equation $\displaystyle x^2y^2-2y=\sin 3x$ at the point (0,0).

(a)Write down and simplify the general term in the binomial expansion of $\displaystyle \left(3x-\frac{2}{x^2}\right)^8.$

(b)Hence, or otherwise, determine the coefficient of $\displaystyle x^{-1}.$

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 703 and 399.

(b)Find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle d=703a+399b.$

(c)Hence find integers $\displaystyle p$ and $\displaystyle q$ such that $\displaystyle 76=703p+399q.$

(a)Solve the differential equation

$\displaystyle \frac{dy}{dx}-2y=6e^{5x}$

given that when $\displaystyle x=0, y=-1.$ Express $\displaystyle y$ in terms of $\displaystyle x.$

(b)The solution of the differential equation in (a) is also a solution of

$\displaystyle \frac{d^3y}{dx^3}-5\frac{d^2y}{dx^2}=ke^{2x}$, $\displaystyle k \in \mathbb{R}.$

Find the value of $\displaystyle k.$

The fourth and seventh terms of a geometric sequence are 9 and 243 respectively.

(a)Find the:
(i) common ratio
(ii) first term.

(b)Show that $\displaystyle \frac{S_{2n}}{S_n} = 1+3^n$ where $\displaystyle S_n$ represents the sum of the first $\displaystyle n$ terms of this geometric sequence.

Express $\displaystyle 572_{10}$ in base 9.

A curve is defined by $\displaystyle y=x^{5x^2}$, where $\displaystyle x>0.$
Find $\displaystyle \frac{dy}{dx}$ in terms of $\displaystyle x.$

On a building site, water is stored in a container. The container is a cone with diameter 180 cm at its widest point and height of 150 cm.

(a)Show that when the water level is at a height of $\displaystyle h$ cm, $\displaystyle 0 \le h \le 150$, the volume of water in the container can be written as

$\displaystyle V = \frac{3\pi h^3}{25}$

[The volume of a cone is given by $\displaystyle V = \frac{1}{3}\pi r^2 h.$]

Water is pumped into the container at a constant rate of 10 litres per second.

(b)Find the rate at which the height is increasing when $\displaystyle h = 125.$

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

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

Points scored in the long jump element of the decathlon can be calculated using a solution of the differential equation

$\displaystyle \frac{dP}{dm} = \frac{1.4P}{m-220}$, $\displaystyle m > 220$

where $\displaystyle m$ is the distance jumped in centimetres and $\displaystyle P$ the points scored.

Given that a jump of 807 centimetres scores 1079 points, find an expression for $\displaystyle P$ in terms of $\displaystyle m.$

A complex number is defined by $\displaystyle w=a+ib$, where $\displaystyle a$ and $\displaystyle b$ are positive real numbers.

Given $\displaystyle w^2=8+6i$, determine the values of $\displaystyle a$ and $\displaystyle b.$

A function $\displaystyle f(x)$ has the following properties:

$\displaystyle f'(x) = \frac{x+1}{1+(x+1)^4}$

the first term in the Maclaurin expansion of $\displaystyle f(x)$ is 1.

(a)Find the Maclaurin expansion of $\displaystyle f(x)$ up to and including the term in $\displaystyle x^2.$

(b)Use the substitution $\displaystyle u=(x+1)^2$ to find $\displaystyle \int \frac{x+1}{1+(x+1)^4}dx.$

(c)Determine an expression for $\displaystyle f(x).$