2022 Paper 1

Given $\displaystyle y = \frac{1-3x}{x^2+4}$, find $\displaystyle \frac{dy}{dx}.$ Simplify your answer.

Given $\displaystyle f(x) = \text{cosec } 5x$, find $\displaystyle f'(x).$

Use Gaussian elimination to solve the following system of equations:

$\displaystyle x - 2y + z = 4$
$\displaystyle 2x + y - 3z = 3$
$\displaystyle x - 7y - 4z = 9$

Given that $\displaystyle z_1 = 5 + 3i$ and $\displaystyle z_2 = 6 + 2i$, express $\displaystyle z_1\overline{z_2}$ in the form $\displaystyle a + ib$ where $\displaystyle a$ and $\displaystyle b$ are real numbers.

A curve is defined by the equation $\displaystyle y^3 + 4y = 2xy + 1.$

(a)Use implicit differentiation to find an expression for $\displaystyle \frac{dy}{dx}.$

(b)Find the gradient of the tangent to the curve when $\displaystyle y = -1.$

(c)Show that the curve has no stationary point.

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

(b)Hence find the first four terms of the Maclaurin expansion of $\displaystyle \frac{3+2x}{e^{4x}}.$

Consider the statement: For all odd numbers $\displaystyle n$, $\displaystyle n^2 + 4$ is prime.

(a)Find a counterexample to show that the statement is false.

(b)Prove directly that the difference between the cubes of any two consecutive integers is not divisible by 3.

(a)Use the substitution $\displaystyle u = y^2 + 1$, or otherwise, to find the exact value of

$\displaystyle \int_0^5 \frac{4y}{\sqrt{y^2+1}} \,dy.$

Student engineers are using a 3D printer to make a model. Relative to a suitable set of axes, the cross-section of the model is symmetrical about the $\displaystyle y$-axis and is represented in the first quadrant by the curve $\displaystyle x = \frac{4y}{\sqrt{y^2+1}}$, $\displaystyle 0 \le y \le 5.$

(b)State the area of the cross-section.

(c)Express $\displaystyle \frac{y^2}{y^2+1}$ in the form $\displaystyle a + \frac{b}{y^2+1}$ where $\displaystyle a$ and $\displaystyle b$ are real numbers.

The curve $\displaystyle x = \frac{4y}{\sqrt{y^2+1}}$, $\displaystyle 0 \le y \le 5$ will be rotated through $\displaystyle 2\pi$ radians about the $\displaystyle y$-axis to make the model.

(d)Find the volume of the model.