PYQs Based Test - 24 : Taylor and Laurent's Series

The Taylor series expansion of $\frac{1}{z-2},|z|<1$ is

We have f(x) = x${}^{2+e}$${}^{2x}$. Which of the following is the correct expansion of f(x + 1) about 1?

For a function f(x, y) at (1, 2), the following are given fₓ = 20, f${}_y$ = 24, f(1, 2) = 30, fₓₓ = 22, f${}_{xy}$ = 8, f${}_{yy}$ = 10 using this information and the 2${}^{nd}$ order Taylor expansion of f(x, y) write the coefficient of x.

(The subscripts denote partial derivatives $f_x=\frac{\partial f}{\partial x},f_{xx}=\frac{{\partial }^{2}f}{\partial x^2}$)

What is the expansion of y = sin⁻¹ x?

Consider the following complex function:

$f\left(z\right)=\frac{9}{\left(z-1\right){\left(z+2\right)}^2}$

Which of the following is one of the residues of the above function?