PYQs Based Test - 18 : Engineering Mathematics

The vector function F(r) = -x î + yĵ is defined over a circular are C shown in the figure.



The line integral of ∫${}_C$ F(r) ⋅ dr is

A function f(x) = 1 - x² + x³ is defined in the closed interval [-1, 1]. The value of x, in the open interval (-1, 1) for which the mean value theorem is satisfied, is

The contour C given below is on the complex plane z = x + jy, where $\mathrm{j}=\sqrt{-1}$



The value of the integral $\frac{1}{\pi \mathrm{j}}\underset{\mathrm{c}}{\oint }\frac{\mathrm{dz}}{{\mathrm{z}}^2-1}$ is _______

Which one of the following functions is analytic over the entire complex plane?

Consider the matrix

$J_6=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right]$

Which is obtained by reversing the order of columns of the Identity matrix I₆.

Let P = I₆ + α J₆, where α is a non-negative real number. The value of α for which det (P) = 0 is _________.