GATE 2021 EC question paper PDF and answer key

Mirror every shaded square across P-Q and count the mirror positions that are empty. Each of the six squares that has no partner on the other side of the line needs one added square, so 6 squares are required.

Both purple and black lie inside green, but nothing links them to each other, so neither conclusion follows on its own. However, I and II are complementary (either some black are purple or none are), so exactly one of them must hold: either I or II.

$\mathbf{F} = \nabla\phi$ with $\phi = \frac{-x^2 + y^2}{2}$. The arc goes from $(1, 0)$ to $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$: $\int_C \mathbf{F}\cdot d\mathbf{r} = \phi\left(\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}}\right) - \phi(1, 0) = 0 - \left(-\frac{1}{2}\right) = \frac{1}{2}$

$x\left(1 + \frac{t}{2}\right)$ is $x$ stretched in time by 2, so its bandwidth is 5 kHz. Multiplication convolves the spectra, giving a bandwidth of $10 + 5 = 15$ kHz and a Nyquist rate of 30 kHz.

The determinant must be zero: $\begin{vmatrix} 1 & -1 & 2 \\ 7 & 3 & x \\ 2 & 3 & 1 \end{vmatrix} = (3 - 3x) + (7 - 2x) + 2(21 - 6) = 40 - 5x = 0 \Rightarrow x = 8$