GATE EC 2023 Question Paper (05-Feb-2023) (Shift 2)

Let ${\mathrm{v}}_1=\left[\begin{array}{c}1\\ 2\\ 0\end{array}\right]$ and ${\mathrm{v}}_2=\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right]$ be two vectors. The value of the coefficient 𝛼 in the expression v₁ = αv₂ + e, which minimizes the length of the error vector 𝒆, is

The rate of increase, of a scalar field 𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧, in the direction 𝒗 = (2, 1, 2) at a point (0, 2, 1) is

Let 𝑤⁴ = 16𝑗. Which of the following cannot be a value of 𝑤?

The value of the contour integral, ${\oint }_{\mathrm{c}}\left(\frac{\mathrm{z}+2}{{\mathrm{z}}^2+2z+2}\right)\mathrm{dz}$, where the contour C is $\left\{\mathrm{z}:\left|\mathrm{z}+1-\frac{3}{2}\mathrm{j}\right|=1\right\}$ taken in the counter clockwise direction, is

Let the sets of eigenvalues and eigenvectors of a matrix 𝐵 be {𝜆${}_{\text{k}}$ |1 ≤ 𝑘 ≤ 𝑛} and {𝒗${}_{\text{k}}$ |1 ≤ 𝑘 ≤ 𝑛}, respectively. For any invertible matrix 𝑃, the sets of eigenvalues and eigenvectors of the matrix 𝐴, where 𝐵 = 𝑃⁻¹𝐴𝑃, respectively, are