1
GATE ECE 2023
MCQ (Single Correct Answer)
+1
-0.33

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

A
$${2 \over 3}$$
B
$${4 \over 3}$$
C
2
D
4
2
GATE ECE 2023
MCQ (Single Correct Answer)
+1
-0.33

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

A
$$ - \pi (1 + j)$$
B
$$\pi (1 + j)$$
C
$$\pi (1 - j)$$
D
$$ - \pi (1 - j)$$
3
GATE ECE 2023
MCQ (Single Correct Answer)
+1
-0.33

Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are

A
$$\{ {\lambda _k}\,\mathrm{det}(A)|1 \le k \le n\} $$ and $$\{ P{v_k}|1 \le k \le n\} $$
B
$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$
C
$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ P{v_k}|1 \le k \le n\} $$
D
$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {P^{ - 1}}{v_k}|1 \le k \le n\} $$
4
GATE ECE 2023
MCQ (Single Correct Answer)
+2
-0.67

The value of the line integral $$\int_P^Q {({z^2}dx + 3{y^2}dy + 2xz\,dz)} $$ along the straight line joining the points $$P(1,1,2)$$ and $$Q(2,3,1)$$ is

A
20
B
24
C
29
D
$$-$$5
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