1
GATE EE 2024
Numerical
+1
-0.33

Consider the complex function $f(z) = \cos z + e^{z^2}$. The coefficient of $z^5$ in the Taylor series expansion of $f(z)$ about the origin is ______ (rounded off to 1 decimal place).

Your input ____
2
GATE EE 2024
Numerical
+1
-0.33

The sum of the eigenvalues of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ______ (rounded off to the nearest integer).

Your input ____
3
GATE EE 2024
MCQ (Single Correct Answer)
+2
-1.33

Consider a vector $\vec{u} = 2\hat{x} + \hat{y} + 2\hat{z}$, where $\hat{x}$, $\hat{y}$, $\hat{z}$ represent unit vectors along the coordinate axes $x$, $y$, $z$ respectively. The directional derivative of the function $f(x, y, z) = 2\ln(xy) + \ln(yz) + 3\ln(xz)$ at the point $(x, y, z) = (1, 1, 1)$ in the direction of $\vec{u}$ is

A

0

B

$\frac{7}{5\sqrt{2}}$

C

7

D

21

4
GATE EE 2024
MCQ (More than One Correct Answer)
+2
-1.33

Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty < t < \infty$. Which of the following statements is/are always true?

A

$f(t)$ has at least one local minimum.

B

$f(t)$ cannot have two distinct local minima.

C

$f(t)$ has at least one local maximum.

D

The minimum value of $f(t)$ cannot be negative.

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