GATE Overflow for GATE XE - Questions without answers in Complex variables

GATE XE 2026 | Question: 1

Tue, 24 Feb 2026 15:50:38 +0000

For the complex function

$$f(z)=f(x+i y)=x^{3}-4 x y^{2}+i\left(4 x^{2} y-2 y^{3}\right)$$

consider the following two statements

$P: f$ is not analytic at $(0,0)$.
$Q: f$ does not satisfy the Cauchy-Riemann equations along the $x$-axis.

Then, which one of the following statements is true?

$P$ is false and $Q$ is true
$P$ is true and $Q$ is false
Both $P$ and $Q$ are true, and $Q$ implies $P$
Both $P$ and $Q$ are true, but $Q$ does not imply $P$

GATE XE 2026 | Question: 2

Tue, 24 Feb 2026 15:50:34 +0000

Let $C$ be the circle $(x-1)^{2}+y^{2}=25$ oriented counterclockwise. Then, the value of the line integral

$$\oint_{C}\left[\left(x^{5}-3 y\right) d x+\left(-2 x+e^{y^{2}}\right) d y\right]$$

$10 \pi$
$15 \pi$
$20 \pi$
$25 \pi$

GATE XE 2025 | Question: 18

Sun, 04 May 2025 19:08:33 +0000

Let $f(z)$ be an analytic function such that

$$ {Re}\left(f^{\prime}(z)\right)=3 x^{2}-4 y-3 y^{2}, f(i)=0, f^{\prime}(0)=0, $$

where $i=\sqrt{-1}$. Then the value of $f(1)$ is equal to

$4+2 i$
$1+5 i$
$1-i$
$4-2 i$

GATE XE 2024 | Question: 8

Sun, 21 Jul 2024 16:42:39 +0000

Consider $f(z)=e^{z}$, where $z=x+i y$ and $i=\sqrt{-1}$. Which of the following statements is correct?

$f$ is periodic
$f$ is not periodic
$|f|=1$
$\arg (f)=y \pm n \pi$ for all $n=0,1,2, \ldots$

GATE XE 2023 | Question: 5

Wed, 14 Feb 2024 18:10:13 +0000

Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?

$f(z)=z|z|, \quad z \in \mathbb{C}$
$f(z)=\sin (z), \quad z \in \mathbb{C}$
$f(z)=\left\{\begin{array}{r}e^{\frac{1}{z}}, z \neq 0 \\
0, z=0\end{array} \quad\right.$ for $\quad z \in \mathbb{C}$
$f(z)=e^{-z^{2}}, \quad z \in \mathbb{C}$

GATE XE 2022 | Question: 2

Fri, 17 Feb 2023 06:52:37 +0000

Let $\mathbb{C}=\{z=x+i y: x$ and $y$ are real numbers, $i=\sqrt{-1}\}$ be the set of complex numbers. Let the function $f(z)=u(x, y)+i v(x, y)$ for $z=x+i y \in \mathbb{C}$ be analytic in C, where
$$
u(x, y)=x y^3-y x^3 \quad \text { and } \quad v(x, y)=\frac{x^4}{4}+\frac{y^4}{4}-\frac{3}{2} x^2 y^2 .
$$
If $f^{\prime}(z)$ denotes the derivative of $f(z)$, then

$\left|f^{\prime}(-1+i)\right|^2=1$
$\left|f^{\prime}(-1+i)\right|^2=7$
$\left|f^{\prime}(-1+i)\right|^2=8$
$\left|f^{\prime}(-1+i)\right|^2=10$

GATE XE 2022 | Question: 4

Fri, 17 Feb 2023 06:52:35 +0000

Let $\Gamma$ be the positively oriented circle $x^2+y^2=9$ in the $x y$-plane. If
$$
\oint_{\Gamma}\left(3 y+e^{x \sin x}\right) d x+\left(7 x+\sqrt{e^y+2}\right) d y=\alpha \pi,
$$
where $\alpha$ is a real constant, then $\alpha$ is equal to ___________.