GATE Overflow for GATE XE - Recent questions in Functions of Single Variable

GATE XE 2024 | Question: 1

Sun, 21 Jul 2024 16:42:45 +0000

Let
\[
f(x)=\left\{\begin{array}{cr}
\pi+x, & -\pi \leq x<0, \\
0, & 0 \leq x<\pi,
\end{array}\right.
\]
with $f(x+2 \pi)=f(x)$. If $F(x)$ represents the Fourier series of $f(x)$, then the value of $F\left(-\frac{\pi}{2}\right)+F(0)$ is

$0$
$\frac{\pi}{2}$
$\pi$
$\frac{3 \pi}{2}$

GATE XE 2024 | Question: 4

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

Assume that $f:[0,1] \rightarrow \mathbb{R}$ is continuous on $[0,1]$ and differentiable on $(0,1)$ such that $f(x+h)=f(x)+h f^{\prime}(x+\theta h)$ for some $0<\theta<1$. If $f(x)=x^{2}(1+x)$, and $\theta$ is expressed in terms of $x$ and $h$, then the value of
\[
\lim _{h \rightarrow 0} \theta(x, h)
\]
is

$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{4}{5}$

GATE XE 2023 | Question: 6

Wed, 14 Feb 2024 18:10:12 +0000

If the polynomial

$P(x)=a_{0}+a_{1} x+a_{2} x(x-1)+a_{3} x(x-1)(x-2)$

interpolates the points $(0,2),(1,3),(2,2)$, and $(3,5)$, then the value of $P\left(\frac{5}{2}\right)$ is _______$\text{(round off to 2 decimal places)}$.

GATE XE 2023 | GA Question: 9

Tue, 13 Feb 2024 18:01:43 +0000

Which one of the given figures $P, Q, R$ and $S$ represents the graph of the following function?
$$
f(x)=|| x+2|-| x-1||
$$

$\text{P}$
$\text{Q}$
$\text{R}$
$\text{S}$

GATE XE 2022 | Question: 1

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

The value of

$\lim _{x \rightarrow 0} \frac{1}{x} \int_{2}^{2+x}\left(t+\sqrt{t^{2}+5}\right) d t$

GATE XE 2022 | Question: 8

Fri, 17 Feb 2023 06:52:33 +0000

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be the continuous function such that $f(x)=2+\frac{g(x)}{x}$ for all $x>0$, where $g(x)=\int_{1}^{x} f(t) d t$ for all $x>0$. Then $f(2)$ is equal to

$2+\log _{e} 2$
$2-\log _{e} 2$
$2+\log _{e} 4$
$2-\log _{e} 4$