1
GATE ECE 2025
MCQ (More than One Correct Answer)
+1
-0

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as

$$ f(x)=2 x^3-3 x^2-12 x+1 $$

Which of the following statements is/are correct?

(Here, $\mathbb{R}$ is the set of real numbers.)

A
$f$ has no global maximizer
B
$f$ has no global minimizer
C
$x=-1$ is a local minimizer of $f$
D
$x=2$ is a local maximizer of $f$
2
GATE ECE 2025
Numerical
+1
-0

The function $y(t)$ satisfies

$$ t^2 y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0 $$

where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is ___________ (rounded off to two decimal places).

Your input ____
3
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval.

Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality

$$ \beta \leq \int_2^8 f(x) d x \leq \alpha $$

is guaranteed to hold.

A
$\beta=5 \mathrm{~m}, \alpha=7 \mathrm{M}$
B
$\beta=6 \mathrm{~m}, \alpha=5 \mathrm{M}$
C
$\beta=7 \mathrm{~m}, \alpha=6 \mathrm{M}$
D
$\beta=7 \mathrm{~m}, \alpha=5 \mathrm{M}$
4
GATE ECE 2025
MCQ (More than One Correct Answer)
+2
-0

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

A
$\oint_C e^z d z=0$
B
$\oint_C z^n d z=0$, where $n$ is an even integer
C
$\oint_C \cos z d z \neq 0$
D
$\oint_C \sec z d z \neq 0$
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