1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $I=\int_{e^{-\pi / 2}}^{e^{\pi / 2}}\left(\sin ^2(\log (x))+\sin \left(\log \left(x^2\right)\right)\right) d x$. What is the value of $I$ ?
A
0
B
$\frac{\pi e^{\frac{\pi}{2}}}{2}$
C
$e^{\pi / 2}-e^{-\pi / 2}$
D
$e^\pi-1$
2
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1

Consider the following subset of the $X Y$-plane.

$$ S=\left\{\left(|z-\mathrm{i} z|,|z|^2\right): \quad z \text { is a complex number }\right\} $$

Which one of the following statements is correct?

A
$S$ is a circle
B
$S$ is a parabola.
C
$S$ is an ellipse but not a circle
D
$S$ is a hyperbola.
3
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
A ship sets off on a voyage with three engines, labelled $A, B$, and $C$, which work independently. The ship can complete the voyage only if at least two of these engines keep working. The probability that engine $A$ breaks down is $1 / 4$, that engine $B$ breaks down is $1 / 4$, and that engine $C$ breaks down is $1 / 2$. What is the probability that the ship can complete the voyage?
A
3/4
B
$1 / 2$
C
$1 / 32$
D
$1 / 4$
4
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Consider the differential equation $\cos (y) \frac{d y}{d x}+\frac{1}{x} \sin (y)=x, \quad(x>0)$; given that, $y=\frac{\pi}{2}$ at $x=\sqrt{3}$. Which one of the following is the value of $y$ at $x=\sqrt{\frac{3}{2}}$ ?
A
$\frac{\pi}{6}$
B
$\frac{\pi}{3}$
C
$\frac{\pi}{2}$
D
$\frac{\pi}{4}$
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