1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $f: Q \rightarrow Q$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbf{Q}$, and $f(1)=$ 10. Which one of the following statements is Correct?
A
$f$ is neither injective nor surjective
B
$f$ is injective but not surjective
C
$f$ is surjective but not injective
D
$f$ is bijective
2
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
$e^\pi-1$
3
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.
4
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
$1 / 4$
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