1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
What is the value of ${ }^{23} C_0+{ }^{23} C_2+{ }^{23} C_4+\cdots+{ }^{23} C_{22}$ ?
A
$2^{23}$
B
$2^{22}-1$
C
$2^{23}+1$
D
$2^{22}$
2
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
3
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
$e^\pi-1$
4
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.
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