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TIFR Mathematics 2022 | Part A | Question: 20
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What is the number of real solutions of the equation
\[e^{\sin x}=\pi ?\]
$0$
$1$
Countably infinite
Uncountable
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TIFR Mathematics 2022 | Part B | Question: 20
Answer whether the following statements are True or False. The set $\{f(x) \in \mathbb{R}[x] \mid f(n) \in \mathbb{Z}$ for all $n \in \mathbb{Z}\}$ is uncountable.
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TIFR Mathematics 2022 | Part A | Question: 1
Consider the following properties of a metric space $(X, d)$ : $(X, d)$ is complete as a metric space. For any sequence $\left\{Z_{n}\right\}_{n \in \mathbb{N}}$ of closed nonempty subsets of $X$, such that $Z_{1} \supseteq Z_{2} \supseteq$ ... (I). (I) does not imply (II) but (II) implies (I). (I) does not imply (II) and (II) does not imply (I).
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TIFR Mathematics 2022 | Part A | Question: 2
Consider the following assertions: $\left\{(x, y) \in \mathbb{R}^{2} \mid x y=1\right\}$ is connected. $\left\{(x, y) \in \mathbb{C}^{2} \mid x y=1\right\}$ is connected. Which of the following sentences is true? Both (I) and (II) are true. (I) is true but (II) is false. (I) is false but (II) is true. Both (I) and (II) are false.
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TIFR Mathematics 2022 | Part A | Question: 3
What is the number of solutions of: \[ x=\frac{x^{2}}{50}-\cos \frac{x}{2}+2 \] in $[0,10]$ ? $0$ $1$ $2$ $\infty$
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