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TIFR Mathematics 2024 | Part A | Question: 1

admin asked in Others Jan 19 edited Jan 21 by makhdoom ghaya

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What is the number of even positive integers $n$ such that every group of order $n$ is abelian?

$1$
$2$
Greater than $2$, but finite
Infinite

tifrmaths2024

admin asked in Others Jan 19 edited Jan 21 by makhdoom ghaya

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TIFR Mathematics 2024 | Part A | Question: 2
Let $n$ be a positive integer, and let \[ S=\{g \in \mathbb{R}[x] \mid g \text { is a polynomial of degree at most } n\}. \] For $g \in S$, let $A_{g}=\left\{x \in \mathbb{R} \mid e^{x}=g(x)\right\} \subset \mathbb{R}$. Let \[ m=\min \left\{\# A_{g} \mid ... \left\{\# A_{g} \mid g \in S\right\} . \] Then $m=0, M=n$ $m=0, M=n+1$ $m=1, M=n$ $m=1, M=n+1$

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TIFR Mathematics 2024 | Part A | Question: 3
Let $V, W$ be nonzero finite dimensional vector spaces over $\mathbb{C}$. Let $m$ be the dimension of the space of $\mathbb{C}$-linear transformations $V \rightarrow W$, viewed as a real vector space. Let $n$ ... transformations $V \rightarrow W$, viewed as a real vector space. Then $n=m$ $2 n=m$ $n=2 m$ $4 n=m$

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TIFR Mathematics 2024 | Part A | Question: 4
Consider the real vector space of infinite sequences of real numbers \[ S=\left\{\left(a_{0}, a_{1}, a_{2}, \ldots\right) \mid a_{k} \in \mathbb{R}, k=0,1,2, \ldots\right\} . \] Let $W$ be the subspace of $S$ ... 2}=2 a_{k+1}+a_{k}, \quad k=0,1,2, \ldots \] What is the dimension of $W$ ? $1$ $2$ $3$ $\infty$

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TIFR Mathematics 2024 | Part A | Question: 5
Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a continuous function. If \[ \lim _{n \rightarrow \infty} \int_{0}^{1} f(x+n) d x=2, \] then which of the following statements about the limit \[ \lim _{n \rightarrow \infty} ... equals $0$ The limit exists and equals $\frac{1}{2}$ The limit exists and equals $2$ None of the remaining three options is correct

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tifrmaths2024

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