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TIFR-2011-Maths-A-24

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makhdoom ghaya asked Dec 9, 2015
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TIFR-2011-Maths-A-1
Consider the sequence $\left \{x_{n} \right \}$ defined by $x_{n}=\frac{\left[nx\right]}{n}$ for $x \in \mathbb{R}$ where $[·]$ denotes the integer part. Then $\left \{x_{n} \right \}$ Converges to $x.$ Converges but not to $x.$ Does not converge Oscillates
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makhdoom ghaya asked Dec 9, 2015
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TIFR-2011-Maths-A-6
The function $f_{n}(x)= n \sin (x/n)$ Does not converge for any $x$ as $n \rightarrow \infty$ Converges to the constant function $1$ as $n \rightarrow \infty$ Converges to the function $x$ as $n \rightarrow \infty$ Does not converge for all $x$ as $n \rightarrow \infty$
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makhdoom ghaya asked Dec 9, 2015
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Suppose $f_{n}(x)$ is a sequence of continuous functions on the closed interval $[0, 1]$ converging to $0$ point wise. Then the integral $\int_{0}^{1} f_{n}(x) \text{d}x$ converges to 0.
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makhdoom ghaya asked Dec 9, 2015
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TIFR-2011-Maths-A-22
There exists a group with a proper subgroup isomorphic to itself.
There exists a group with a proper subgroup isomorphic to itself.
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