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Questions by soujanyareddy13

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941
TIFR-2019-Maths-A: 4
Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have: $f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$ Then which one of the following ... $a,b \in \mathbb{R}$
Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have:$$f\left ( b \right )-f\left ( a \righ...
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943
TIFR-2019-Maths-A: 6
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
The limit$$\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$$is equal to$1$$0$$+\infty$$1/2$
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946
TIFR-2019-Maths-A: 9
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ is bounded and uniformly continuous bounded but not uniformly continuous not bounded but uniformly continuous not bounded and not uniformly continuous
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ isbounded and uniformly continu...
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947
TIFR-2019-Maths-A: 10
Let $S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$ Then which of the following describes $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
Let $$S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$$Then which of the following describes $S$?$S=\left...
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948
TIFR-2019-Maths-A: 11
Let $f$ be a continuous function on $\left [ 0,1 \right ]$. Then the limit $\underset{n\rightarrow \infty }{lim}\int ^{1}_{0}nx^{n} f\left ( x \right )dx$ is equal to $f(0)$ $f(1)$ $\underset{x\in\left [ 0,1 \right ]}{sup} f\left ( x\right )$ The limit need not exist
Let $f$ be a continuous function on $\left [ 0,1 \right ]$. Then the limit $\underset{n\rightarrow \infty }{lim}\int ^{1}_{0}nx^{n} f\left ( x \right )dx$ is equal to $f(...
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953
TIFR-2019-Maths-A: 16
The number of ring homomorphisms from $\mathbb{Z}\left [ x,y \right ]$ to $\mathbb{F}_{2}\left [ x \right ]/\left ( x^{3}+x^{2}+x+1 \right )$ equals $2^{6}$ $2^{18}$ $1$ $2^{9}$
The number of ring homomorphisms from $\mathbb{Z}\left [ x,y \right ]$ to $\mathbb{F}_{2}\left [ x \right ]/\left ( x^{3}+x^{2}+x+1 \right )$ equals $2^{6}$$2^{18}$$1$$2^...
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956
TIFR-2019-Maths-A: 19
Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ ... given by $f \mapsto f+\frac{df}{dx}$ is injective but not surjective surjective but not injective neither injective nor surjective both injective and surjective
Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ that have derivatives of all orders. Then the map $C^{\inft...
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957
TIFR-2019-Maths-A: 20
A stick of length $1$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece? $1/8$ $1/4$ $1/e$ $1/\pi$
A stick of length $1$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece?$1/8$$1/4$$1/e$$1/\pi$
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958
TIFR-2020-Maths-B: 1
True/False Question : There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.
True/False Question :There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.
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960
TIFR-2020-Maths-B: 3
True/False Question : Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a bounded sequence of positive real numbers. Then: $\underset{n\rightarrow\infty }{lim sup}\:\frac{1}{a_{n}}=\frac{1}{\underset{n\rightarrow \infty }{lim\:inf \:a_{n}}}.$
True/False Question :Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a bounded sequence of positive real numbers. Then: $$\underset{n\rightarrow\infty }{lim sup}\:\frac...
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