TIFR-2015-Maths-B-9

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+2 votes

1 answer

1

TIFR-2015-Maths-B-12
Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$? $[0, 1]$ $(0, 1)$ $[0, 1)$ $\mathbb{R}$

asked Dec 21, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 164 views

+4 votes

1 answer

2

TIFR-2015-Maths-B-6
Let $f: [0, 1]\rightarrow \mathbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0, 1)$ and $f(0) = f(1) = 0$. Then the equation $f(x) = f' (x)$ admits. No solution $x \in (0, 1)$ More than one solution $x \in (0, 1)$ Exactly one solution $x \in (0, 1)$ At least one solution $x \in (0, 1)$

asked Dec 21, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 138 views

+1 vote

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3

TIFR-2015-Maths-B-3
Let $f$ be a function from $\left \{ 1, 2,....10 \right \}$ to $\mathbb{R}$ such that $(\sum_{i=1}^{10}\frac{|f(i)|}{2^{i}})^{2} = (\sum_{i=1}^{10} |f(i)|^{2}) (\sum_{i=1}^{10}\frac{1}{4^{i}})$ ... statement. There are uncountably many $f$ with this property. There are only countably infinitely many $f$ with this property. There is exactly one such $f$ There is no such $f$.

asked Dec 20, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 75 views

+3 votes

0 answers

4

TIFR-2015-Maths-A-9
Let $\left\{a_{n}\right\}$ be a sequence of real numbers such that $|a_{n+1}-a_{n}|\leq \frac{n^{2}}{2^{n}}$ for all $n \in \mathbb{N}$. Then The sequence $\left\{a_{n}\right\}$ may be unbounded. The sequence $\left\{a_{n}\right\}$ is bounded but may not converge. The sequence $\left\{a_{n}\right\}$ has exactly two limit points. The sequence $\left\{a_{n}\right\}$ is convergent.

asked Dec 19, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 72 views

+2 votes

0 answers

5

TIFR-2015-Maths-B-7
A complex number $\alpha \in \mathbb{C}$ is called algebraic if there is a non-zero polynomial $P(x) \in \mathbb{Q}\left[x\right]$ with rational coefficients such that $P(\alpha)=0$. Which of the following statements is true? There are only finitely many algebraic ... . All complex numbers are algebraic. $\sin \frac{\pi}{3}+ \cos \frac{\pi}{4}$ is algebraic. None of the above.

asked Dec 21, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 75 views

+2 votes

1 answer

6

TIFR-2015-Maths-A-7
Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct? If $g$ is continuous, then $f ∘ g$ is continuous. If $f$ is continuous, then $f ∘ g$ is ... $f ∘ g$ are continuous, then $g$ is continuous. If $g$ and $f ∘ g$ are continuous, then $f$ is continuous.

asked Dec 19, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 122 views

+2 votes

0 answers

7

TIFR-2015-Maths-A-2
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Which one of the following sets cannot be the image of $(0, 1]$ under $f$? {0} (0, 1) [0, 1) [0, 1]

asked Dec 19, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (29.5k points) | 65 views

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