Recent questions tagged tifrmaths2022 / GATE Overflow for GATE CSE

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TIFR Mathematics 2022 | Part A | Question: 11
Consider the real polynomial \[ f(x)=x^{11}-x^{7}+x^{2}-1 \] Which of the following sentences is correct? $f(x)$ has exactly one positive root. $f(x)$ has exactly two positive roots. $f(x)$ has at least three positive roots. None of the other three options.

Consider the real polynomial\[f(x)=x^{11}-x^{7}+x^{2}-1\]Which of the following sentences is correct?$f(x)$ has exactly one positive root.$f(x)$ has exactly two positive ...

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admin asked Sep 9, 2022

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32

TIFR Mathematics 2022 | Part A | Question: 12
Consider polynomials \[ f_{1}(x, y)=\sum_{i, j=0}^{\infty} a_{i j} x^{i} y^{j} \quad \text { and } \quad f_{2}(x, y)=\sum_{i, j=0}^{\infty} b_{i j} x^{i} y^{j} \in \mathbb{R}[x, y] \] $($where $a_{i j}=b_{i j}=0$ ... $(i, j)$ with $i+j=3$. $a_{i j}=b_{i j}$ if $i+j \leq 3$, but we may not have $f_{1}=f_{2}$.

Consider polynomials\[ f_{1}(x, y)=\sum_{i, j=0}^{\infty} a_{i j} x^{i} y^{j} \quad \text { and } \quad f_{2}(x, y)=\sum_{i, j=0}^{\infty} b_{i j} x^{i} y^{j} \in \mathbb...

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admin asked Sep 9, 2022

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33

TIFR Mathematics 2022 | Part A | Question: 13
What is the number of bijections $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ such that, for all distinct $i, j \in\{1, \ldots, 9\}$, whenever the squares labelled $i$ and $j$ in the diagram below share an edge, the squares labelled $f(i)$ ... $8$ $9$ $4$ $24$

What is the number of bijections $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ such that, for all distinct $i, j \in\{1, \ldots, 9\}$, whenever the squares labelle...

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admin asked Sep 9, 2022

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34

TIFR Mathematics 2022 | Part A | Question: 14
Let $m$ be the number of positive integers $n$ such that $1 \leq n \leq 2022$ and such that $n$ has an odd number of (positive integer) divisors. Then $m$ is $22$ $33$ $44$ $55$

Let $m$ be the number of positive integers $n$ such that $1 \leq n \leq 2022$ and such that $n$ has an odd number of (positive integer) divisors. Then $m$ is$22$$33$$44$$...

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admin asked Sep 9, 2022

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35

TIFR Mathematics 2022 | Part A | Question: 15
Let $S$ be the set of nonnegative continuous functions $f$ on $[0,1]$ satisfying \[ \int_{0}^{1} \sin ^{2}(x) f(x) d x=\int_{0}^{1} \sin (x) \cos (x) f(x) d x=\int_{0}^{1} \cos ^{2}(x) f(x) d x=1 . \] Then $S$ is: an uncountable set a countably infinite set a finite and nonempty set the empty set

Let $S$ be the set of nonnegative continuous functions $f$ on $[0,1]$ satisfying\[ \int_{0}^{1} \sin ^{2}(x) f(x) d x=\int_{0}^{1} \sin (x) \cos (x) f(x) d x=\int_{0}^{1}...

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admin asked Sep 9, 2022

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36

TIFR Mathematics 2022 | Part A | Question: 16
Let $f(x)=1-\sin x$ for $x \in \mathbb{R}$. Define \[ a_{n}=\sqrt[n]{f\left(\frac{1}{n}\right) f\left(\frac{2}{n}\right) \ldots f\left(\frac{n}{n}\right)}. \] Then $\left\{a_{n}\right\}_{n}$ converges to $0$ ... $\left\{a_{n}\right\}_{n}$ converges and $\displaystyle{}\lim _{n \rightarrow \infty} a_{n}>0$ none of the other three options is correct

Let $f(x)=1-\sin x$ for $x \in \mathbb{R}$. Define\[ a_{n}=\sqrt[n]{f\left(\frac{1}{n}\right) f\left(\frac{2}{n}\right) \ldots f\left(\frac{n}{n}\right)}. \]Then$\left\{a...

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admin asked Sep 9, 2022

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37

TIFR Mathematics 2022 | Part A | Question: 17
Which of the following is true for every function $u: \mathbb{R} \rightarrow \mathbb{R}$ which is continuously differentiable on $\mathbb{R} \;\text{(i.e}., u$ is diffferentiable on $\mathbb{R}$ and its derivative $u^{\prime}$ is continuous ... attains a minimum at some $x \in \mathbb{R}$. $u^{\prime}$ is nondecreasing. $u^{\prime}$ is nonincreasing.

Which of the following is true for every function $u: \mathbb{R} \rightarrow \mathbb{R}$ which is continuously differentiable on $\mathbb{R} \;\text{(i.e}., u$ is difffer...

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admin asked Sep 9, 2022

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38

TIFR Mathematics 2022 | Part A | Question: 18
Let $\left\{x_{n}\right\}$ be a sequence of positive numbers such that $\displaystyle{}\lim _{n \rightarrow \infty} x_{n}=x$. Define \[ z_{n}=\frac{1}{n}\left[x_{1}\left(1+\frac{x}{n}\right)^{n}+x_{2}\left(1+\frac{x}{n-1}\right)^{n-1}+\ ... $e^{x}$ $\left\{z_{n}\right\}$ does not have a limit $\left\{z_{n}\right\}$ converges to $x e^{x}$

Let $\left\{x_{n}\right\}$ be a sequence of positive numbers such that $\displaystyle{}\lim _{n \rightarrow \infty} x_{n}=x$. Define\[ z_{n}=\frac{1}{n}\left[x_{1}\left(1...

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admin asked Sep 9, 2022

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39

TIFR Mathematics 2022 | Part A | Question: 19
The value of \[\lim _{n \rightarrow \infty} \int_{0}^{1} \frac{n e^{x}}{1+n^{2} x^{2}} d x\] is $\pi e$ $\frac{\pi}{2}$ $\frac{\pi}{2} e$ $\pi$

The value of\[\lim _{n \rightarrow \infty} \int_{0}^{1} \frac{n e^{x}}{1+n^{2} x^{2}} d x\]is$\pi e$$\frac{\pi}{2}$$\frac{\pi}{2} e$$\pi$

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admin asked Sep 9, 2022

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40

TIFR Mathematics 2022 | Part A | Question: 20
What is the number of real solutions of the equation \[e^{\sin x}=\pi ?\] $0$ $1$ Countably infinite Uncountable

What is the number of real solutions of the equation\[e^{\sin x}=\pi ?\]$0$$1$Countably infiniteUncountable

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admin asked Sep 9, 2022

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