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282
TIFR Mathematics 2023 | Part A | Question: 4
Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ ... . None of the remaining three options is correct.
Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ be its inverse function. Which of the fol...
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284
TIFR Mathematics 2023 | Part A | Question: 6
For every positive integer $n,$ define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by $f_{n}(x)=\dfrac{\sin \left(n^{2} x\right)+\cos \left(e^{n} x\right)}{1+n^{2} x^{2}}$. Then \[ \lim _{n \rightarrow \infty} \int_{0}^{1-\sin (1 / n)} f_{n}(x) d x \] equals $1$. $0.$ $\infty$. $1 / 2$.
For every positive integer $n,$ define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by $f_{n}(x)=\dfrac{\sin \left(n^{2} x\right)+\cos \left(e^{n} x\right)}{1+n^{2} x^{2}}$. Then...
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287
TIFR Mathematics 2023 | Part A | Question: 9
Let $T: \mathrm{M}_{3}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be the linear map defined by $T(A)=A\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$. Then the dimension of the kernel of $T$ equals $2$. $8$. $1$. None of the remaining three options.
Let $T: \mathrm{M}_{3}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be the linear map defined by $T(A)=A\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$. Then the dimensi...
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292
TIFR Mathematics 2023 | Part A | Question: 14
The number of finite groups, up to isomorphism, with exactly two conjugacy classes, equals $1$. $2$. Greater than $2$, but finite. Infinite.
The number of finite groups, up to isomorphism, with exactly two conjugacy classes, equals$1$.$2$.Greater than $2$, but finite.Infinite.
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294
TIFR Mathematics 2023 | Part A | Question: 16
The number of elements of finite order in the group \[ \left\{\left(\begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right) \mid a, b, c \in \mathbb{R}\right\} \] is $1$. Finite, but not $1$. Countably infinite. Uncountably infinite.
The number of elements of finite order in the group\[\left\{\left(\begin{array}{ccc}1 & a & b \\0 & 1 & c \\0 & 0 & 1\end{array}\right) \mid a, b, c \in \mathbb{R}\right\...
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297
TIFR Mathematics 2023 | Part A | Question: 19
The number of (nondegenerate Euclidean) triangles with sides of integer length and perimeter $8,$ up to congruence, is $1$ . $2$. $3$. $4$.
The number of (nondegenerate Euclidean) triangles with sides of integer length and perimeter $8,$ up to congruence, is$1$ .$2$.$3$.$4$.
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1 answer
298
TIFR Mathematics 2023 | Part A | Question: 20
Let ... $A$ is infinite. $A$ is empty. $A$ is singleton. $A$ is finite, but neither empty nor singleton.
Let $$\begin{align} & A=\left\{(\alpha, \beta) \in \mathbb{Z}^{2} \mid \text { the roots } r_{1}, r_{2}, r_{3}\right. \text { of the polynomial } \\ & \qquad \left.p(x)=x...
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299
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Hi Is it worth doing M.Tech. CSE from old IITs / IISc after having 12 years of IT work experience at the age of 35+?
HiIs it worth doing M.Tech. CSE from old IITs / IISc after having 12 years of IT work experience at the age of 35+?
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