Limits - General Aptitude - General Aptitude (GA) Previous Year Questions

1

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(f: [0, \infty) \to \mathbb{R}\) be defined by \(f(x) = \int_0^x \sin^2(t^2) dt\). Then the function \(f\) is

A

uniformly continuous on \([0, 1]\) but NOT on \((0, \infty)\)

B

uniformly continuous on \((0, \infty)\) but NOT on \([0, 1]\)

C

uniformly continuous on both \([0, 1]\) and \((0, \infty)\)

D

neither uniformly continuous on \([0, 1]\) nor uniformly continuous on \((0, \infty)\)

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2

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Consider the power series \(\sum_{n=0}^{\infty} a_n z^n\), where \(a_n = \begin{cases} \frac{1}{2^n} & \text{if } n \text{ is even} \\ \frac{1}{5^n} & \text{if } n \text{ is odd} \end{cases}\). The radius of convergence of the series is equal to ________

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3

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(V = C^1[0,1]\), \(X = (C[0,1], \| \cdot \|_\infty)\) and \(Y = (C[0,1], \| \cdot \|_2)\). Then \(V\) is

A

dense in \(X\) but NOT in \(Y\)

B

dense in \(Y\) but NOT in \(X\)

C

dense in both \(X\) and \(Y\)

D

neither dense in \(X\) nor dense in \(Y\)

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4

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(T : (C[0,1], \| \cdot \|_\infty) \to \mathbb{R}\) be defined by \(T(f) = \int_0^1 2x f(x) dx\) for all \(f \in C[0,1]\). Then \(\|T\|\) is equal to ________

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5

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(d_1\) and \(d_2\) denote the usual metric and the discrete metric on \(\mathbb{R}\), respectively. Let \(f : (\mathbb{R}, d_1) \to (\mathbb{R}, d_2)\) be defined by \(f(x) = x\), \(x \in \mathbb{R}\). Then

A

\(f\) is continuous but \(f^{-1}\) is NOT continuous

B

\(f^{-1}\) is continuous but \(f\) is NOT continuous

C

both \(f\) and \(f^{-1}\) are continuous

D

neither \(f\) nor \(f^{-1}\) is continuous

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6

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(S = \{ (x, \sin \frac{1}{x}) : 0 < x \le 1 \}\) and \(T = S \cup \{(0,0)\}\). Under the usual metric on \(\mathbb{R}^2\),

A

\(S\) is closed but \(T\) is NOT closed

B

\(T\) is closed but \(S\) is NOT closed

C

both \(S\) and \(T\) are closed

D

neither \(S\) nor \(T\) is closed

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7

2015 · General Aptitude (GA) · General Aptitude · Limits

Mathematics (MA) 2015

Let \(H = \{ (x_n) \in \ell_2 : \sum_{n=1}^\infty \frac{x_n}{n} = 1 \}\). Then \(H\)

A

is bounded

B

is closed

C

is a subspace

D

has an interior point

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