Metric Spaces and Continuity - Real Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Let \(E\) be a connected subset of \(\mathbb{R}\) with at least two elements. Then the number of elements in \(E\) is

A

exactly two

B

more than two but finite

C

countably infinite

D

uncountable

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2

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Two (distinguishable) fair coins are tossed simultaneously. Given that ONE of them lands up head, the probability of the OTHER to land up tail is equal to

A

\(\frac{1}{3}\)

B

\(\frac{1}{2}\)

C

\(\frac{2}{3}\)

D

\(\frac{3}{4}\)

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3

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Which one of the following does NOT hold for all continuous functions \(f: [-\pi, \pi] \to \mathbb{C}\)?

A

If \(f(-t) = f(t)\) for each \(t \in [-\pi, \pi]\), then \(\int_{-\pi}^{\pi} f(t) dt = 2 \int_0^{\pi} f(t) dt\)

B

If \(f(-t) = -f(t)\) for each \(t \in [-\pi, \pi]\), then \(\int_{-\pi}^{\pi} f(t) dt = 0\)

C

\(\int_{-\pi}^{\pi} f(-t) dt = -\int_{-\pi}^{\pi} f(t) dt\)

D

There is an \(\alpha\) with \(-\pi < \alpha < \pi\) such that \(\int_{-\pi}^{\pi} f(t) dt = 2\pi f(\alpha)\)

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4

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Which one of the following subsets of \(\mathbb{R}\) (with the usual metric) is NOT complete?

A

\([1, 2] \cup [3, 4]\)

B

\([0, \infty)\)

C

\([0, 1]\)

D

\(\{0\} \cup \left\{\frac{1}{n} : n \in \mathbb{N}\right\}\)

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5

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

For two random variables \(X\) and \(Y\), the regression lines are given by \(Y = 5X - 15\) and \(Y = 10X - 35\). Then the regression coefficient of \(X\) on \(Y\) is

A

0.1

B

0.2

C

5

D

10

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6

2010 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2010

The set \( X = \mathbb{R} \) with the metric \( d(x,y) = \frac{|x-y|}{1+|x-y|} \) is

A

bounded but not compact

B

bounded but not complete

C

complete but not bounded

D

compact but not complete

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7

2010 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2010

Which of the following options is the closest in meaning to the word below:
Circuitous

A

cyclic

B

indirect

C

confusing

D

crooked

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8

2011 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2011

Given that \( f(y) = | y | / y \), and q is any non-zero real number, the value of \( | f(q) - f(-q) | \) is

A

0

B

-1

C

1

D

2

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9

2012 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2012

A simple random sample of size 10 from \(N(\mu, \sigma^2)\) gives 98% confidence interval \((20.49, 23.51)\). Then the null hypothesis \(H_0 : \mu = 20.5\) against \(H_A : \mu \neq 20.5\)

A

can be rejected at 2% level of significance

B

cannot be rejected at 5% level of significance

C

can be rejected at 10% level of significance

D

cannot be rejected at any level of significance

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10

2012 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2012

Which of the following statements are TRUE?
P : The set \(\{x\in R:\cos x\le\frac{1}{2}\}\) is compact.
Q : The set \(\{x\in R:\tan x\text{ is not differentiable}\}\) is complete.
R : The set \(\{x\in R:\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!}\text{ is convergent}\}\) is bounded.
S : The set \(\{x\in R:f(x)=\cos x\text{ has a local maxima}\}\) is closed.

A

P and Q

B

R and S

C

Q and S

D

P and S

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11

2012 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2012

If a random variable \(X\) assumes only positive integral values, with the probability \(P(X=x)=\frac{2}{3}\left(\frac{1}{3}\right)^{x-1},\ x=1,2,3,\ldots,\) then \(E(X)\) is

A

2/9

B

2/3

C

1

D

3/2

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12

2012 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2012

Choose the most appropriate word from the options given below to complete the following sentence:
Given the seriousness of the situation that he had to face, his __ was impressive.

A

beggary

B

nomenclature

C

jealousy

D

nonchalance

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13

2013 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2013

Let \(X\) be an arbitrary random variable that takes values in \(\{0, 1, ..., 10\}\). The minimum and maximum possible values of the variance of \(X\) are

A

0 and 30

B

1 and 30

C

0 and 25

D

1 and 25

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14

2013 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2013

Which of the following is NOT an unbiased estimate of \(\mu\)?

A

\(X_1\)

B

\(\frac{1}{n-1} (X_2 + X_3 + \cdots + X_n)\)

C

\(n \cdot (\min\{X_1, X_2, ..., X_n\})\)

D

\(\frac{1}{n} \max\{X_1, X_2, ..., X_n\}\)

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15

2013 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2013

Consider the problem of estimating \(\mu\). The m.s.e (mean square error) of the estimate \(T(X) = \frac{X_1 + X_2 + \cdots + X_n}{n + 1}\) is

A

\(\mu^2\)

B

\(\frac{1}{n+1} \mu^2\)

C

\(\frac{1}{(n+1)^2} \mu^2\)

D

\(\frac{n^2}{(n+1)^2} \mu^2\)

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16

2014 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2014

Let d₁, d₂ and d₃ be metrics on a set X with at least two elements. Which of the following is NOT a metric on X?

A

min(d₁, 2)

B

max(d₂, 2)

C

d₃/(1+d₃)

D

(d₁+d₂+d₃)/3

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17

2015 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2015

Let \(X_1, ..., X_n\) be a random sample from \(N(\mu, 1)\) distribution, where \(\mu \in \{0, \frac{1}{2}\}\). For testing the null hypothesis \(H_0 : \mu = 0\) against the alternative hypothesis \(H_1 : \mu = \frac{1}{2}\), consider the critical region \(R = \left\{ (x_1, x_2, ..., x_n) : \sum_{i=1}^n x_i > c \right\}\), where \(c\) is some real constant. If the critical region \(R\) has size \(0.025\) and power \(0.7054\), then the value of the sample size \(n\) is equal to ______

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18

2016 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2016

Let S = [0,1) ∪ [2,3] and f : S → ℝ be a strictly increasing function such that f(S) is connected. Which of the following statements is TRUE?

A

f has exactly one discontinuity

B

f has exactly two discontinuities

C

f has infinitely many discontinuities

D

f is continuous

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19

2016 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2016

\(f : [0,1] \to [0,1]\) is called a shrinking map if \(|f(x) - f(y)| < |x - y|\) for all \(x,y \in [0,1]\) and a contraction if there exists an \(\alpha < 1\) such that \(|f(x) - f(y)| \leq \alpha|x - y|\) for all \(x,y \in [0,1]\). Which of the following statements is TRUE for the function \(f(x) = x - \frac{x^2}{2}\)?

A

\(f\) is both a shrinking map and a contraction

B

\(f\) is a shrinking map but NOT a contraction

C

\(f\) is NOT a shrinking map but a contraction

D

\(f\) is Neither a shrinking map nor a contraction

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20

2016 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2016

Suppose \(X\) and \(Y\) are two random variables such that \(aX + bY\) is a normal random variable for all \(a, b \in \mathbb{R}\). Consider the following statements P, Q, R and S:
(P) : \(X\) is a standard normal random variable.
(Q) : The conditional distribution of \(X\) given \(Y\) is normal.
(R) : The conditional distribution of \(X\) given \(X + Y\) is normal.
(S) : \(X - Y\) has mean 0.
Which of the above statements ALWAYS hold TRUE?

A

both P and Q

B

both Q and R

C

both Q and S

D

both P and S

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