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

Open complete paper

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

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

Open complete paper

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 \(\int_{-\pi}^{\pi} f(t) dt = 2\pi f(\alpha)\)

Open complete paper

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

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

Open complete paper

5

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Let \(X_1, X_2, \cdots, X_n\) be a random sample from uniform distribution on \([0, \theta]\). Then the maximum likelihood estimator (MLE) of \(\theta\) based on the above random sample is

A

\(\frac{2}{n} \sum_{i=1}^{n} X_i\)

B

\(\frac{1}{n} \sum_{i=1}^{n} X_i\)

C

\(\min \{X_1, X_2, \cdots, X_n\}\)

D

\(\max \{X_1, X_2, \cdots, X_n\}\)

Open complete paper

6

2008 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2008

Then

A

\(F\) and \(f\) are continuous

B

\(F\) is continuous and \(f\) is discontinuous

C

\(F\) is discontinuous and \(f\) is continuous

D

\(F\) and \(f\) are discontinuous

Open complete paper

7

2010 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2010

Let \( X \) have a binomial distribution with parameters \( n \) and \( p \), where \( n \) is an integer greater than 1 and \( 0 < p < 1 \). If \( P(X = 0) = P(X = 1) \), then the value of \( p \) is

A

\(\frac{1}{n-1} \)

B

\(\frac{n}{n+1} \)

C

\(\frac{1}{n+1} \)

D

\(\frac{1}{1+n^{n-1}} \)

Open complete paper

8

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

Open complete paper

9

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

Open complete paper

10

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

Open complete paper

11

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

Open complete paper

12

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

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

Open complete paper

13

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

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

Open complete paper

14

2014 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2014

If \(X_1, X_2\) is a random sample of size 2 from an \(N(0,1)\) population, then \(\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2}\) follows

A

\(\chi^2_{(2)}\)

B

\(F_{2,2}\)

C

\(F_{2,1}\)

D

\(F_{1,1}\)

Open complete paper

15

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 ______

Check answer

Open complete paper

16

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

Open complete paper

17

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

Check answer

Open complete paper

18

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

Open complete paper

19

2016 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2016

Let \( x_1 = x_2 = x_3 = 1, \; x_4 = x_5 = x_6 = 2 \) be a random sample from a Poisson random variable with mean \( \theta \), where \( \theta \in \{1, 2\} \). Then, the maximum likelihood estimator of \( \theta \) is equal to ______________

Check answer

Open complete paper

20

2017 · Mathematics · Real Analysis · Metric Spaces and Continuity

Mathematics (MA) 2017

Let \(F(x)\) be the distribution function of a random variable \(X\). Consider the functions: \(G_1(x) = (F(x))^3\), \(x \in \mathbb{R}\), \(G_2(x) = 1 - (1 - F(x))^5\), \(x \in \mathbb{R}\). Which of the above functions are distribution functions?

A

Neither \(G_1\) nor \(G_2\)

B

Only \(G_1\)

C

Only \(G_2\)

D

Both \(G_1\) and \(G_2\)

Open complete paper