## 09 Nov 15 questions on Real Analysis for NET and GATE aspirants

**Find the correct option:**

1. Let be a continuous function such that and The most we can say about the set is that

A. It is a set which contains [3,6].

B. It is a closed interval.

C. It is a set which contains 3 and 6.

D. It is a closed interval which contains [3,6].

2. Let be a continuous function such that and The most we can say about the set is that

A. It is an interval which contains [3,6].

B. It is an open interval which contains [3,6].

C. It is a bounded set which contains [3,6].

D. It is a bounded interval which contains [3,6].

3. Let be a uniformly continuous function such that and The most we can say about the set is that

A. It is a bounded set which contains [3,6].

B. It is an open interval which contains [3,6].

C. It is a bounded interval which contains [3,6].

D. It is an open bounded interval which contains [3,6].

4. Let be a set. What does it mean for to be finite?

A. is a proper subset of the natural numbers.

B. There exists a natural number and a bijection from to

C. There is a bijection from to a proper subset of the natural numbers.

D. There exists a natural number and a bijection from to

5. Let be a set. What does it mean for to be countable?

A. One can assign a different element of to each natural number in

B. There is a way to assign a natural number to every element of such that each natural number is assigned to exactly one element of

C. is of the form for some sequence

D. One can assign a different natural number to each element of

6. Let be a set. What does it mean for to be uncountable?

A. There is no way to assign a distinct element of to each natural number.

B. There exist elements of which cannot be assigned to any natural number at all.

C. There is no way to assign a distinct natural number to each element of

D. There is a bijection from to the real numbers

7. and be bounded non-empty sets. Following are two groups of statements:

(i)

(ii)

(iii)

(iv)

(p) For every & s.t.

(q) For every and s.t.

(r) For every and s.t.

(s) For every and

Find the correct option from the following:

A.

B.

C.

D.

8. The radius of convergence of the power series is and be a positive integer. Then the radius of convergent of the power series is

A.

B.

C. not depend on

D.

9. Let s.t. x\neq0x=0

And

s.t. x\neq0x=0

Then,

A. f and g both are continuous at x=0.

B. Neither f nor g is continuous at x=0.

C. f is continuous at x=0, but g is not.

D. g is continuous at x=0, but f is not.

** **

10. Let x\in Qx\in Q^c

Then,

A. f is not continuous.

B. f is continuous at x=0.

C. f is continuous at x=2.

D. f is continuous at both x=0 and x=2.

11. x\in Qx\in Q^c

Then,

A. f is not continuous.

B. f is continuous at x=1, but not continuous at x=-1.

C. f is continuous at both x=1 and x=-1.

D. f is continuous at x=-1, but not continuous at x=1.

12. Let be continuous and Then equals to

A.

B. 0.

C. Neither nor

D. None of these.

13. Let be continuous, and Then,

(i) There exist such that

(ii) There exist such that f(d)=d.

A. (i) is true, but (ii) is not true.

B. (ii) is true, but (i) is not true.

C. Both (i) and (ii) are true.

D. None of above.

14. be onto. Then

A. f is not continuous.

B. f is continuous.

C. f is differentiable everywhere.

D. f is continuous, but not differentiable anywhere.

15. The sequence

A. is convergent.

B. is divergent.

C. converges to 0.

D. converges to 1.

**The answers are here.**

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## Manpreet Kaur Bhatia

Posted at 12:25h, 08 MayRegarding question 1, since the function is continuous over an interval, then as per Preservation of Intervals Theorem, the co-domain i.e. f([2,4]) is also an interval. So the deduction leads us to option d as the correct one.

Kindly correct us if we are missing on anything.

## Gonit Sora

Posted at 17:03h, 10 MayThe answers are given here (http://gonitsora.com/answers/) and your answer is correct.

## Ramkrishna

Posted at 08:56h, 03 Julyplease post possible question for bed 2 year

## Answers - Gonit Sora (গণিত চ'ৰা)

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