## 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 0" /> & s.t.

(q) For every and 0" /> s.t.

(r) For every and 0" /> 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.

\frac{|x|}{x} & \mbox{if " />x\neq0

1 & \mbox{if " />x=0

\end{array} \right}." />

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

8x & \mbox{for " />x\in Q

2x^2+8 & \mbox{for " />x\in Q^c

\end{array} \right}." />

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 1." /> 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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#### Pankaj Jyoti Mahanta

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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