15 questions on Real Analysis for NET and GATE aspirants

Find the correct option:


1. Let f:[2,4]\rightarrow R be a continuous function such that f(2)=3 and f(4)=6. The most we can say about the set f([2,4]) 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 f:]1,5[\rightarrow R be a continuous function such that f(2)=3 and f(4)=6. The most we can say about the set f(]1,5[) 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 f:]1,5[\rightarrow R be a uniformly continuous function such that f(2)=3 and f(4)=6. The most we can say about the set f(]1,5[) 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 A be a set. What does it mean for A to be finite?

A. is a proper subset of the natural numbers.

B. There exists a natural number n and a bijection f from {i\in N:i<n} to A.

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

D. There exists a natural number n and a bijection f from {i\in N:i\leq n} to A.

 

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

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

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

C. A is of the form {a_1,a_2,a_3,\dots} for some sequence a_1,a_2,a_3,\dots

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

 

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

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

B. There exist elements of A 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 A.

D. There is a bijection f from A to the real numbers R.

 

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

(i) \inf(A)\leq \inf(B)

(ii) \inf(A)\leq \sup(B)

(iii) \sup(A)\leq \inf(B)

(iv) \sup(A)\leq \sup(B)

(p) For every \epsilon  data-recalc-dims=0" /> \exists a\in A & b\in B s.t. a<b+\epsilon.

(q) For every b\in B and \epsilon  data-recalc-dims=0" /> \exists a\in A s.t. a<b+\epsilon.

(r) For every a\in A and \epsilon  data-recalc-dims=0" /> \exists b\in B s.t. a<b+\epsilon.

(s) For every a\in A and b\in B, a\leq b.

 

Find the correct option from the following:

A. (i)\Rightarrow (p), (ii)\Rightarrow (s), (iii)\Rightarrow (q), (iv)\Rightarrow (r).

B. (i)\Rightarrow (q), (ii)\Rightarrow (r), (iii)\Rightarrow (p), (iv)\Rightarrow (s).

C. (i)\Rightarrow (q), (ii)\Rightarrow (p), (iii)\Rightarrow (s), (iv)\Rightarrow (r).

D. (i)\Rightarrow (s), (ii)\Rightarrow (q), (iii)\Rightarrow (r), (iv)\Rightarrow (s).

 

8. The radius of convergence of the power series \sum a_nx^n is R and k be a positive integer. Then the radius of convergent of the power series \sum a_nx^{kn} is

A. \frac{R}{k}.

B. R.

C. not depend on k.

D. R^{\frac{1}{k}}.

 

9. Let f:R\rightarrow R s.t. f(x)=\left\{\begin{array}{ll} \frac{|x|}{x} & \mbox{if x\neq0};\\ 0 & \mbox{if x=0}. \end{array} \right}.

And

g:R\rightarrow R s.t. g(x)=\left\{\begin{array}{ll}</p data-recalc-dims=

\frac{|x|}{x} & \mbox{if " />x\neq0};\\</p data-recalc-dims=

1 & \mbox{if " />x=0}.</p data-recalc-dims=

\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 f(x)=\left{\begin{array}{ll}</p data-recalc-dims=

8x & \mbox{for " />x\in Q};\\</p data-recalc-dims=

2x^2+8 & \mbox{for " />x\in Q^c}.</p data-recalc-dims=

\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. f(x)=\left{\begin{array}{ll} X^2-1 & \mbox{if x\in Q};\\ 0 & \mbox{if x\in Q^c}. \end{array} \right}.

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 f:\Rightarrow R be continuous and f(x)=\sqrt{2} \forall x\in Q. Then f(\sqrt{2}) equals to

A. \sqrt{2}.

B. 0.

C. Neither \sqrt{2} nor 0.

D. None of these.

 

13. Let f:\Rightarrow R be continuous, f(0)<0 and f(1) data-recalc-dims=1." /> Then,

(i) There exist c\in (0,1) such that f(c)=c^2.

(ii) There exist d\in (0,1) 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. f:\Rightarrow {-1,1} 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 \{\frac{\sin{\frac{n\pi}{2}}}{n}\}_{n=1}^{\infty}

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

3 Comments
  • Manpreet Kaur Bhatia
    Posted at 12:25h, 08 May

    Regarding 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.

  • Ramkrishna
    Posted at 08:56h, 03 July

    please post possible question for bed 2 year