## 09 Nov 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 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?

READ:   Ramanujan and More: A conversation with Ken Ono

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 $\epsilon >0$ $\exists a\in A$ & $b\in B$ s.t. $a

(q) For every $b\in B$ and $\epsilon >0$ $\exists a\in A$ s.t. $a

(r) For every $a\in A$ and $\epsilon >0$ $\exists b\in B$ s.t. $a

(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.

\frac{|x|}{x} & \mbox{if " />$g(x)=\left\{\begin{array}{ll}

\frac{|x|}{x} & \mbox{if$

x\neq0

1 & \mbox{if " />$};\\

1 & \mbox{if$

x=0

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

\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 " />$f(x)=\left{\begin{array}{ll}

8x & \mbox{for$

x\in Q

2x^2+8 & \mbox{for " />$};\\

2x^2+8 & \mbox{for$

x\in Q^c

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

\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 $f(1)>1.$ 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.

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#### Latest posts by Pankaj Jyoti Mahanta (see all)

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• ##### 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.

• ##### Gonit Sora
Posted at 17:03h, 10 May