Real Analysis Test - T 3

Rsquared Mathematics

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Time: 45:00

Topic: Real Analysis

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

(1.) Let \(f:[0,1] \rightarrow \mathbf{R}\) such that \(f(x)=\left\{\begin{array}{l}x \sin \frac{\pi}{x} ; x \neq 0 \\ 0 ; x=0\end{array}\right.\) and \(S=\left\{\alpha \mid f^{\prime}(\alpha)=0\right\}\) then

(a.) \(S\) is an empty set
(b.) \(S\) is a finite set
(c.) \(S\) is a countably infinite set
(d.) \(S\) is an uncountable set

Explanation:

Use Rolle's theorem.

(2.) Let \(f:(0,1) \rightarrow \mathbb{R}\) be a one-to-one function then

(a.) Range of \(f\) must contain rational numbers
(b.) Range of \(f\) must contain irrational numbers
(c.) Range of \(f\) must contain both rational and irrational numbers
(d.) None of these

Explanation:

The range must be uncountable, and every uncountable subset must contain irrational numbers.

(3.) Which of the following is true

(a.) If the series \(\sum a_n\) is of positive terms, then \(\sum a_n < \infty\) if and only if the corresponding sequence of partial sums (\(S_n\)) is bounded
(b.) The series \(\sum_{k=1}^{\infty} a_k\) converges if and only if \(\lim_{n \rightarrow \infty} (a_n+a_{n+1}+\ldots+a_{n+p}) = 0\) for some positive \(p\)
(c.) \(a_n\) be a sequence of positive terms such that \(\sum a_n < \infty\), then \(\lim_{n \rightarrow \infty} n a_n = 0\)
(d.) None of these

Explanation:

Series of positive terms converge iff it has a bounded sequence of partial sums.

(4.) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a periodic function and \(P=\{t: f(x+t)=f(x)\}\)

(a.) If \(t\) is the smallest positive period then \(f\) is uniformly continuous
(b.) If \(P=\mathbb{R}\) then \(f\) is uniformly continuous
(c.) If \(f\) is uniformly continuous then the smallest positive period \(t\) exists
(d.) None of these

Explanation:

If \(P=\mathbb{R}\) then \(f\) is constant.

(5.) Let \(a_n\) be a sequence, and \(l \in \mathbf{R}\). Then \(l\) is the limit of \(a_n\) if

(a.) \(\forall \epsilon>0, (l-\epsilon, l+\epsilon)\) contains finitely many terms of sequence \(a_n\)
(b.) \(\forall \epsilon>0, (l-\epsilon, l+\epsilon)\) contains infinitely many terms of sequence \(a_n\)
(c.) \(\forall \epsilon>0, (l-\epsilon, l+\epsilon)^c\) contains infinitely many terms of sequence \(a_n\)
(d.) \(\forall \epsilon>0, (l-\epsilon, l+\epsilon)^c\) contains finitely many terms of sequence \(a_n\)

Explanation:

Definition of limit.

Part - C

(6.) Let \(f\) be a non-decreasing function on \(\mathbb{R}\), and \(S\) be the set of points of discontinuity of \(f\). Then \(S\) can be similar to

(a.) \(\left\{(x, y) \in \mathbb{R}^2: x^2+y^2=\frac{1}{n^2}\right.\) for \(n\) fixed \(\)
(b.) \(\left\{\frac{1}{m}+\frac{1}{n}: m, n \in \mathbb{N}\right\}\)
(c.) \(\bigcup_{n=1}^{\infty} U_n\), where \(U_n=\left(-\frac{1}{n}, \frac{1}{n}\right)\)
(d.) \(\{\sqrt{n}: n\) is a prime number greater than or equal to 8\(\}\)

Explanation:

\(f\) can have a countable number of discontinuities.

(7.) If \(a_n=\frac{e^\pi \pi^\pi+e^{2 \pi} \pi^{\pi / 2}+e^{3 \pi} \pi^{\pi / 3}+\ldots+e^{n \pi} \pi^{\pi / n}}{e^\pi+e^{2 \pi}+\ldots+e^{n \pi}}\) then

(a.) The sequence is convergent
(b.) The sequence is bounded but not convergent
(c.) Sequence is divergent
(d.) Sequence is of positive terms and bounded

Explanation:

Use Cauchy-Stolz theorem.

(8.) Consider the functions \(f: \mathbb{R} \setminus\left\{-\frac{d}{c}\right\} \rightarrow \mathbb{R}\) such that

\[f(x)=\frac{a x+b}{c x+d}, \quad ad-bc \neq 0\]

Then

(a.) \(f(x)\) is increasing if \(ad-bc<0\)
(b.) \(f(x)\) is increasing if \(ad-bc>0\)
(c.) \(f(x)\) is decreasing if \(ad-bc<0\)
(d.) \(f(x)\) is decreasing if \(ad-bc>0\)

Explanation:

\(f^{\prime}(x)=\frac{ad-bc}{(cx+d)^2}\)

(9.) If \(f: I \rightarrow \mathbb{R}\) is a one-to-one function

(a.) If \(f\) satisfies the intermediate value property then \(f\) is monotone
(b.) If \(f\) satisfies the intermediate value property then \(f\) is continuous
(c.) If \(f\) satisfies the intermediate value property then \(f\) is onto
(d.) If \(f\) is onto, then \(f\) is continuous

Explanation:

Results

(10.) Which of the following is/are correct -

(a.) The set of all circles with rational radius is countable
(b.) The set of all circles with rational center is countable
(c.) The set of all circles with rational centers and area greater than \(\pi^2\) unit is uncountable
(d.) The set of all circles with areas less than 1 unit is uncountable

Explanation:

For (a) and (b), the choice for center and radius is uncountable.