Real Analysis Test - T 1

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. Consider the statement

(i.) \(l \in \mathbb{R}\) is a limit point of \(A \cup B\) iff \(l\) is a limit point of \(A\) or a limit point of \(B\).

(ii.) \(l \in \mathbb{R}\) is a limit point of \(A \cap B\) iff \(l\) is a limit point of \(A\) and a limit point of \(B\).

(a.) 1 correct 2 incorrect
(b.) 2 correct 1 incorrect
(c.) Both are correct
(d.) Both are incorrect

Exp: Let \(\ell\) be a limit point of \(A \cup B\) but it is neither a limit point of \(A\) nor of \(B\). So \(\exists \delta^{\prime}, \delta^{\prime \prime} > 0\) such that \(N_{\delta^{\prime}}(\ell) \cap A = \phi\) and \(N_{\delta^{\prime \prime}}(\ell) \cap B = \phi\).

Choose \(\delta = \min \{\delta^{\prime}, \delta^{\prime \prime}\}\) then \(N_{\delta^{\prime}}(\ell) \cap (A \cup B) = \phi\)

\(\Rightarrow \ell\) is not a limit point of \(A \cup B\), which is a contradiction. Then (i) is true.

(ii) Take \(A = [0, \frac{1}{2})\) and \(B = [-1, 0]\)

\(\Rightarrow A \cap B = \{0\}\)

Here 0 is a limit point of both \(A\) and \(B\) but not of \(A \cap B\).

2. Consider the following statements:

1. Let \(A = \{x: x \leq a\) and \(x, a \in \mathbb{R}\}\) then \(a\) is a limit point of \(A\)

2. Let \(B = \{y: y \geq b\) and \(y, b \in \mathbb{R}\}\) then \(b\) is a limit point of \(B\)

(a.) 1 is true and 2 is false
(b.) 2 is true and 1 is false
(c.) Both 1 & 2 are true
(d.) Both 1 & 2 are false

Exp: \(A = (-\infty, a], A^{\prime} = (-\infty, a]\) i.e. \(a \in A\)

\(B = [b, \infty), B^{\prime} = [b, \infty)\) i.e. \(b \in B\)

3. The Sum of the Series \(\frac{1}{2}+\frac{1}{3}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\ldots \ldots \ldots \ldots\)

(a.) 2
(b.) 1
(c.) \(\frac{3}{2}\)
(d.) \(\frac{2}{3}\)

Exp: \(\sum \frac{1}{2^{n}}+\sum \frac{1}{3^{n}} ; n \in \mathbb{N}\)

Use the formula for the sum of an infinite GP.

4. Let \(f(x) = x^{\alpha}|x|\) be a function and \(S = \{\alpha \in \mathbb{R} \mid f \text{ is differentiable at } 0\}\) then

(a.) \(S\) is an empty set
(b.) \(S\) is a finite set
(c.) \(S\) is a countable set
(d.) \(S \sim\) Set of All differentiable functions on \(\mathbb{R}\)

Exp: \(\phi(x) = \frac{x^{\alpha}|x| - 0}{x - 0} = x^{\alpha}\)

\(\lim_{{x \to 0}} \phi(x)\) exists if \(\alpha \in \mathbb{R}^{+}\)

5. The function defined by \(f(x) = \left\{\begin{array}{c}a x + b ; x \leq 1 \\ x^{2} + 3x + 3 ; x \geq 1\end{array}\right.\) is differentiable if

(a.) For \(a\) a unique value of \(a\) and infinite values of \(b\)
(b.) For \(a\) a unique value of \(b\) and infinite values of \(a\)
(c.) For infinite values of \(a\) and \(b\)
(d.) None of the above

Exp: \(Rf^{\prime}(1) = 5\) and \(Lf^{\prime}(1) = a\)

Now, \(f\) is differentiable at \(x = 1\) if \(a = 5\).

Part - C

6. Choose correct statements:

(a.) The derived set of a set is always closed.
(b.) The closure set of a set is always closed.
(c.) The supremum and infimum of a bounded non-empty set \(S\) is always a member of its closure \(\overline{S}\)
(d.) Supremum and infimum of a bounded infinite set is always a limit point of a set.

Exp:

(a) Let \(S\) be a set; if \(S\) is empty or finite, then its derived set \(S^{\prime}\) is \(\phi\), which is a closed set.

For (b), (c), and (d), detailed explanations are provided.

(b) \(\overline{\bar{S}}=\overline{S \cup S^{\prime}}=\bar{S} \cup \overline{S^{\prime}}=\bar{S}\) as \(S^{\prime}\) is a closed set, so \(\overline{S^{\prime}}=S\).

(c) Let \(M\) be the supremum of a set \(S\) (say). Now if \(m \in S\), then \(m \in S \Rightarrow m \in S \cup S^{\prime} \Rightarrow m \in \bar{S}\).

When \(m \notin S\), then \(m \notin S \Rightarrow m \in S^{\prime} \Rightarrow m \in S \cup S^{\prime}=\bar{S}\).

So \(m \in \bar{S}\). Similarly for infimum can be proved.

(d) Consider the set \(S=\{\pm 1,1,-1 \frac{1}{2}, 1 \frac{1}{2},-1 \frac{1}{3}, 1 \frac{1}{3}, \ldots\}\)

Let \(\inf S=-1 \frac{1}{2}\) and \(\sup S=1 \frac{1}{2}\).

\(S^{\prime}=\{-1,1\}\)

7. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a map, \(\phi \neq A \subseteq \mathbb{R}\) and define \(f\) as \(f(x)=\left\{\begin{array}{lll}1 & \text{ if } & x \in A \\ 0 & \text{ if } & x \notin A\end{array}\right.\)

Then

(a.) If \(A\) is open, then \(\forall G\) open in \(\mathbb{R}\), \(f^{-1}(G)\) is open.
(b.) If \(A\) is closed, then \(\forall F\) closed in \(\mathbb{R}\), \(f^{-1}(F)\) is closed.
(c.) If \(A\) is open and closed both in \(\mathbb{R}\) then \(\left(f^{-1}(G)\right)^{\prime} = f^{-1}(G)\), \(\forall G\) open in \(\mathbb{R}\).
(d.) If \(A\) is closed and open both in \(\mathbb{R}\) then \(\left(f^{-1}(G)\right)^{\circ} = f^{-1}(G)\), \(\forall G\) open in \(\mathbb{R}\).

Exp:

(c) If \(A\) is open and closed...

(d) If \(A\) is closed and open...

8. Which of the following is/are true

(a.) Every Subsequence of a convergent sequence is convergent and conversely
(b.) \(\lim _{n \rightarrow \infty} a_{n}=a \Leftrightarrow \lim _{n \rightarrow \infty} \frac{1}{n} \sum a_{n}\)
(c.) A monotonic sequence can have more than one limit point in \(\mathbb{R}\)
(d.) \(l \in \mathbb{R}\) is a limit point of a sequence \(a_{n} \Leftrightarrow l\) is a limit point of Range \(\{a_{n}\}\)

Exp:

For (b) and (d), take \(a_{n}=(-1)^{n}\)

9. \(f(x)=\lim _{n \rightarrow \infty} \frac{x^{2 n}+\phi(x)}{x^{2 n}+\psi(x)}\) where \(\phi(x) \& \psi(x)\) are continuous functions then which of the following is/are incorrect -

(a.) \(f\) can be continuous on \(\mathbb{R}\)
(b.) \(f\) is not continuous on \(\mathbb{R}\)
(c.) \(f\) is not continuous on \(\mathbb{R}\) if \(\phi(x) \neq \psi(x)\)
(d.) Points of discontinuity of \(f\) are at most finite

Exp:

Compute \(f(x)\) in case of \(|x|>1, |x|<1, |x|=1\)

10. For given \(f_n(x)\), Which of the following series is/are uniformly convergent -

(a.) \(\left(x^{2}+n^{2}\right)^{-1}, x \in \mathbb{R}\)
(b.) \(\frac{\cos n x}{n}, x \in \mathbb{R}\)
(c.) \(x^{n}, 0 \leq x<1\)
(d.) \(\frac{(-1)^{n-1} x^{n}}{n\left(1+x^{n}\right)}, x \in[0,1]\)

Exp:

(a) and (d) by \(M\) test.