Real Analysis Test - T 2

Rsquared Mathematics

Name: *

Email: *

Phone: *

Time: 45:00

Topic: Real Analysis

Note : Welcome to Rsquared. We will continue with such quizzes in the future and this is just beginning. We request you to forward this quiz to your knowns so that it can spread more.
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 \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\left\{\begin{array}{c}x, x<1 \\ 2-x, 1 \leq x \leq 2 \\ x^{2}-3 x+2, x>2\end{array}\right.\)

(a.) \(f^{\prime}\) is continuous
(b.) \(f^{\prime}(x)\) exist for any real \(x\)
(c.) \(f^{\prime}\) exists except finite points
(d.) None of the above

Explanation: Try left and right-hand derivatives and connecting points, i.e., \(x=1,2\). Elsewhere function is continuous.

2. \(\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n !}}{n}\) is

(a.) \(e^{-1}\)
(b.) \(e\)
(c.) 0
(d.) None of the above

Explanation: \(\log L=\lim _{n \rightarrow \infty} \frac{1}{-}\{\log n !\}-\log n\). And use the inequality \(\log n !=n \log n\) as \(n\) tends to infinity.

3. Which of the following are uniformly continuous on the given domain -

(a.) \(\log \sin x ; x \in\left(0, \frac{\pi}{2}\right)\)
(b.) \(\cot x ; x \in\left(0, \frac{\pi}{2}\right)\)
(c.) \(e^{\sin \left(x^{2}\right)} ; x \in[0, \infty)\)
(d.) \(\sin \sqrt{x} ; x \in[0, \infty)\)

Explanation: \(|\sin \sqrt{x}-\sin \sqrt{y}| \leq\left|2 \sin \frac{\sqrt{x}-\sqrt{y}}{2} \cdot \cos \frac{\sqrt{x}+\sqrt{y}}{2}\right| \leq|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|}<\varepsilon=\sqrt{\delta}\)

4. Let \(\mathrm{f}\) and \(\mathrm{g}\) be two functions on \(S \subseteq \mathbb{R}\) and \(\mathrm{f} \circ \mathrm{g}\) denote composition of functions. Let \(a \in \mathbb{R}\) such that \(\lim _{x \rightarrow a}(\mathrm{f} \circ \mathrm{g})(x)\) exists then -

(a.) Both \(\lim _{x \rightarrow a} f(x)\) and \(\lim _{x \rightarrow a} g(x)\) exist
(b.) Either \(\lim _{x \rightarrow a} f(x)\) or \(\lim _{x \rightarrow a} g(x)\) exist
(c.) It is possible that both functions do not have a limit at \(x=a\)
(d.) None of the above

Explanation: Take \(\mathrm{f}=\mathrm{g}=\frac{1}{x}\) on the interval \((0, \infty)\) and \(\mathrm{a}=0\).

5. Let \((X, d)\) be a metric space. Let \(A \subseteq X\) and \(x \in X\). Define \(d^{\prime}(x, A)=\inf \{d(x, a) \mid a \in A\}\) and \(\mathrm{d}^{\prime}(\mathrm{x}, \mathrm{a})=0 \forall x \in X\) then which of the following are correct -

(a.) A is compact
(b.) A is closed
(c.) \(A\) is dense in \(X\)
(d.) \(\mathrm{A}=\mathrm{X}\)

Explanation: As the distance between any element of \(A\) and \(x\) can be made as small as we please, hence \(A\) must be dense in \(X\).

Part - C

6. Let \(\mathfrak{I}\) be the set of all irrational numbers which are not transcendental numbers, then choose correct -

(a.) \(\mathfrak{I}\) is countable
(b.) \(\mathfrak{I}^{c}\) is countable
(c.) \(\mathfrak{I} \cup \mathbb{Q}\) is countable
(d.) \(\mathfrak{I}\) is similar to \((0,1)\)

Explanation: \(\mathfrak{I}\) is countable as it is a subset of the set of algebraic numbers.

7. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x+y)=f(x)+f(y) \forall x, y \in \mathbb{R}\) and if \(t \geq 0 \Rightarrow f(t) \geq 0\). Then

(a.) \(f\) is continuous on \(\mathbb{R}\)
(b.) \(f\) need not be continuous on \(\mathbb{R}\)
(c.) \(f\) is monotonic on \(\mathbb{R}\)
(d.) \(f\) is continuous on \(\mathbb{R}^{+} \cup\{0\}\)

Explanation: Function satisfies Cauchy functional equation.

8. In which of the following cases there exist a 1-1 continuous function from \(S\) to \(T-\)

(a.) \(S=(0,1], T=(0,1)\)
(b.) \(S=\mathbb{R}, T=[0,1]\)
(c.) \(S=\mathbb{R}, T=(0,1)\)
(d.) \(S=[0,1], T=(0,1)\)

Explanation: For a and d, modify the equation of the line, and for b, c, modify \(\operatorname{arc}(\tan )\) function.

9. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function such that \(f(\mathbb{Q}) \subset \mathbb{Q}^{c} \cup S\) and \(f\left(\mathbb{Q}^{c}\right) \subset \mathbb{Q} \cup S\)

(a.) If no such \(f\) exist then \(S=\phi\)
(b.) If \(S=\phi\) then no such \(f\) exist
(c.) Such \(f\) exist and non-constant if \(S\) is uncountable
(d.) Such \(f\) exist and non-constant then \(S\) is uncountable.

Explanation: Use the identity theorem to claim that \(S\) must be the set of all real numbers for \(f\) to be non-constant.

10. Let

\[f_{n}(x)=\frac{n^{2} x}{n^{2} x^{2}+1}; x \in \mathbb{R}\]

Then

(a.) \(\lim _{n \rightarrow \infty} f_{n}(x)=f(x)\) has a finite number of discontinuity
(b.) Each \(f_{n}(x)\) is continuous
(c.) Convergence of \(f_{n}(x)\) is uniform.
(d.) Convergence of \(f_{n}(x)\) cannot be uniform in any interval containing 0.

Explanation: \(\lim f_{n}(x)=f(x)=\frac{1}{x}\), which is not continuous at \(x=0\).