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| msc:mcqs_short_questions:real_analysis [2021/10/11 17:48] – Administrator | msc:mcqs_short_questions:real_analysis [2023/04/03 04:06] (current) – [Series of Numbers] Administrator | ||
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| - Give an example to prove that bounded sequence may not convergent. | - Give an example to prove that bounded sequence may not convergent. | ||
| - Prove that every convergent sequence is bounded. | - Prove that every convergent sequence is bounded. | ||
| - | - | ||
| ===== Multiple choice questions (MCQs) ===== | ===== Multiple choice questions (MCQs) ===== | ||
| - | - What is not true about number zero. | + | ==== Real Number System==== |
| + | < | ||
| + | 1. | ||
| * (A) Even | * (A) Even | ||
| * (B) Positive | * (B) Positive | ||
| * (C) Additive identity | * (C) Additive identity | ||
| * (D) Additive inverse of zero \\ <btn type=" | * (D) Additive inverse of zero \\ <btn type=" | ||
| - | - Which one of them is not interval. | + | </ |
| + | < | ||
| + | | ||
| * (A) $(1,2)$ | * (A) $(1,2)$ | ||
| * (B) $\left(\frac{1}{2}, | * (B) $\left(\frac{1}{2}, | ||
| * (C) $[3. \pi]$ | * (C) $[3. \pi]$ | ||
| - | * (D) $(2\pi, | + | * (D) $(2\pi, |
| - | | + | </ |
| + | < | ||
| + | | ||
| * (A) 0 | * (A) 0 | ||
| * (B) 2 | * (B) 2 | ||
| * (C) $2n$ such that $n \in \mathbb{Z}$ | * (C) $2n$ such that $n \in \mathbb{Z}$ | ||
| - | * (D) $2\pi$ \\ <btn type=" | + | * (D) $2\pi$ \\ <btn type=" |
| - | | + | </ |
| + | < | ||
| + | | ||
| * (A) 0 | * (A) 0 | ||
| * (B) 1 | * (B) 1 | ||
| * (C) $\pi$ | * (C) $\pi$ | ||
| - | * (D) None of these | + | * (D) None of these \\ <btn type=" |
| - | | + | </ |
| + | | ||
| * (A) natural numbers | * (A) natural numbers | ||
| * (B) integers | * (B) integers | ||
| * (C) rational numbers | * (C) rational numbers | ||
| - | * (D) real numbers | + | * (D) real numbers |
| - | | + | </ |
| + | | ||
| * (A) integer | * (A) integer | ||
| * (B) algebraic number | * (B) algebraic number | ||
| * (C) irrational number | * (C) irrational number | ||
| - | * (D) complex numbers | + | * (D) complex numbers |
| - | | + | </ |
| + | | ||
| * (A) $\pi$ | * (A) $\pi$ | ||
| * (B) $\sqrt{2}$ | * (B) $\sqrt{2}$ | ||
| * (C) $\sqrt{3}$ | * (C) $\sqrt{3}$ | ||
| - | * (D) 7 | + | * (D) 7 \\ <btn type=" |
| - | | + | </ |
| + | | ||
| * (A) $f$ is bijective | * (A) $f$ is bijective | ||
| * (B) $f$ is surjective | * (B) $f$ is surjective | ||
| * (C) $f$ is identity map | * (C) $f$ is identity map | ||
| - | * (D) None of these | + | * (D) None of these \\ <btn type=" |
| - | | + | </ |
| + | | ||
| * (A) 7 | * (A) 7 | ||
| * (B) 3 | * (B) 3 | ||
| - | * (C) does not exist | + | * (C) 2 |
| - | * (D) 0 | + | * (D) does not exist \\ <btn type=" |
| - | | + | </ |
| + | ==== Sequence of Numbers ==== | ||
| + | < | ||
| + | 1. A convergent sequence has only ................ limit(s). | ||
| * (A) one | * (A) one | ||
| * (B) two | * (B) two | ||
| * (C) three | * (C) three | ||
| - | * (D) None of these | + | * (D) None of these \\ <btn type=" |
| - | | + | </ |
| + | | ||
| * (A) there exists | * (A) there exists | ||
| * (B) there exists real number $p$ such that $|s_n|< | * (B) there exists real number $p$ such that $|s_n|< | ||
| - | * (C) there exists positive real number $s$ such that $|s_n|< | + | * (C) there exists positive real number $s$ such that $|s_n|< |
| - | * (D) the term of the sequence lies in a vertical strip of finite width. | + | * (D) the term of the sequence lies in a vertical strip of finite width. |
| - | | + | </ |
| + | 3. If the sequence is convergent then | ||
| * (A) it has two limits. | * (A) it has two limits. | ||
| * (B) it is bounded. | * (B) it is bounded. | ||
| * (C) it is bounded above but may not be bounded below. | * (C) it is bounded above but may not be bounded below. | ||
| - | * (D) it is bounded below but may not be bounded above. | + | * (D) it is bounded below but may not be bounded above. |
| - | | + | </ |
| + | 4. A sequence $\{(-1)^n\}$ is | ||
| * (A) convergent. | * (A) convergent. | ||
| * (B) unbounded. | * (B) unbounded. | ||
| * (C) divergent. | * (C) divergent. | ||
| - | * (D) bounded. | + | * (D) bounded. |
| - | | + | </ |
| + | < | ||
| + | 5. A sequence $\left\{\dfrac{1}{n} | ||
| * (A) bounded. | * (A) bounded. | ||
| * (B) unbounded. | * (B) unbounded. | ||
| * (C) divergent. | * (C) divergent. | ||
| - | * (D) None of these. | + | * (D) None of these. |
| - | | + | </ |
| + | < | ||
| + | 6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon> | ||
| * (A) $|s_n-s_m|< | * (A) $|s_n-s_m|< | ||
| * (B) $|s_n-s_m|< | * (B) $|s_n-s_m|< | ||
| * (C) $|s_n-s_m|< | * (C) $|s_n-s_m|< | ||
| - | * (D) $|s_n-s_m|< | + | * (D) $|s_n-s_m|< |
| - | | + | </ |
| + | 7. Every Cauchy sequence has a ............... | ||
| * (A) convergent subsequence. | * (A) convergent subsequence. | ||
| * (B) increasing subsequence. | * (B) increasing subsequence. | ||
| * (C) decreasing subsequence. | * (C) decreasing subsequence. | ||
| - | * (D) positive subsequence. | + | * (D) positive subsequence. |
| - | | + | </ |
| + | 8. A sequence of real number is Cauchy iff | ||
| * (A) it is bounded | * (A) it is bounded | ||
| * (B) it is convergent | * (B) it is convergent | ||
| * (C) it is positive term sequence | * (C) it is positive term sequence | ||
| - | * (D) it is convergent but not bounded. | + | * (D) it is convergent but not bounded. |
| - | | + | </ |
| + | 9. Let $\{s_n\}$ be a convergent sequence. If $\lim_{n\to\infty}s_n=s$, | ||
| * (A) $\lim_{n\to\infty}s_{n+1}=s+1$ | * (A) $\lim_{n\to\infty}s_{n+1}=s+1$ | ||
| * (B) $\lim_{n\to\infty}s_{n+1}=s$ | * (B) $\lim_{n\to\infty}s_{n+1}=s$ | ||
| * (C) $\lim_{n\to\infty}s_{n+1}=s+s_1$ | * (C) $\lim_{n\to\infty}s_{n+1}=s+s_1$ | ||
| - | * (D) $\lim_{n\to\infty}s_{n+1}=s^2$. | + | * (D) $\lim_{n\to\infty}s_{n+1}=s^2$. |
| - | | + | </ |
| + | 10. Every convergent sequence has ................. one limit. | ||
| * (A) at least | * (A) at least | ||
| * (B) at most | * (B) at most | ||
| * (C) exactly | * (C) exactly | ||
| - | * (D) none of these | + | * (D) none of these \\ <btn type=" |
| - | | + | </ |
| + | 11. If the sequence is decreasing, then it ................ | ||
| * (A) converges to its infimum. | * (A) converges to its infimum. | ||
| * (B) diverges. | * (B) diverges. | ||
| * (C) may converges to its infimum | * (C) may converges to its infimum | ||
| - | * (D) is bounded. | + | * (D) is bounded. |
| - | | + | </ |
| + | 12. If the sequence is increasing, then it ................ | ||
| * (A) converges to its supremum. | * (A) converges to its supremum. | ||
| * (B) diverges. | * (B) diverges. | ||
| * (C) may converges to its supremum. | * (C) may converges to its supremum. | ||
| - | * (D) is bounded. | + | * (D) is bounded. |
| - | | + | </ |
| + | 13. If a sequence converges to $s$, then .............. of its sub-sequences converges to $s$. | ||
| * (A) each | * (A) each | ||
| * (B) one | * (B) one | ||
| * (C) few | * (C) few | ||
| - | * (D) none | + | * (D) none \\ <btn type=" |
| - | | + | </ |
| + | 14. If two sub-sequences of a sequence converge to two different limits, then a sequence ............... | ||
| * (A) may convergent. | * (A) may convergent. | ||
| * (B) may divergent. | * (B) may divergent. | ||
| * (C) is convergent. | * (C) is convergent. | ||
| - | * (D) is divergent. | + | * (D) is divergent. |
| - | | + | </ |
| - | * (A) $\{\sum_{k=1}^{\infty}a_k \}$ | + | ==== Series of Numbers ==== |
| - | * (B) $\{\sum_{k=1}^{n}a_k \}$ | + | < |
| - | * (C) $\{\sum_{n=1}^{\infty}a_k \}$ | + | 1. A series $\sum_{n=1}^\infty |
| - | * (D) $\{ a_n \}$ | + | * (A) $s_n=\sum_{n=1}^\infty |
| - | | + | * (B) $s_n=\sum_{k=1}^n a_k$ is convergent. |
| - | * (A) it is bounded. | + | * (C) $s_n=\sum_{k=1}^n a_n$ is convergent. |
| - | * (B) it may bounded. | + | * (D) $s_n=\sum_{k=1}^n a_k$ is divergent. \\ <btn type=" |
| - | * (C) it is unbounded. | + | </ |
| - | * (D) it may unbounded. | + | 2. If $\sum_{n=1}^\infty a_n$ converges then ........................... |
| - | - If $\lim_{n\to\infty} a_n=0$, then $\sum a_n$ ................ | + | * (A) $\lim_{n\to \infty} a_n=0$. |
| + | * (B) $\lim_{n\to \infty} a_n=1$. | ||
| + | * (C) $\lim_{n\to \infty} a_n \neq 0$ | ||
| + | * (D) $\lim_{n\to \infty} a_n$ exists. \\ <btn type=" | ||
| + | </ | ||
| + | < | ||
| + | 3. If $\lim_{n\to \infty} a_n \neq 0$, then $\sum_{n=1}^\infty | ||
| + | * (A) is convergent. | ||
| + | * (B) may convergent. | ||
| + | * (C) is divergent | ||
| + | * (D) is bounded. \\ <btn type=" | ||
| + | </ | ||
| + | < | ||
| + | 4. A series $\sum_{n=1}^\infty \left( 1+\frac{1}{n} \right)$ is .................... | ||
| + | * (A) convergent. | ||
| + | * (B) divergent. | ||
| + | * (C) constant. | ||
| + | * (D) none of these \\ <btn type=" | ||
| + | </ | ||
| + | 5. Let $\sum a_n$ be a series of non-negative terms. Then it is convergent if its sequence of partial sum ............... | ||
| + | * (A) is bounded. | ||
| + | * (B) may bounded. | ||
| + | * (C) is unbounded. | ||
| + | * (D) is divergent. \\ <btn type=" | ||
| + | </ | ||
| + | | ||
| * (A) is convergent. | * (A) is convergent. | ||
| * (B) is divergent. | * (B) is divergent. | ||
| * (C) may or may not convergent | * (C) may or may not convergent | ||
| - | * (D) none of these | + | * (D) none of these \\ <btn type=" |
| - | - A series $\sum \frac{1}{n^p}$ is convergent if | + | </ |
| - | * (A) $p\geq 1$. | + | 7. A series $\sum \frac{1}{n^p}$ is convergent if |
| - | * (B) $p\leq 1$. | + | * (A) $p\leq 1$. |
| - | * (C) $p>1$. | + | * (B) $p\geq 1$. |
| - | * (D) $p<1$. | + | * (C) $p<1$. |
| - | - If a sequence $\{a_n\}$ is convergent then the series $\sum a_n$ ................ | + | * (D) $p>1$. \\ <btn type=" |
| + | </ | ||
| + | 8- If a sequence $\{a_n\}$ is convergent then the series $\sum a_n$ ................ | ||
| * (A) is convergent. | * (A) is convergent. | ||
| * (B) is divergent. | * (B) is divergent. | ||
| * (C) may or may not convergent | * (C) may or may not convergent | ||
| - | * (D) none of these | + | * (D) none of these \\ <btn type=" |
| - | | + | </ |
| + | | ||
| * (A) $\{a_n\}$ is convergent. | * (A) $\{a_n\}$ is convergent. | ||
| * (B) $\{a_n\}$ is decreasing. | * (B) $\{a_n\}$ is decreasing. | ||
| * (C) $\{a_n\}$ is bounded. | * (C) $\{a_n\}$ is bounded. | ||
| - | * (D) $\{a_n\}$ is decreasing and $\lim a_n=0$. | + | * (D) $\{a_n\}$ is decreasing and $\lim a_n=0$. |
| - | | + | </ |
| + | 10. An series $\sum a_n$ is said to be absolutely convergent if | ||
| * (A) $\left| \sum a_n \right|$ is convergent. | * (A) $\left| \sum a_n \right|$ is convergent. | ||
| * (B) $\left| \sum a_n \right|$ is convergent but $\sum a_n$ is divergent. | * (B) $\left| \sum a_n \right|$ is convergent but $\sum a_n$ is divergent. | ||
| * (C) $\sum |a_n|$ is convergent. | * (C) $\sum |a_n|$ is convergent. | ||
| - | * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. | + | * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. |
| - | | + | </ |
| - | | + | 11. A series $\sum a_n$ is convergent if and only if ..................... is convergent |
| - | | + | * (A) $\{\sum_{k=1}^{\infty}a_k \}$ |
| - | | + | * (B) $\{\sum_{k=1}^{n}a_k \}$ |
| - | | + | * (C) $\{\sum_{n=1}^{\infty}a_k \}$ |
| - | - If $\lim_{x \to c}f(x)=L$, then ............ sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$. | + | * (D) $\{ a_n \}$ \\ <btn type=" |
| - | | + | </ |
| - | | + | ==== Limit of functions ==== |
| - | | + | < |
| - | | + | 1. A number $L$ is called limit of the function $f$ when $x$ approaches to $c$ if for all $\varepsilon> |
| - | | + | |
| - | | + | |
| - | | + | |
| - | | + | |
| - | | + | </ |
| - | | + | 2. If $\lim_{x \to c}f(x)=L$, then .............. sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$. |
| - | | + | |
| - | | + | |
| - | | + | |
| - | | + | |
| - | | + | </ |
| - | | + | 3. Let $f(x)=\frac{x^2-5x+6}{x-3}$, |
| - | | + | |
| - | | + | |
| - | | + | |
| + | | ||
| + | </ | ||
| + | ==== Riemann Integrals ==== | ||
| + | < | ||
| + | 1. Which one is not partition of interval $[1,5]$. | ||
| + | * (A) $\{1,2,3,5 \}$ | ||
| + | * (B) $\{1, | ||
| + | * (C)$\{1,1.1,5 \}$ | ||
| + | * (D) $\{1,2.1,3,4,5.5 \}$ \\ <btn type=" | ||
| + | </ | ||
| + | 2. What is norm of partition $\{0, | ||
| + | * (A) $10$ | ||
| + | * (B) $3$ | ||
| + | * (C) $3.8$ | ||
| + | * (D) $0.1$ \\ <btn type=" | ||
| + | </ | ||