Test 2: Real Analysis

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom

Total Marks: 75

1) The value of $\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{n}}(1+2^{1/2}+3^{1/3}+\dots ..+n^{1/n})=\dots$ is given by:

a) $\mathrm{\infty }$
b) 0
c) 1
d) limit does not exist

2) If $x\ge 0$, the series ${\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{x^2}{4}}+{\displaystyle \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\cdot {\displaystyle \frac{x^4}{8}}+{\displaystyle \frac{1\cdot 3\cdot 5\cdot 7\cdot 9}{2\cdot 4\cdot 6\cdot 8\cdot 10}}\cdot {\displaystyle \frac{x^6}{12}}+\dots$ converges for:

a) $x\in [0,1]$
b) $x\in [0,1)$
c) $x\in [1,\mathrm{\infty })$
d) $x\in (1,\mathrm{\infty })$

3) Find the values of $p$ for which the series ${\displaystyle \frac{1}{(\log 2{)}^p}}+{\displaystyle \frac{1}{(\log 3{)}^p}}+{\displaystyle \frac{1}{(\log 4{)}^p}}+\dots$ converges:

a) $p\in [0,1)$
b) $p\in (1,\mathrm{\infty })$
c) $p\in (0,\mathrm{\infty })$
d) The series is divergent for all values of $p$

4) The series ${\displaystyle \frac{1}{2}}x+{\displaystyle \frac{1\cdot 3}{2\cdot 4}}{x}^2+{\displaystyle \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}{x}^3+\dots ..,x>0$ converges for:

a) $x\in (0,1]$
b) $x\in (0,1)$
c) $x\in [1,\mathrm{\infty })$
d) $x\in (1,\mathrm{\infty })$

5) The series $1+{\displaystyle \frac{\alpha \cdot \beta }{1\cdot \gamma }}x+{\displaystyle \frac{\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma \cdot (\gamma +1)}}{x}^2+{\displaystyle \frac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma \cdot (\gamma +1)\cdot (\gamma +2)}}{x}^3+\dots$, $x\ne 1$ converges for:

a) $x<1$
b) $x>1$
c) $x\in R$
d) The series does not converge for any values of $x$

6) The series $\sum {\displaystyle \frac{1\cdot 3\cdot 5\cdot \dots \cdot \cdot (2n-I)}{2\cdot 4\cdot 6\dots \cdot 2n}}\cdot {\displaystyle \frac{x^{2n}}{2n}}$ converges for:

a) $x^2<1$
b) $x^2\le 1$
c) $x^2>1$
d) The series does not converge for any values of $x$

7) For Riemann integrability, condition of continuity is

a) necessary
b) sufficient
c) necessary and sufficent
d) None of these

8) Let $f_n(x)=(\sin nx)/n,0\le x\le 1.$ Find $m\in N$ such that $|f_n(x)-0|<1/10$ for $n>m$ for all $x\in [0,1]$.

a) 1
b) 10
c) 11
d) 12

9) The integral ${\int }_0^{\pi /2}\sin x\log \sin x$ converges to:

a) $\log (6/\mathrm{e})$
b) $\log (4/\mathrm{e})$
c) $\log (3/\mathrm{e})$
d) $\log (2/\mathrm{e})$

10) The integral ${\int }_0^{\pi /2}{\displaystyle \frac{\sin x}{x^p}}dx$ converges for:

a) $p<2$
b) $p<1$
c) $p<-1$
d) $p>1$

11) If $\alpha \ge 0,\beta \ge 0,$ then show that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x^{\beta }dx}{1+x^{\alpha }{\sin }^{2}x}}$ converges for:

a) $\alpha >2(\beta +1)$
b) $\alpha <2(\beta +1)$
c) $\beta >2(\alpha +1)$
d) $\beta ><(\alpha +1)$

12) If $F(x)=\left(1/x^2\right)\times {\int }_4^x\{4t^2-3F^{\mathrm{\prime }}(t)\}dt,$ then what is $F^{\mathrm{\prime }}(4)$? a) $32/19$
b) $64/3$
c) $64/19$
d) $16/3$

13) The value of $\underset{x\to 0}{lim}{\displaystyle \frac{x{e}^{x^2}}{{\int }_0^x{e}^{t^2}dt}}$ is given by:

a) 0 b) 1
c) does not exist
d) -1

14) The maximum value of ${\left(1/x^2\right)}^{2x^2}$ is given by:

a) $\mathrm{\infty }$
b) $e^{1/e}$
c) $e^{2/e}$
d) $e^{4/e}$

15) The the dimensions of a right circular cone of maximum volume which can be circumscribed about a sphere of radius $a$ are given by:

a) radius= $a\sqrt{2}$, height = $4a$
b) radius= $a/\sqrt{2}$, height = $2a$
c) radius= $a\sqrt{2}$, height = $8a$
d) radius= $a/\sqrt{2}$, height = $4\sqrt{2}a$