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 \(\lim _{n \rightarrow \infty} \dfrac{1}{n}\left(1+2^{1 / 2}+3^{1 / 3}+\ldots . .+n^{1 / n}\right)=\ldots\) is given by:

a) \(\infty\)
b) 0
c) 1
d) limit does not exist

2) If \(x \geq 0\), the series \(\dfrac{1}{2} \cdot \dfrac{x^{2}}{4}+\dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \dfrac{x^{4}}{8}+\dfrac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 10} \cdot \dfrac{x^{6}}{12}+\ldots\) converges for:

a) \(x \in [0,1]\)
b) \(x \in [0,1)\)
c) \(x \in [1,\infty)\)
d) \(x \in (1,\infty)\)

3) Find the values of \(p\) for which the series \(\dfrac{1}{(\log 2)^{p}}+\dfrac{1}{(\log 3)^{p}}+\dfrac{1}{(\log 4)^{p}}+\ldots\) converges:

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

4) The series \(\dfrac{1}{2} x+\dfrac{1 \cdot 3}{2 \cdot 4} x^{2}+\dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3}+\ldots . ., x>0\) converges for:

a) \(x \in (0,1]\)
b) \(x \in (0,1)\)
c) \(x \in [1,\infty)\)
d) \(x \in (1,\infty)\)

5) The series \(1+\dfrac{\alpha \cdot \beta}{1 \cdot \gamma} x+\dfrac{\alpha(\alpha+1) \beta(\beta+1)}{1 \cdot 2 \cdot \gamma \cdot(\gamma+1)} x^{2}+\dfrac{\alpha(\alpha+1)(\alpha+2) \beta(\beta+1)(\beta+2)}{1 \cdot 2 \cdot 3 \cdot \gamma \cdot(\gamma+1) \cdot(\gamma+2)} x^{3}+\ldots\), \(x \neq 1\) 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 \(\sum \dfrac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot \cdot(2 n-I)}{2 \cdot 4 \cdot 6 \ldots \cdot 2 n} \cdot \dfrac{x^{2 n}}{2 n}\) converges for:

a) \(x^2<1\)
b) \(x^2 \leq 1\)
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 \(f_{n}(x)=(\sin n x) / n, 0 \leq x \leq 1 .\) Find \(m \in N\) such that \(\vert f_{n}(x)-0\vert <1 / 10\) 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 \(\int_{0}^{\pi / 2} \dfrac{\sin x}{x^{p}} d x\) converges for:

a) \(p<2\)
b) \(p<1\)
c) \(p<-1\)
d) \(p>1\)

11) If \(\alpha \geq 0, \beta \geq 0,\) then show that \(\int_{0}^{\infty} \dfrac{x^{\beta} d x}{1+x^{\alpha} \sin ^{2} x}\) converges for:

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

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

13) The value of \(\lim _{x \rightarrow 0} \dfrac{x e^{x^{2}}}{\int_{0}^{x} e^{t^{2}} d t}\) is given by:

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

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

a) \(\infty\)
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 = \(4 \sqrt{2} a\)