21). Any finite subset of a metric space has
| A). limit points |
| B). no limit points |
| C). both (A) and (B) are true |
| D). none of these |
|
22). Examine the convergence of \(\Large\int\limits_{0}^{1}\frac{dx}{x^{2}}\)
| A). convergent |
| B). divergent |
| C). converges to 1 |
| D). converges to 0 |
|
23). Examine the convergence \(\Large\int\limits_{0}^{2}\frac{dx}{2x^{2}-x^{2}}\)
| A). converges |
| B). diverges |
| C). converges to |
| D). none of these |
|
24). The integral \(\Large\int\limits_{1}^{\infty}\frac{dx}{x(x+1)}\)
| A). log 2 |
| B). log 3 |
| C). log 5 |
| D). log 7 |
|
25). Examine the convergence of \(\Large\int\limits_{0}^{\pi}\frac{dx}{1+cosx}\).
| A). convergent |
| B). converges to 1 |
| C). converges to 0 |
| D). diverges |
|
26). Let \(f(x)=x,\ 0\le x\le 1\) and \(p=0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\) be the partition of [0, 1]. Find U(p, f) and L(p, f).
| A). \(\Large \frac{1}{8},\frac{1}{8}\) |
| B). \(\Large \frac{3}{8},\frac{7}{8}\) |
| C). \(\Large \frac{5}{8},\frac{3}{8}\) |
| D). \(\Large \frac{7}{8},\frac{9}{8}\) |
|
27). Let f(x) be defined on [0, 1] as follows:\[ f(x) = 1\begin{cases}\text{1 when } x \text{ is rational}\\\text{-1 when } x\text{ is irrational}\end{cases}\]
| A). Riemann integrable |
| B). not Rimann integrable |
| C). continuous |
| D). none of these |
|
28). Let \(f(x)=\frac{1}{x}\ 0\le x\le 2\ P=\{ 1,1.2,1.4,1.6,1.8,2 \}\) Find \(w(P,f)\)
| A). 0,1 |
| B). 0.2 |
| C). 0.3 |
| D). none of these |
|
29). \(f(x)=x^{2}\). Find \(\int\limits_{\overline{0}}^{a}\) and \(\int\limits_{n}^{\overline{0}}f\)
| A). \(\Large \frac{a}{3},\frac{a}{3}\) |
| B). \(\Large \frac{a^{2}}{3},\frac{a^{2}}{3}\) |
| C). \(\Large \frac{a^{3}}{3},\frac{a^{3}}{3}\) |
| D). none of these |
|
30). Any countable infinite Set is equivalent to a
| A). subset |
| B). proper subset |
| C). null set |
| D). none of these |
|
