21. The value of the integral \[\int\limits_0^2 {\frac{{{{\left( {{\text{x}} - 1} \right)}^2}\sin \left( {{\text{x}} - 1} \right)}}{{{{\left( {{\text{x}} - 1} \right)}^2} + \cos \left( {{\text{x}} - 1} \right)}}} {\text{dx}}\] is
22. Stokes theorem connects
23. Which of the following integrals is unbounded?
24. Changing the order of the integration in the double integral \[{\text{I}} = \int\limits_0^8 {\int\limits_{\frac{{\text{x}}}{4}}^2 {{\text{f}}\left( {{\text{x,}}\,{\text{y}}} \right){\text{dydx}}} } \] leads to \[{\text{I}} = \int\limits_{\text{r}}^{\text{s}} {\int\limits_{\text{p}}^{\text{q}} {{\text{f}}\left( {{\text{x,}}\,{\text{y}}} \right){\text{dx dy}}} .} \] What is q?
25. The divergence of the vector field \[\overrightarrow {\rm{A}} = {\rm{x}}{{{\rm{\hat a}}}_{\rm{x}}} + {\rm{y}}{{{\rm{\hat a}}}_{\rm{y}}} + {\rm{z}}{{{\rm{\hat a}}}_{\rm{z}}}\] is
26. Compute \[\mathop {\lim }\limits_{{\text{x}} \to 3} \frac{{{{\text{x}}^4} - 81}}{{2{{\text{x}}^2} - 5{\text{x}} - 3}}\]
27. The angle (in degree) between two planes vectors \[\overrightarrow {\rm{a}} = \frac{{\sqrt 3 }}{2}{\rm{\hat i}} + \frac{1}{2}{\rm{\hat j}}\] and \[\overrightarrow {\rm{b}} = \frac{{ - \sqrt 3 }}{2}{\rm{\hat i}} + \frac{1}{2}{\rm{\hat j}}\] is
28. The length of the curve \[{\text{y}} = \frac{2}{3}{{\text{x}}^{\frac{3}{2}}}\] between x = 0 and x = 1 is
29. Consider the function f(x) = x2 - x - 2. The maximum value of f(x) in the closed interval [-4, 4] is
30. \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\log }_{\text{e}}}\left( {1 + 4{\text{x}}} \right)}}{{{{\text{e}}^{3{\text{x}}}} - 1}}\] is equal to
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