21. The value of the quantity P, where $${\text{P}} = \int\limits_0^1 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx,}}} $$ is equal to
22. What is the value of $$\mathop {\lim }\limits_{{\text{x}} \to \frac{\pi }{4}} \frac{{\cos {\text{x}} - \sin {\text{x}}}}{{{\text{x}} - \frac{\pi }{4}}}$$
23. The value of $$\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + {{\text{x}}^2}} \right)^{{{\text{e}}^{ - {\text{x}}}}}}$$ is
24. For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is
25. The value of $$\mathop {\lim }\limits_{{\text{x}} \to 8} \frac{{{{\text{x}}^{\frac{1}{3}}} - 2}}{{\left( {{\text{x}} - 8} \right)}}$$
26. The vector that is NOT perpendicular to the vectors (i + j + k) and (i + 2j + 3k) is . . . . . . . .
27. Given $${\text{i}} = \sqrt { - 1} ,$$ what will be the evaluation of the definite integral $$\int_0^{\frac{\pi }{2}} {\frac{{\cos {\text{x}} + {\text{i}}\sin {\text{x}}}}{{\cos {\text{x}} - {\text{i}}\sin {\text{x}}}}} {\text{dx}}\,?$$
28. At x = 0, the function $${\text{f}}\left( {\text{x}} \right) = \left| {\sin \frac{{2\pi {\text{x}}}}{{\text{L}}}} \right|$$ , (-$$\infty $$ < x < $$\infty $$, L > 0) is
29. The value of $$\int_0^\infty {\frac{1}{{1 + {{\text{x}}^2}}}} {\text{dx}} + \int_0^\infty {\frac{{\sin {\text{x}}}}{{\text{x}}}} {\text{dx}}$$ is
30. The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {\frac{{{\text{x}} - {\text{y}}}}{{{{\left( {{\text{x}} + {\text{y}}} \right)}^3}}}{\text{dy}}} } \right){\text{dx}}} $$ and $$\int\limits_0^1 {\left( {\int\limits_0^1 {\frac{{{\text{x}} - {\text{y}}}}{{{{\left( {{\text{x}} + {\text{y}}} \right)}^3}}}{\text{dx}}} } \right){\text{dy}}} $$ are
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