31. What is the area common to the circles r = a and r = 2a cos θ?
32. What should be the value of \[\lambda \] such that the function defined below is continuous at \[{\text{x}} = \frac{\pi }{2}?\]
\[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}}
{\frac{{\lambda \cos {\text{x}}}}{{\frac{\pi }{2} - {\text{x}}}}}&{{\text{if x}} \ne \frac{\pi }{2}} \\
1&{{\text{if x}} = \frac{\pi }{2}}
\end{array}} \right.\]
\[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\lambda \cos {\text{x}}}}{{\frac{\pi }{2} - {\text{x}}}}}&{{\text{if x}} \ne \frac{\pi }{2}} \\ 1&{{\text{if x}} = \frac{\pi }{2}} \end{array}} \right.\]
33. For the function f(x) = x2e-x, the maximum occurs when x is equal to
34. The value of integral \[\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left( {{\text{x}}\cos {\text{x}}} \right){\text{dx}}} \] is
35. For the two functions, f(x, y) = x3 - 3xy2 and g(x, y) = 3x2y - y2, which one of the following options is correct?
36. A function f(x) is continuous in the interval [0, 2]. It is known that f(0) = f(2) = -1 and f(1) = 1.
Which one of the following statements must be true?
Which one of the following statements must be true?
37. How many distinct values of x satisfy the equation sin(x) = \[\frac{{\text{x}}}{2}\], where x is in radians?
38. The \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{\sin \left[ {\frac{2}{3}{\text{x}}} \right]}}{{\text{x}}}\] is
39. The value expression \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{\sin {\text{x}}}}{{{{\text{e}}^{\text{x}}}{\text{x}}}}\] is
40. Assuming \[{\text{i}} = \sqrt { - 1} \] and t is a real number, \[\int\limits_0^{\frac{\pi }{3}} {{{\text{e}}^{{\text{it}}}}} {\text{dt}}\] is
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