11. By a change of variable x(u, v) = uv, y(u, v) = v/u is double integral, the integrand f(x, y) changes to f(uv, v/u) \[\phi \] (u, v). Then, \[\phi \] (u, v) is
12. Which one of the following describes the relationship among the three vectors, \[{\rm{\hat i}} + {\rm{\hat j}} + {\rm{\hat k}},\,2{\rm{\hat i}} + 3{\rm{\hat j}} + {\rm{\hat k}}\] and \[{\rm{5\hat i}} + 6{\rm{\hat j}} + 4{\rm{\hat k}}\,{\rm{?}}\]
13. Let the function
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}}
{\sin \theta }&{\cos \theta }&{\tan \theta } \\
{\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\
{\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)}
\end{array}} \right|\]
where \[\theta \in \left[ {\frac{\pi }{6},\,\frac{\pi }{3}} \right]\] and \[{\text{f'}}\left( \theta \right)\] denote the derivative of f with respect to \[\theta \]. Which of the following statements is/are TRUE?
I. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that \[{\text{f'}}\left( \theta \right) = 0.\]
II. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that
\[{\text{f'}}\left( \theta \right) \ne 0\]
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}} {\sin \theta }&{\cos \theta }&{\tan \theta } \\ {\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\ {\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)} \end{array}} \right|\]
where \[\theta \in \left[ {\frac{\pi }{6},\,\frac{\pi }{3}} \right]\] and \[{\text{f'}}\left( \theta \right)\] denote the derivative of f with respect to \[\theta \]. Which of the following statements is/are TRUE?
I. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that \[{\text{f'}}\left( \theta \right) = 0.\]
II. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that \[{\text{f'}}\left( \theta \right) \ne 0\]
14. The value of the integral \[\int\limits_0^2 {\int\limits_0^{\text{x}} {{{\text{e}}^{{\text{x}} + {\text{y}}}}} } {\text{dy dx}}\]
15. For the scalar field \[{\text{u}} = \frac{{{{\text{x}}^2}}}{2} + \frac{{{{\text{y}}^2}}}{3},\] magnitude of the gradient at the point (1, 3) is
16. A series expansion for the function sin θ is
17. Directional derivative of \[\phi \] = 2xz - y2 at the point (1, 3, 2) becomes maximum in the direction of:
18. The area enclosed between the straight line y = x and the parabola y = x2 in the x - y plane is
19. Consider function f(x) = (x2 - 4)2 where x is a real number. Then the function has
20. The local minima of function f(x) = x2 - x4 in the range -0.8 ≤ x ≤ 0.8 is located at
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