Calculus MCQs Question and Answer in Engineering Maths

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

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 \[{\rm{5\hat i}} + 6{\rm{\hat j}} + 4{\rm{\hat k}}\,{\rm{?}}\]

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 \[{\text{f'}}\left( \theta \right) \ne 0\]

The value of the integral \[\int\limits_0^2 {\int\limits_0^{\text{x}} {{{\text{e}}^{{\text{x}} + {\text{y}}}}} } {\text{dy dx}}\]

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

A series expansion for the function sin θ is

Directional derivative of \[\phi \] = 2xz - y² at the point (1, 3, 2) becomes maximum in the direction of:

The area enclosed between the straight line y = x and the parabola y = x² in the x - y plane is

Consider function f(x) = (x² - 4)² where x is a real number. Then the function has

The local minima of function f(x) = x² - x⁴ in the range -0.8 ≤ x ≤ 0.8 is located at