51. If Z = eax + by F(ax - by); the value of \[{\text{b}} \cdot \frac{{\partial {\text{Z}}}}{{\partial {\text{x}}}} + {\text{a}} \cdot \frac{{\partial {\text{Z}}}}{{\partial {\text{y}}}}\] is
52. Curl of vector V(x, y, z) = 2x2i + 3z2j + y3k at x = y = z = 1 is
53. For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point P(1, 2, -1) in the direction of a vector \[\overrightarrow {\text{i}} - \overrightarrow {\text{j}} + 2\overrightarrow {\text{k}} \] is
54. The directional derivative of the function f(x, y) = x2 + y2 along a line directed from (0, 0) to (1, 1), evaluated at point x = 1, y = 1 is
55. Value of the integral \[\oint\limits_{\text{c}} {\left( {{\text{xydy}} - {{\text{y}}^2}{\text{dx}}} \right)} \] , where c is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (Use Green's theorem to change the line integral into double integral)
56. The tangent to the curve represented by y = x $$l$$nx is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be
57. Consider the function f(x) = sin (x) in the interval \[{\text{x}} \in \left[ {\frac{\pi }{4},\,\frac{{7\pi }}{4}} \right].\] The number and location(s) of the local minima of this function are
58. Euclidean norm (length) of the vector [4 -2 -6]T is
59. ∇ × ∇ × P where P is a vector is equal to
60. A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is \[{\text{y}} = 4{\text{h}}\left( {\frac{{{{\text{x}}^2}}}{{{{\text{L}}^2}}}} \right)\] , where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is
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