31. Let $${\text{f}}\left( {{\text{x, y}}} \right) = \frac{{{\text{a}}{{\text{x}}^2} + {\text{b}}{{\text{y}}^2}}}{{{\text{xy}}}},$$ where a and b are constants. If $$\frac{{\partial {\text{f}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{f}}}}{{\partial {\text{y}}}}$$ at x = 1 and y = 2, then the relation between a and b is
32. The curve y = f(x) is such that the tangent to the curve at every point (x, y) has a Y-axis intercept c, given by c = -y. Then f(x) is proportional to
33. If f(x) is an even function and a is a positive real number, then $$\int_{ - {\text{a}}}^{\text{a}} {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$ equals
34. Curl of vector \[\overrightarrow {\rm{F}} = {{\rm{x}}^2}{{\rm{z}}^2}{\rm{\hat i}} - 2{\rm{x}}{{\rm{y}}^2}{\rm{z\hat j}} + 2{{\rm{y}}^2}{{\rm{z}}^3}{\rm{\hat k}}\] is
35. The directional derivative of the scalar function f(x, y, z) = x2 + 2y2 + z at the point P = (1, 1, 2) in the direction of the vector \[\overrightarrow {\rm{a}} = 3{\rm{\hat i}} - 4{\rm{\hat j}}\] is
36. The curl of the gradient of the scalar field defined by V = 2x2y + 3y2z + 4z2x is
37. The value of $$\mathop {\lim }\limits_{\left( {{\text{x,}}\,{\text{y}}} \right) \to \left( {0,\,0} \right)} \frac{{{{\text{x}}^2} - {\text{xy}}}}{{\sqrt {\text{x}} - \sqrt {\text{y}} }}$$ is
38. If $${{\text{e}}^{\text{y}}} = {{\text{x}}^{\frac{1}{{\text{x}}}}},$$ then y has a
39. $$\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\sin }^2}{\text{x}}}}{{\text{x}}}$$ is equal to
40. Equation of the line normal to function $${\text{f}}\left( {\text{x}} \right) = {\left( {{\text{x}} - 8} \right)^{\frac{2}{3}}} + 1$$ at P(0, 5) is
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