Calculus MCQs Question and Answer in Engineering Maths

$$\overrightarrow {\text{a}} ,\,\overrightarrow {\text{b}} ,\,\overrightarrow {\text{c}} $$   are three orthogonal vectors, Given that \[\overrightarrow {\rm{a}} = {\rm{\hat i}} + 2{\rm{\hat j}} + 5{\rm{\hat k}}\]    and \[\overrightarrow {\rm{b}} = {\rm{\hat i}} + 2{\rm{\hat j}} - {\rm{\hat k}}\]    , the vector $$\,\overrightarrow {\text{c}} $$ is parallel to

Given $$\overrightarrow {\text{F}} = \left( {{{\text{x}}^2} - 2{\text{y}}} \right)\overrightarrow {\text{i}} - 4{\text{yz}}\overrightarrow {\text{j}} + 4{\text{x}}{{\text{z}}^2}\overrightarrow {\text{k}} ,$$       the value of the line integral $$\int\limits_{\text{c}} {\overrightarrow {\text{F}} \cdot d\overrightarrow l } $$   along the straight line c from (0, 0, 0) to (1,1,1) is

At x = 0, the function f(x) = |x| has

A parabola x = y² with 0 ≤ x ≤ 1 is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360° around the x-axis is

The line integral $$\int\limits_{{{\text{P}}_1}}^{{{\text{P}}_2}} {\left( {{\text{ydx}} + {\text{xdy}}} \right)} $$   for P₁(x₁, y₁) to P₂(x₂, y₂) along the semicircle P₁, P₂ shown in the figure is

The angle between two unit-magnitude coplanar vectors P(0.866, 0.500, 0) and Q(0.259, 0.966, 0) will be

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
$$\iint\limits_{\text{S}} {\frac{1}{4}\left( {{\text{F}} \cdot {\text{n}}} \right){\text{dA}}}$$    where S is the sphere, x² + y² + z² = 1 and n is the outward unit normal vector to the sphere.
The value of the surface integral is

If a function is continuous at a point,

The vector that is normal to the surface 2xz² - 3xy - 4x = 7 at the point (1, -1, 2) is

The derivative of the symmetric function drawn in given figure will look like