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The following Boolean expression $$Y = A \cdot \overline B \cdot \overline C \cdot \overline D + \overline A \cdot B \cdot \overline C \cdot D + \overline A \cdot \overline B \cdot \overline C \cdot D + \overline A \cdot \overline B \cdot \overline C \cdot D + \overline A \cdot B \cdot C \cdot D + A \cdot \overline B \cdot \overline C \cdot D$$                 can be simplified to
W, X and Y are the intermediates in a biochemical pathway as shown below:
$$S \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \to W \to X \to Y \to Z$$
Mutants auxotrophic for Z are found in four different complementation groups, namely Z1, Z2, Z3 and Z4. The growth of these mutants on media supplemented with W, X, Y or Z is shown below (Yes: growth observed: No: growth not observed):
Evaluate the following:
$$\frac{{\cos 2\theta \cdot \cos 3\theta - \cos 2\theta \cdot \cos 7\theta + \cos \theta \cdot \cos 10\theta }}{{\sin 4\theta \cdot \sin 3\theta - \sin 2\theta \cdot \sin 5\theta + \sin 4\theta \cdot \sin 7\theta }}$$
Let $$\nabla \cdot \left( {{\text{f}}\overrightarrow {\text{v}} } \right) = {{\text{x}}^2}{\text{y}} + {{\text{y}}^2}{\text{z}} + {{\text{z}}^2}{\text{x}},$$      where f and v are scalar and vector fields respectively. If $$\overrightarrow {\text{v}} = {\text{y}}\overrightarrow {\text{i}} + {\text{z}}\overrightarrow {\text{j}} + {\text{x}}\overrightarrow {\text{k}} ,$$     then $$\overrightarrow {\text{v}} \cdot \nabla {\text{f}}$$   is
For a vector potential $$\overrightarrow {\bf{A}} ,$$ the divergence of $$\overrightarrow {\bf{A}} $$ is $$\overrightarrow \nabla \cdot \overrightarrow {\bf{A}} = - \frac{{{\mu _0}}}{{4\pi }} \cdot \frac{Q}{{{r^2}}},$$    where Q is a constant of appropriate dimension. The corresponding scalar potential $$\phi \left( {r,\,t} \right)$$  that makes $$\overrightarrow {\bf{A}} $$ and $$\phi $$ Lorentz gauge invariant is
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
Consider a system described by ẋ = Ax + Bu y = Cx + Du The system is completely output controllable if and only if Where: x = State vector (n-vector) u = Control vector (r-vector) y = Output vector (m-vector) A = n × n matrix B = n × r matrix C = m × n matrix D = m × r matrix
Velocity of a man is 25 units in the positive X direction and 75 units in negative Y direction starting from the origin. Velocity of a woman is 25 units in the positive Y direction and 75 units in the negative X direction starting from the origin. What is the relative velocity of the man with respect to the woman?
Velocity of a dog is 15 units in the positive X direction and 3 units in positive Y direction starting from the origin. Velocity of a pig is 7 units in the positive Y direction and 17 units in the positive X direction starting from the origin. What is the relative velocity of the dog with respect to the pig?
A vector, 7 units from the origin, along the X axis, is added to vector 11 units from the origin along the Y axis. What is the resultant vector?