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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
A
$$\frac{3}{{16}}$$
B
0
C
$$\frac{{ - 5}}{{12}}$$
D
-1
Correct Answer:
-1
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
A
$$\overline A \cdot \overline B \cdot C + A \cdot \overline D $$
B
$$\overline A \cdot B \cdot \overline C + A \cdot \overline D $$
C
$$A \cdot \overline B \cdot \overline C + \overline A \cdot D$$
D
$$A \cdot \overline B \cdot C + \overline A \cdot D$$
The Hamiltonian of a particle is given by $$H = \frac{{{p^2}}}{{2m}} + V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right) + \phi \left( { + \left| {\overrightarrow {\bf{r}} } \right|} \right)\overrightarrow {\bf{L}} .\overrightarrow {\bf{S}} ,$$ where $$\overrightarrow {\bf{S}} $$ is the spin, $$V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ and $$\phi \left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$ are potential functions and $$\overrightarrow {\bf{L}} \left( { = \overrightarrow {\bf{r}} \times \overrightarrow {\bf{p}} } \right)$$ is the angular momentum. The Hamiltonian does not commute with
A
$$\overrightarrow {\bf{L}} + \overrightarrow {\bf{S}} $$
B
$$\overrightarrow {{{\bf{S}}^2}} $$
C
$${L_z}$$
D
$$\overrightarrow {{{\bf{L}}^2}} $$
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):
A
<table class="table-style-1"> <tr> <td rowspan="2"><strong>Mutants</strong></td> <td colspan="4"><strong>Media Supplemented with Mutants</strong></td> </tr> <tr> <td>W</td> <td>X</td> <td>Y</td> <td>Z</td> </tr> <tr> <td>Z1</td> <td>No</td> <td>No</td> <td>Yes</td> <td>Yes</td> </tr> <tr> <td>Z2</td> <td>No</td> <td>Yes</td> <td>Yes</td> <td>Yes</td> </tr> <tr> <td>Z3</td> <td>No</td> <td>No</td> <td>No</td> <td>Yes</td> </tr><tr> <td>Z4</td> <td>Yes</td> <td>Yes</td> <td>Yes</td> <td>Yes</td> </tr></table><br>What is the order of the four complementation groups in terms of the step they block? \
B
\
C
\
D
\
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
A
x<sup>2</sup>y + y<sup>2</sup>z + z<sup>2</sup>x
B
2xy + 2yz + 2zx
C
x + y + z
D
0
Evaluate the following:
$$\frac{{\cos 2\theta \cdot \cos 3\theta - \cos 2\theta \cdot \cos 7\theta + \cos \theta \cdot \cos 10\theta }}{{\sin 4\theta \cdot \sin 3\theta - \sin 2\theta \cdot \sin 5\theta + \sin 4\theta \cdot \sin 7\theta }}$$
A
cot6θ.cot5θ
B
cos6θ.cos5θ
C
cos6θ.cot5θ
D
-cot6θ.cot5θ
Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below
The value of $$I$$ is
A
$$\frac{1}{2}{\text{i}}$$
B
$$\frac{2}{3}{\text{i}}$$
C
$$\frac{3}{4}{\text{i}}$$
D
$$\frac{4}{5}{\text{i}}$$
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
A
$$\frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{Q}{r}$$
B
$$\frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{{Qt}}{r}$$
C
$$\frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{Q}{{{r^2}}}$$
D
$$\frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{{Qt}}{{{r^2}}}$$
In a cubic system with cell edge a, two phonons with wave vectors $${\overrightarrow {\bf{q}} _1}$$ and $${\overrightarrow {\bf{q}} _2}$$ collide and produce a third phonon with a wave. vector $${\overrightarrow {\bf{q}} _3}$$ such that $${\overrightarrow {\bf{q}} _1} + {\overrightarrow {\bf{q}} _2} = {\overrightarrow {\bf{q}} _3} + \overrightarrow {\bf{R}} $$ where, $$\overrightarrow {\bf{R}} $$ is a lattice vector. Such a collision process will lead to(a)
A
finite thermal resistance
B
zero thermal resistance
C
an infinite thermal resistance
D
a finite thermal resistance for certain $$\left| {\overrightarrow {\bf{R}} } \right|$$ only
The primitive translation vectors of the body centred cubic lattice are $$\overrightarrow {\bf{a}} = \frac{a}{2}\left( {{\bf{\hat x}} + {\bf{\hat y}} - {\bf{\hat z}}} \right),\,\overrightarrow {\bf{b}} = \frac{a}{2}\left( { - {\bf{\hat x}} + {\bf{\hat y}} + {\bf{\hat z}}} \right)$$ and $$\overrightarrow {\bf{c}} = \frac{a}{2}\left( {{\bf{\hat x}} - {\bf{\hat y}} + {\bf{\hat z}}} \right)$$ . The primitive translation vectors $$\overrightarrow {\bf{A}} ,\,\overrightarrow {\bf{B}} $$ and $$\overrightarrow {\bf{C}} $$ of the reciprocal lattice are
A
$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} + {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat z}}} \right)$$
B
$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} - {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat z}}} \right)$$
C
$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} + {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat z}}} \right)$$
D
$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} + {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat z}}} \right)$$
If for a system of N particles of different masses m
1
, m
2
, . . . m
N
with position vectors $${\overrightarrow {\bf{r}} _1},\,{\overrightarrow {\bf{r}} _2},\,.\,.\,.\,{\overrightarrow {\bf{r}} _N}$$ and corresponding velocities $${\overrightarrow {\bf{v}} _1},\,{\overrightarrow {\bf{v}} _2},\,.\,.\,.\,{\overrightarrow {\bf{v}} _N}$$ respectively such that $$\sum\limits_i {\overrightarrow {{{\bf{v}}_i}} = 0,} $$ then
A
total momentum must be zero
B
total angular momentum must be independent of the choice of the origin
C
the total force on the system must be zero
D
the total torque on the system must be zero