71
The inverse of the complex number $$\frac{{3 + 4i}}{{3 - 4i}}$$ is
A)
$$\frac{7}{{25}} + i\frac{{24}}{{25}}$$
B)
$$ - \frac{7}{{25}} + i\frac{{24}}{{25}}$$
C)
$$\frac{7}{{25}} - i\frac{{24}}{{25}}$$
D)
$$ - \frac{7}{{25}} - i\frac{{24}}{{25}}$$
Answer & Solution
Answer: Option
D
72
A 3 × 3 matrix has elements such that its trace is 11 and its determinant is 36. The eigen values of the matrix are all known to be positive integers. The largest eigen value of the matrix is
A)
18
B)
12
C)
9
D)
6
Answer & Solution
Answer: Option
C
73
The unit vector normal to the surface x2 + y2 - z = 1 at the point P (1, 1, 1) is
A)
$$\frac{{{\bf{\hat i}} + {\bf{\hat j}} - {\bf{\hat k}}}}{{\sqrt 3 }}$$
B)
$$\frac{{2{\bf{\hat i}} + {\bf{\hat j}} - {\bf{\hat k}}}}{{\sqrt 6 }}$$
C)
$$\frac{{{\bf{\hat i}} + 2{\bf{\hat j}} - {\bf{\hat k}}}}{{\sqrt 6 }}$$
D)
$$\frac{{2{\bf{\hat i}} + 2{\bf{\hat j}} - {\bf{\hat k}}}}{3}$$
Answer & Solution
Answer: Option
D
74
The value of $$\int_{ - i}^i {\pi \left( {z + 1} \right)dz} $$ is
A)
zero
B)
2πi
C)
-2πi
D)
(-1 + 2i)π
Answer & Solution
Answer: Option
D
75
For arbitrary matrices E, F, G and H, if EF - FE = 0 then Trace (EFGH) is equal to
A)
Trace (HGFE)
B)
Trace (E).Trace (F).Trace (G).Trace (H)
C)
Trace (GFEH)
D)
Trace (EGHF)
Answer & Solution
Answer: Option
A
76
Consider the differential equation $$\frac{{{d^2}x}}{{d{t^2}}} + 2\frac{{dx}}{{dt}} + x = 0$$
A)
$$\frac{1}{e}$$
B)
$$\frac{2}{e}$$
C)
1
D)
$$\frac{3}{e}$$
Answer & Solution
Answer: Option
B
77
Given the four vectors \[{u_1} = \left[ {\begin{array}{*{20}{c}}
1 \\
2 \\
3
\end{array}} \right],\,{u_2} = \left[ {\begin{array}{*{20}{c}}
3 \\
{ - 5} \\
1
\end{array}} \right],\,{u_3} = \left[ {\begin{array}{*{20}{c}}
2 \\
4 \\
{ - 8}
\end{array}} \right].\,{u_4} = \left[ {\begin{array}{*{20}{c}}
3 \\
6 \\
{ - 12}
\end{array}} \right]\]
A)
u1 , u2
B)
u1 , u3
C)
u1 , u4
D)
u3 , u4
Answer & Solution
Answer: Option
D
78
Eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}}
0&1&0&0 \\
1&0&0&0 \\
0&0&0&{ - 2i} \\
0&0&{2i}&0
\end{array}} \right]\] are
A)
-2, -1, 1, 2
B)
-1, 1, 0, 2
C)
1, 0, 2, 3
D)
-1, 1, 0, 3
Answer & Solution
Answer: Option
A
79
The eigen values of the matrix \[A = \left[ {\begin{array}{*{20}{c}}
0&i \\
i&0
\end{array}} \right]\] are
A)
real and distinct
B)
complex and distinct
C)
complex and coinciding
D)
real and coinciding
Answer & Solution
Answer: Option
B
80
If \[\overrightarrow {\mathbf{r}} = x{\mathbf{\hat i}} + y{\mathbf{\hat j}},\] then
A)
\[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{r}} = 0{\text{ and }}\overrightarrow \nabla \left| {\overrightarrow {\mathbf{r}} } \right| = \overrightarrow {\mathbf{r}} \]
B)
\[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{r}} = 2{\text{ and }}\overrightarrow \nabla \left| {\overrightarrow {\mathbf{r}} } \right| = \overrightarrow {\mathbf{r}} \]
C)
\[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{r}} = 2{\text{ and }}\overrightarrow \nabla \left| {\overrightarrow {\mathbf{r}} } \right| = \frac{{{\mathbf{\hat r}}}}{r}\]
D)
\[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{r}} = 3{\text{ and }}\overrightarrow \nabla \left| {\overrightarrow {\mathbf{r}} } \right| = \frac{{{\mathbf{\hat r}}}}{r}\]
Answer & Solution
Answer: Option
C
