The value of $\underset{\mathrm{n}\to \infty }{\lim } \frac{{\left(\mathrm{n}!\right)}^{{\displaystyle \frac{1}{\mathrm{n}}}}}{\mathrm{n}}$

• 1

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

The points representing the complex number z for which arg $\left(\frac{\mathrm{z} - 2}{\mathrm{z} + 2}\right) = \frac{\mathrm{\pi }}{3}$ lie on

• a circle

• a straight line

• an ellipse

• a parabola

Let a, b, c, p, q and r be positive real numbers such that a, band c are in GP and a^p = b^q = c^r . Then,

• p, q, r are in GP

• p, q, r are in AP

• p, q, r are in HP

• p², q², r² are in AP

Let S_k be the sum of an infinite GP series whose first term is k and common ratio is $\frac{\mathrm{k}}{\mathrm{k} + 1}$(k > 0). Then, the value of $\sum _{\mathrm{k} = 1}^{\infty } \frac{\left(- 1\right)}{{\mathrm{S}}_{\mathrm{k}}}$ is equal to

• ${\log }_{\mathrm{e}}\left(4\right)$

• ${\log }_{\mathrm{e}}\left(2\right) - 1$

• 1 - ${\log }_{\mathrm{e}}\left(2\right)$

• 1 - ${\log }_{\mathrm{e}}\left(4\right)$

55.

The quadratic equation 2x² - (a³ + 8a - 1)x + a² - 4a = 0 posses roots of opposite sign. Then,

• a $\le$ 0

• 0 < a < 4

• $4 \le \mathrm{a} < 8$

• $\mathrm{a} \ge 8$

If ${\log }_{\mathrm{e}}\left({\mathrm{x}}^2 - 16\right) \le {\log }_{\mathrm{e}}\left(4\mathrm{x} - 11\right)$, then

• $4 < \mathrm{x} \le 5$

• x < - 4 or x > 4

• $- 1 < \mathrm{x} \le 5$

• x < - 1 or x > 5

If P = $\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]$, Q = PP^T, then the value of the determinant of Q is

• 2

• - 2

• 1

• 0

If P, Q and R are angles of $△$PQR, then the value of

$\left|\begin{array}{ccc}- 1& \cos \left(\mathrm{R}\right)& \cos \left(\mathrm{Q}\right)\\ \cos \left(\mathrm{R}\right)& - 1& \cos \left(\mathrm{P}\right)\\ \cos \left(\mathrm{Q}\right)& \cos \left(\mathrm{P}\right)& - 1\end{array}\right|$ is equal to

• - 1

• 0

• $\frac{1}{2}$

• 1

The number of real values of a for which the system of equations

x + 3y + 5z = $\mathrm{\alpha x}$

5x + y + 3z = $\mathrm{\alpha y}$

3x + 5y + z = $\mathrm{\alpha z}$

has infinite number of solutions is

• 1

• 2

• 4

• 6

The total number of injections (one-one into mappings) from {a₁, a₂, a₃, a₄} to {b₁, b₂, b₃, b₄, b₅, b_6, b₇} is

• 400

• 420

• 800

• 840