If \(\alpha\) is the angle made by the perpendicular drawn from origin to the line \(12 x-5 y+13=0\) with the positive \(X\)-axis in anti-clockwise direction, then \(\alpha=\)

\((\sqrt{\sqrt{2}+1}+i \sqrt{\sqrt{2}-1})^8=\)

A unit vector perpendicular to the vectors \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(\vec{b}=3 \vec{j}+2 \vec{k}\) is

If 5 letters are to be placed in 5 -addressed envelopes, then the probability that at least one letter is placed in the wrongly addressed envelope is

Two particles $P$ and $Q$ located at the points $P\left(t,t^3-16t-3\right),Q\left(t+1,t^3-6t-6\right)$ are moving in a plane, the minimum distance between the points in their motion is

\[
\lim _{n \rightarrow \infty} \sum_{n=5}^{5 n+5} \frac{n^2}{2^{\frac{3}{2}}}=
\]

\(\lim _{n \rightarrow \infty} n^4\left[\frac{1}{n^5}+\frac{1}{\left(n^2+1\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{5}{2}}}+\ldots+\right]=\)

Two circles each of radius $5$ units touch each other at $(1,2)$ and $4x+3y=10$ is their common tangent. The equation of that circle among the two given circles, such that some portion of it lies in every quadrant is

Number of different bonds present in \(\mathrm{P}_4 \mathrm{O}_{10}\) is

For a common-emitter transistor amplifier, the current gain is 60 . If the emitter current is \(6.6 \mathrm{~mA}\), then its base current is

The number of electrons present in bonding and antibonding orbitals in \(\mathrm{O}_2^{2-}\) is respectively

The numbers \(a_n=6^n-5 n\) for \(n=1,2,3, \ldots\) when divided by 25 leave the remainder

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is