If \(z_1 , z_2\) and \(z_3, z_4\) are \(2\) pairs of complex conjugate numbers, then \(\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)\) equals 

Number of solutions of \( \sqrt{3}\cos 2\theta+8\cos \theta+3\sqrt{3}=0 \), \( \theta \in [-3\pi, 2\pi] \) is:

The total number of \(4-\)digit numbers whose greatest common divisor with \(54\) is \(2\) , is \(..........\).

Let the line \(L\) pass through the point \((0,1,2)\), intersect the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and be parallel to the plane \(2 x+y-3 z=4\). Then the distance of the point \(P(1,-9,2)\) from the line \(L\) is

 Let a vertical tower \(AB\) have its end \(A\)  on the level ground. Let \(C\)  be the mid-point of \(AB\) and \(P\) be apoint on the ground such that \(AP=2AB\) . If \(\angle BPC = \beta \) then \(\tan \beta \) is equal to :

The axis of a parabola is the line \(y=x\) and its vertex and focus are in the first quadrant at distances \(\sqrt{2}\) and \(2 \sqrt{2}\) units from the origin, respectively. If the point \((1, \mathrm{k})\) lies on the parabola, then a possible value of \(k\) is :-

If \(\sum \limits_{i=1}^{n}\left(x_{i}-a\right)=n\) and \(\sum \limits_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a>1)\) then the standard deviation of \(n\) observations \(x _{1}, x _{2}, \ldots, x _{ n }\) is

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

The general solution of the differential equation \((y^2 -x^3) dx -xydy = 0\, (x \ne 0)\) is (where \(c\) is a constant of integration)

The interior angles of a polygon with n sides, are in an A.P. with common difference \(6^{\circ}\). If the largest interior angle of the polygon is \(219^{\circ}\), then n is equal to

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:

Let \(\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T\) where each \(X_{i}\) contains \(10\) elements and each \(Y_{i}\) contains \(5\) elements. If each element of the set \(T\) is an element of exactly \(20\) of sets \(X_{i}\) 's and exactly \(6\) of sets \(Y_{i}\) 's, then \(n\) is equal to

The area (in sq. units) of the region bounded by the parabola, \(y = x^2 + 2\) and the lines, \(y = x + 1, x = 0\) and \(x = 3\), is