Let \(A\) and \(B\) be two non -null events such that \(A \subset B\). Then, which of the following statements is always correct?

If \(z\) and \(\omega\) are two complex numbers such that \(|z \omega|=1\) and \(\arg (z)-\arg (\omega)=\frac{3 \pi}{2}\), then \(\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)\) is: (Here arg(z) denotes the principal argument of complex number \(z\) )

If \(15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6,\) for some \(\alpha \in R ,\) then the value of \(27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha\) is equal to ....... .

Let \(P_{1}, P_{2}, \ldots \ldots, P_{15}\) be \(15\) points on a circle. The number of distinct triangles formed by points \(P_{i}, P_{j}, P_{k}\) such that \(i+j+k \neq 15\), is :

Let  \(a,b,c\; \in R.\) If \(f\left( x \right) = a{x^2} + bx + c\) is such that \(a + b + c = 3\) and \(f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy,\) \(\forall x,y \in R,\) then \(\mathop \sum \limits_{n = 1}^{10} f\left( n \right)\) is equal to : 

A circle touches both the \(y\)-axis and the line \(x+y=0\). Then the locus of its center is

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

The area (in sq. units) of an equilateral triangle inscribed in the parabola \(\mathrm{y}^{2}=8 \mathrm{x},\) with one of its vertices on the vertex of this parabola, is 

A data consists of \(n\) observations \({x_1},{x_2},......,{x_n}.\) If \(\sum\limits_{i - 1}^n {{{({x_i} + 1)}^2}}  = 9n\) and \(\sum\limits_{i - 1}^n {{{({x_i} - 1)}^2}}  = 5n,\) then the standard deviation of this data is

\(\lim _{x \rightarrow 0} \operatorname{cosec} x\)\(\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)\) is:

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

The angles \(A, B\) and \(C\) of a triangle \(ABC\) are in \(A.P\) and \(a : b = 1 : \sqrt 3 .\) If \(c = 4\, cm,\) then the area (in sq. cm) of this triangle is

Let \(\vec a\, = \,\hat i\, + \,2\hat j\, + 4\hat k\,,\,\vec b\, = \,\hat i\, + \,\lambda \hat j\, + 4\hat k\) and \(\vec c\, = \,2\hat i\, + \,4\hat j\, + ({\lambda ^2} - 1)\hat k\) be coplanar vectors. Then the non -zero vector \(\vec a\times \vec c\) is