Let \(A\) be a non-singular matrix of order \(3\). If \(\det(3 \, \operatorname{adj}(2 \, \operatorname{adj}((\det A)A))) = 3^{-13} \cdot 2^{-10}\) and \(\det(3 \, \operatorname{adj}(2A)) = 2^{m} \cdot 3^{n}\), then \(|3m + 2n|\) is equal to ______.

Different \(A.P.\)'s are constructed with the first term \(100\),the last term \(199\),And integral common differences. The sum of the common differences of all such, \(A.P\)'s having at least \(3\) terms and at most \(33\) terms is.

For the system of linear equations : \(x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R\) consider the following statements : \((A)\) The system has unique solution if \(k \neq 2\), \(k \neq-2\) \((B)\) The system has unique solution if \(k =-2\). \((C)\) The system has unique solution if \(k =2\). \((D)\) The system has no-solution if \(k =2\). \((E)\) The system has infinite number of solutions if \(k \neq-2\) Which of the following statements are correct?

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to

The equations of two sides of a variable triangle are \(x =0\) and \(y =3\), and its third side is a tangent to the parabola \(y^2=6 x\). The locus of its circumcentre is :

The remainder when \(\left((64)^{(64)}\right)^{(64)}\) is divided by 7 is equal to

\(\mathop \smallint \limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} \frac{{dx}}{{1 + \cos x}} = \) . . . .

A tangent line \(\mathrm{L}\) is drawn at the point \((2,-4)\) on the parabola \(\mathrm{y}^{2}=8 \mathrm{x}\). If the line \(\mathrm{L}\) is also tangent to the circle \(x^{2}+y^{2}=a\), then \('a'\) is equal to .... .

A tangent is drawn to the parabola \(y^{2}=6 x\) which is perpendicular to the line \(2 x + y =1\) Which of the following points does \(NOT\) lie on it ?

Let \(f : R \to R\) be a differentiable function satisfying \(f’’(3) + f’(2) = 0\). Then \(\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{1 + f\left( {3 + x} \right) - f\left( 3 \right)}}{{1 + f\left( {2 - x} \right) - f\left( 2 \right)}}} \right)^{\frac{1}{x}}}\) is equal to

Let \(S\) and \(S\,'\) be the foci of an ellipse and \(B\) be any one of the extremities of its minor axis. If \(\Delta S\,'BS\) is a right angled triangle with right angle at \(B\) and area \((\Delta S\,'BS) = 8\,sq.\) units, then the length of a latus rectum of the ellipse is

A coin is tossed \(8\) times. If the probability that exactly \(4\) heads appear in the first six tosses and exactly \(3\) heads appear in the last five tosses is \(p\), then \(96p\) is equal to _____.

Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(\mathrm{x} \in(0,3)\). If \(\mathrm{g}\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) is