Ifa variable plane, at a distance of \(3\, units\) from the origin, intersects the coordinate axes at \(A, B\) and \(C\), then the locus of the centroid of \(\Delta ABC\) is

Let \([t]\) denote the greatest integer less than or equal to \(t.\) Then, the value of the integral \(\int\limits_{0}^{1}\left[-8 x^{2}+6 x-1\right] d x\) is equal to

Let \(\alpha\) be the constant term in the binomial expansion of \(\left(\sqrt{ x }-\frac{6}{ x ^{\frac{3}{2}}}\right)^{ n }, n \leq 15\). If the sum of the coefficients of the remaining terms in the expansion is \(649\) and the coefficient of \(x^{-n}\) is \(\lambda \alpha\), then \(\lambda\) is equal to \(..........\).

Let \(e_{1}\) and \(e_{2}\) be the eccentricities of the ellipse, \(\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1(b<5)\) and the hyperbola \(\frac{ x ^{2}}{16}-\frac{ y ^{2}}{ b ^{2}}=1\) respectively satisfying \(e _{1} e _{2}=1 .\) If \(\alpha\) and \(\beta\) are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair \((\alpha, \beta)\) is equal to

Let two straight lines drawn from the origin \(O\) intersect the line \(3 x+4 y=12\) at the points \(P\) and \(\mathrm{Q}\) such that \(\triangle \mathrm{OPQ}\) is an isosceles triangle and \(\angle \mathrm{POQ}=90^{\circ}\). If \(l=\mathrm{OP}^2+\mathrm{PQ}^2+\mathrm{QO}^2\), then the greatest integer less than or equal to \(l\) is :

If \( \alpha \) and \( \beta \) \( (\alpha < \beta) \) are the roots of the equation \( (-2+\sqrt{3})(|\sqrt{x}-3|) + (x-6\sqrt{x}) + (9-2\sqrt{3}) = 0 \), \( x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta} \) is equal to:

Let the determinant of a square matrix \(A\) of order \(m\) be \(m-n\), where \(m\) and \(n\) satisfy \(4 m+n=22\) and \(17 m +4 n =93\). If \(\operatorname{det}(n \operatorname{adj}(\operatorname{adj}( mA )))=\) \(3^{ a } 5^{ b } 6^{ c }\). then \(a + b + c\) is equal to:

If \(\frac{1^3+2^3+3^3+\ldots \ldots \text {.upto } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \ldots \text { upto } n \text { terms }}=\frac{9}{5}\), then the value of \(n\) is

The probability distribution of random variable \(\mathrm{X}\) is given by:

\(X\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(P(X)\)	\(K\)	\(2K\)	\(2K\)	\(3K\)	\(K\)

Let \(\mathrm{p}=\mathrm{P}(1\,<\mathrm{X}\,<\,4 \mid \mathrm{X}\,<\,3)\). If \(5 \mathrm{p}=\lambda \mathrm{K}\), then \(\lambda\) equal to .... .

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 shortest distance between the line \(x-y=1\) and the curve \(x^{2}=2 y\) is .... .

All possible values of \(\theta \in[0,2 \pi]\) for which \(\sin 2 \theta+\tan 2 \theta>0\) lie in

The square of the distance of the image of the point \((6,1,5)\) in the line \(\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}\), from the origin is .............