Let \(S_{ k }=\frac{1+2+\ldots .+ K }{ K }\) and \(\sum_{j=1}^n S_j^2=\frac{n}{A}\left( Bn ^2+ Cn + D \right)\), where \(A , B , C , D \in N\) and \(A\) has least value. Then 

Let \(z \in C\) with \(Im(z) = 10\) and it satisfies \(\frac{{2z - n}}{{2z + n}} = 2i - 1\) for some natural number \(n\). Then

Let \(A,B\) and \(C\) be the vertices of a variable right angled triangle inscribed in the parabola \(y^2=16x\). Let the vertex \(B\) containing the right angle be \((4,8)\) and the locus of the centroid of \(\triangle ABC\) be a conic \(C_o\). Then three times the length of latus rectum of \(C_o\) is ______

If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c\) where \(c\) is a constant of integration, then \(\lambda f\left(\frac{\pi}{3}\right)\) is equal to

The straight lines \(l_1\) and \(l_2\) pass through the origin and trisect the line segment of the line \(L: 9 x+5 y=\) 45 between the axes. If \(m_1\) and \(m_2\) are the slopes of the lines \(l_1\) and \(1_2\),then the point of intersection of the line \(y =\left( m _1+ m _2\right) x\) with \(L\) lies on

The set of all values of \(a^2\) for which the line \(x + y =0\) bisects two distinct chords drawn from a point \(P\left(\frac{1+a}{2}, \frac{1-a}{2}\right)\) on the circle \(2 x ^2+2 y ^2-(1+ a ) x -(1- a ) y =0\) is equal to:

The range of the function \(f ( x )=\sqrt{3-x}+\sqrt{2+x}\) is

Let the equations of two adjacent sides of a parallelogram \(A B C D\) be \(2 x-3 y=-23\) and \(5 x+4 y\) \(=23\). If the equation of its one diagonal \(AC\) is \(3 x +\) \(7 y=23\) and the distance of A from the other diagonal is \(d\), then \(50 d ^2\) is equal to \(........\).

If the sum of the first 20 terms of the series
\(\frac{4.1}{4+3.1^2+1^4}+\frac{4.2}{4+3.2^2+2^4}+\frac{4.3}{4+3.3^2+3^4}+\frac{4.4}{4+3.4^2+4^4}+\ldots\)
is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to :-

If \(A\, = \,\left[ {\begin{array}{*{20}{c}}
{{e^t}}&{{e^{ - t}}\,\cos \,t}&{{e^{ - t}}\,\sin \,t}\\
{{e^t}}&{ - {e^{ - t}}\,\cos \, - {e^{ - t}}\,\sin \,t}&{ - {e^{ - t}}\,\sin \,t\, + \,{e^{ - t}}\,\cos \,t}\\
{{e^t}}&{2{e^{ - t}}\,\sin \,t}&{2{e^{ - t}}\,\cos \,t}
\end{array}} \right]\) Then \(A\) is

If \(\int {\frac{{dx}}{{{x^3}{{\left( {1 + {x^6}} \right)}^{2/3}}}} = xf\left( x \right){{\left( {1 + {x^6}} \right)}^{\frac{1}{3}}} + C} \) where \(C\) is a constant of integration, then the function \(f(x)\) is equal to

Suppose that a function \(f: R \rightarrow R\) satisfies \(f(x+y)=f(x) f(y)\) for all \(x, y \in R\) and \(f(1)=3 .\) If \(\sum \limits_{i=1}^{n} f(i)=363,\) then \(n\) is 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 : _______