Let \(S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots\) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is \(\sqrt{2026 \mathrm{~S}_{2025}}\), then the absolute difference betwen \(20^{\text {th }}\) and \(15^{\text {th }}\) terms of the A.P. is

The minimum distance of a point on the curve \(y = x^2 - 4\) from the origin 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 : _______

If the equation of the normal to the curve \(y=\frac{x-a}{(x+b)(x-2)}\) at the point \((1,-3)\) is \(x-4 y=13\), then the value of \(a+b\) is equal to \(.......\).

The value of \(\int_{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y\) is :

If in a regular polygon the number of diagonals is \(54\), then the number of sides of this polygon is

Let \(C\) be the locus of the mirror image of a point on the parabola \(y ^{2}=4 x\) with respect to the line \(y = x\). Then the equation of tangent to \(C\) at \(P (2,1)\) is :

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

The system of equations  \(-k x+3 y-14 z=25\)  \(-15 x+4 y-k z=3\)  \(-4 x+y+3 z=4\)  is consistent for all \(k\) in the set

An ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) passes through the vertices of the hyperbola \(H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1\). Let the major and minor axes of the ellipse \(E\) coincide with the transverse and conjugate axes of the hyperbola \(H\). Let the product of the eccentricities of \(E\) and \(H\) be \(\frac{1}{2}\). If \(l\) is the length of the latus rectum of the ellipse \(E\), then the value of \(113\,l\) is equal to \(....\)

If \(\alpha  = {\cos ^{ - 1}}\,\left( {\frac{3}{5}} \right),\beta  = {\tan ^{ - 1}}\,\left( {\frac{1}{3}} \right)\), where  \(0 < \alpha ,\beta  < \frac{\pi }{2}\), then \(\alpha  - \beta \) is equal to

Let \((\lambda, 2,1)\) be a point on the plane which passes through the ponit \((4,-2,2) .\) If the plane is perpendicular to the line joining the points \((-2,-21,29)\) and \((-1,-16,23),\) then \(\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4\) is equal to

A class contains \(b\) boys and \(g\) girls. If the number of ways of selecting \(3\) boys and \(2\) girls from the class is \(168\), then \(b +3\,g\) is equal to.