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 \(....\)

\(\mathop {\lim }\limits_{x \to {1^ - }} \frac{{\sqrt \pi   - \sqrt {2\,{{\sin }^{ - 1}}x} }}{{\sqrt {1 - x} }}\) is equal to

If the number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{8}}\right)^{\text {n }}\) is exactly \(33,\) then the least value of \(n\) is

Let the mid points of the sides of a triangle \(ABC\) be \(\left(\dfrac{5}{2}, 7\right)\), \(\left(\dfrac{5}{2}, 3\right)\) and \((4, 5)\). If its incentre is \((h, k)\), then \(3h + k\) is equal to :

The total number of three-digit numbers, divisible by \(3\) , which can be formed using the digits \(1,3,5,8\) , if repetition of digits is allowed, is:

The number of real roots of the equation \(e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0\) is:

The number of six letter words (with or without meaning), formed using all the letters of the word \('VOWELS',\) so that all the consonants never come together, is ... .

 Let a vertical tower \(AB\) have its end \(A\)  on the level ground. Let \(C\)  be the mid-point of \(AB\) and \(P\) be apoint on the ground such that \(AP=2AB\) . If \(\angle BPC = \beta \) then \(\tan \beta \) is equal to :

If \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\) and \(M=A+A^{2}+A^{3}+\ldots .+A^{20}\), then the sum of all the elements of the matrix \(\mathrm{M}\) is equal to \(.....\)

Let \(f (x) = a^x (a > 0)\) be written as \(f( x) = f_1( x) + f_2( x)\) , where \(f_1( x)\) is an even function and \(f_2( x)\) is an odd function. Then \(f_1( x + y) + f_1( x - y )\) equals

Let \(y=y(x)\) be the solution of the differential equation \(x \log _e x \frac{d y}{d x}+y=x^2 \log _e x,(x > 1)\). If \(y (2)=2\), then \(y ( e )\) 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 : _______

For \(0<\mathrm{c}<\mathrm{b}<\mathrm{a}\), let \((\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}\) \(+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements \((I)\) If \(\alpha \in(-1,0)\), then \(\mathrm{b}\) cannot be the geometric mean of \(\mathrm{a}\) and \(\mathrm{c}\) \((II)\) If \(\alpha \in(0,1)\), then \(\mathrm{b}\) may be the geometric mean of \(a\) and \(c\)