The value of \(\lambda\) and \(\mu\) such that the system of equations \(x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu\) has no solution, are :

The length of the minor axis (along \(y-\)axis) of an ellipse in the standard form is \(\frac{4}{\sqrt{3}} .\) If this ellipse touches the line, \(x+6 y=8 ;\) then its eccentricity is

Let \(a_1, a_2, \ldots \ldots, a_n\) be in A.P. If \(a_5=2 a_7\) and \(a_{11}=18\), then \(12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots . \cdot \frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)\) is equal to \(..........\).

Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

Let \(p(x)\) be a quadratic polynomial such that \(p(0)= 1\) . If \(p(x)\) leaves remainder \(4\) when divided by \(x-1\) and it leaves remainder \(6\) when divided by \(x+ 1\) ; then

Let the sum of the focal distances of the point \(\mathrm{P}(4,3)\) on the hyperbola \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1\) be \(8 \sqrt{\frac{5}{3}}\). If for \(H\), the length of the latus rectum is \(l\) and the product of the focal distances of the point P is m , then \(9 l^2+6 \mathrm{~m}\) is equal to :-

If \(\frac{d y}{d x}=\frac{2^{x+y}-2^{x}}{2^{y}}, y(0)=1\), then \(y(1)\) is equal to :

If one real root of the quadratic equation \(81x^2 + kx  + 256 = 0\) is cube of the other root, then a value of \(k\) is

Let \(\mathrm{n}\) be a non-negative integer. Then the number of divisors of the form " \(4 \mathrm{n}+1\) " of the number \((10)^{10} \cdot(11)^{11} \cdot(13)^{13}\) is equal to \(....\)

The positive value of the determinant of the matrix \(A\), whose \(A d j(A d j(A))=\left(\begin{array}{ccc}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{array}\right)\), is

Let \(a, b, c > 1, a^3, b^3\) and \(c^3\) be in \(A.P.\), and \(\log _a b\), \(\log _c a\) and \(\log _b c\) be in G.P. If the sum of first \(20\) terms of an \(A.P.\), whose first term is \(\frac{a+4 b+c}{3}\) and the common difference is \(\frac{a-8 b+c}{10}\) is \(-444\), then abc is equal to

Let \(f\left( n \right) = \left[ {\frac{1}{3} + \frac{{3n}}{{100}}} \right]n\) , where \([n]\) denotes the greatest integer less than or equal to \(n\). Then \(\sum\limits_{n = 1}^{56} {f\left( n \right)} \) is equal to