If the domain of the function \(f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log _e(3-x)\right)^{-1}\) is \([-\alpha, \beta)-\{y\}\), then \(\alpha+\beta+\gamma\) is equal to :

If the sum of the series \(20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .\) upto \(n ^{ th }\) term is \(488\) and the \(n^{\text {th }}\) term is negative, then

If the constant term in the binomial expansion of \(\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}\) is \(405,\) then \(|k|\) equals 

If \(\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c\) where \(c\) is a constant of integration, then the ordered pair \(( a , b )\) is equal to

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 .... .

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

If \(f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R\), then

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 \(..........\).

Consider the system of linear equations \(x+y+z=5, x+2 y+\lambda^2 z=9\) \(x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in R\). Then, which of the following statement is NOT correct?

A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{i}-\hat{j}, \hat{i}+\hat{k}\). The obtuse angle between \(\vec{a}\) and the vector \(\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}\) is

Consider the system of linear equation \(x+y+z=\) \(4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is \(NOT\) correct?

If \((2,3,9),(5,2,1),(1, \lambda, 8)\) and \((\lambda, 2,3)\) are coplanar, then the product of all possible values of \(\lambda\) is.

If the angle between the line \(2(x + 1)\,= y\, = z + 4\)  and the plane \(2x -\sqrt \lambda  \,z+4\,=0\) is \(\frac{\pi}{6}\), then the value of \(\lambda \), is