If the domain of the function
\(f(x)=\log _e\left(\frac{2 x-3}{5+4 x}\right)+\sin ^{-1}\left(\frac{4+3 x}{2-x}\right) \quad \text { is } \quad[\alpha, \beta)\)
then \(\alpha^2+4 \beta\) is equal to

Let \(f: \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow \mathrm{R}\) be defined as \(f(x)=(1+|\sin x|)^{\frac{3 a}{\sin x \mid}} ,\quad -\frac{\pi}{4}\,<\,x\,<\,0\) \(\quad\quad\quad\quad\quad\quad b ,\quad\quad\quad\quad\quad x=0\) \(\quad\quad\quad\quad e^{\cot 4 x / \cot 2 x} ,\quad\quad\quad 0\,<\,x\,<\,\frac{\pi}{4}\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then the value of \(6 \mathrm{a}+\mathrm{b}^{2}\) is equal to:

Let \(\quad \overrightarrow{ a }=2 \hat{ i }-7 \hat{ j }+5 \hat{ k } \quad, \quad \overrightarrow{ b }=\hat{ i }+\hat{ k } \quad\) and \(\overrightarrow{ c }=\hat{ i }+2 \hat{ j }-3 \hat{ k }\) be three given vectors. If \(\overrightarrow{ r }\) is a vector such that \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ c } \times \overrightarrow{ a }\) and \(\overrightarrow{ r } \cdot \overrightarrow{ b }=0\), then \(|\overrightarrow{ r }|\) is equal to :

\(\int\limits_{0}^{5} \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x\) Where \([t]\) denotes greatest integer less than or equal to \(t\), is equal to

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

In an examination, there are \(10\) true-false type questions. Out of \(10\) , a student can guess the answer of \(4\) questions correctly with probability \(\frac{3}{4}\) and the remaining \(6\) questions correctly with probability \(\frac{1}{4}\). If the probability that the student guesses the answers of exactly \(8\) questions correctly out of \(10\) is \(\frac{27 k }{4^{10}}\), then \(k\) is equal to

Let the length of the latus rectum of an ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \( (a>b) \), be 30. If its eccentricity is the maximum value of the function \( f(t)=-\frac{3}{4}+2t-t^{2}, \) then \( (a^{2}+b^{2}) \) is equal to -

Let \(A\) and \(B\) be two finite sets with \(m\) and \(n\) elements respectively. The total number of subsets of the set \(A\) is 56 more than the total number of subsets of \(B\). Then the distance of the point \(P ( m , n )\) from the point \(Q (-2,-3)\) is

If the absolute maximum value of the function \(f ( x )\) \(=\left( x ^{2}-2 x +7\right) e ^{\left(4 x^{3}-12 x ^{2}-180 x +31\right)}\) in the interval \([-3\), \(0]\) is \(f (\alpha)\), then.

If the mean of the following probability distribution of a random variable \(X\);

\(X\)	\(0\)	\(2\)	\(4\)	\(6\)	\(8\)
\(P(X)\)	\(a\)	\(2a\)	\(a+b\)	\(2b\)	\(3b\)

is \(\frac{46}{9}\) , then the variance of the distribution is

\(\int \limits_{-\pi}^{\pi}|\pi-| x || d x\) is equal to :

A straight line \(L\) at a distance of \(4\) units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of \(60^o\) with the line \(x + y = 0\). Then an equation of the line \(L\) is