Let \(\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T\) where each \(X_{i}\) contains \(10\) elements and each \(Y_{i}\) contains \(5\) elements. If each element of the set \(T\) is an element of exactly \(20\) of sets \(X_{i}\) 's and exactly \(6\) of sets \(Y_{i}\) 's, then \(n\) 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

The number of integral solutions \(x\) of \(\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0\) is

The sum \(\frac{3}{{{1^2}}} + \frac{5}{{{1^2} + {2^2}}} + \frac{7}{{{1^2} + {2^2} + {3^2}}} + ....\) upto \(11-\) terms is

If \(p, q, r\) are \(3\) real numbers satisfying the matrix equation, \([p\,q\,r]\,\left[ {\begin{array}{*{20}{c}}
3&4&1\\
3&2&3\\
2&0&2
\end{array}} \right] = [3\,\,\,0\,\,\,1]\) then \(2p + q - r\) equals

If the line \(\alpha x+2y=1\), where \(\alpha\in\mathbb{R}\), does not meet the hyperbola \(x^{2}-9y^{2}=9\), then a possible value of \(\alpha\) is:

If \(x = \int\limits_0^y {\frac{{dt}}{{\sqrt {1 + {t^2}} }}} \), then \(\frac{{{d^2}y}}{{d{x^2}}}\) is equal to 

For each \(t \in R\) ,let \(\left[ t \right]\) be the greatest interger less than or equal to \(t\) . Then \(\mathop {\lim }\limits_{x \to 0 + } x\left( {\left[ {\frac{1}{x}} \right] + \left[ {\frac{2}{x}} \right] + .\;.\;.\; + \left[ {\frac{{15}}{x}} \right]} \right) =\) . .. . .

Let \(m_{1}, m_{2}\) be the slopes of two adjacent sides of a square of side a such that \(a^{2}+11 a+3\left(m_{2}^{2}+m_{2}^{2}\right)=220\). If one vertex of the square is \((10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))\), where \(\alpha \in\left(0, \frac{\pi}{2}\right)\) and the equation of one diagonal is \((\cos \alpha-\sin \alpha) x +(\sin \alpha+\cos \alpha) y =10\), then  \(72 \left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13\) is equal to.

If \(\vec{a}, \vec{b}, \overrightarrow{ c }\) are three non-zero vectors and \(\hat{ n }\) is a unit vector perpendicular to \(\vec{c}\) such that \(\overrightarrow{ a }=\alpha \overrightarrow{ b }-\hat{ n },(\alpha \neq 0) \quad\) and \(\quad \overrightarrow{ b } \cdot \overrightarrow{ c }=12\), then \(|\overrightarrow{ c } \times(\overrightarrow{ a } \times \overrightarrow{ b })|\) is equal to :

If \(\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p\), then the value of \(\tan p\) is :

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 ...........

The solution of the differential equation \(\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0\) is (where \(C\) is a constant of integration.)