If the coefficients of the middle terms in the binomial expansions of \((1 + \alpha x)^{26}\) and \((1 - \alpha x)^{28}\), \(\alpha \neq 0\), are equal, then the value of \(\alpha\) 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 ...........

For \(x\,\, \in \,R\,,x\, \ne \,0,\) let \({f_0}(x) = \frac{1}{{1 - x}}\) and \({f_{n + 1}}(x) = {f_0}({f_n}(x)),\) \(n\, = 0,1,2,....\)  Then the value of \({f_{100}}(3) + {f_1}\left( {\frac{2}{3}} \right) + {f_2}\left( {\frac{3}{2}} \right)\) is equal to

If \(P=\left[\begin{array}{ll}1 & 0 \\ 1 / 2 & 1\end{array}\right]\), then \(P^{50}\) is:

The number of numbers, strictly between \(5000\) and \(10000\) can be formed using the digits \(1,3,5,7,9\) without repetition, is \(..........\).

For \(x > 0,\) if \(f(x)=\int_{1}^{x} \frac{\log _{e} t}{(1+t)} d t,\) then \(f(e)+f\left(\frac{1}{e}\right)\) is equal to ...... .

Let \(\alpha=\frac{-1+i \sqrt{3}}{2} .\) If \(a=(1+\alpha) \sum\limits_{k=0}^{100} \alpha^{2 k}\) and \(\mathrm{b}=\sum\limits_{\mathrm{k}=0}^{100} \alpha^{3 \mathrm{k}},\) then a and \(\mathrm{b}\) are the roots of the quadratic equation

Let \(A\) and \(B\) be two \(3 \times 3\) real matrices such that \(\left(A^{2}-B^{2}\right)\) is invertible matrix. If \(A^{5}=B^{5}\) and \(A^{3} B^{2}=A^{2} B^{3}\), then the value of the determinant of the matrix \(A^{3}+B^{3}\) is equal to:

The sum of the following series \(1 + 6 + \frac{{9({1^2} + {2^2} + {3^2})}}{7} + \frac{{12({1^2} + {2^2} + {3^2} + {4^2})}}{9} + \frac{{15({1^2} + {2^2} + .... + {5^2})}}{{11}} + ...\) up to \(15\) terms, is

The principal value of \({\tan ^{ - 1}}\left( {\cot \frac{{43\pi }}{4}} \right)\) is

The probability that a randomly chosen \(2 \times 2\) matrix with all the entries from the set of first \(10\) primes, is singular, is equal to

Let \(|\cos \theta \cos (60-\theta) \cos (60-\theta)| \leq \frac{1}{8}, \theta \in[0,2 \pi]\) Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :

Let \([ x ]\) denote the greatest integer function and \(f ( x )=\max \{1+ x +[ x ], 2+ x , x +2[ x ]\}, 0 \leq x \leq 2\). Let \(m\) be the number of points in \([0,2]\), where \(f\) is not continuous and \(n\) be the number of points in \((0,2)\), where \(f\) is not differentiable. Then \((m+n)^2+2\) is equal to: