For \(\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), if \(y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x\) and \(\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0\) then \(y\left(\frac{\pi}{4}\right)\) is equal to

The number of elements in the set \(\left\{A=\left(\begin{array}{ll}a & b \\ 0 & d\end{array}\right): a, b, d \in\{-1,0,1\}\right.\) and \(\left.(I-A)^{3}=I-A^{3}\right\}\) where \(I\) is \(2 \times 2\) identity matrix, is :

If \([ t ]\) denotes the greatest integer \(\leq t\), then the value of \(\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] d x\) is.

Let \(f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right] .\) Then, \(f\) is \(.....\)

If \(\left( {2 + \sin x} \right)\frac{{dy}}{{dx}} + \left( {y + 1} \right)\cos x = 0\) and \(y\left( 0 \right) = 1\) then \(y\left( {\frac{\pi }{2}} \right) = \) . . . . 

Let \(x=x(y)\) be the solution of the differential equation \(2 y \,e^{x / y^{2}} d x+\left(y^{2}-4 x e^{x / y^{2}}\right) d y=0\) such that \(x(1)=0\). Then, \(x(e)\) is equal to

The integral \(\int_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}\,x\,\,\cos e{c^{4/3}}\,x\,dx} \) is equal to

Let an ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}\), passes through \(\left(\sqrt{\frac{3}{2}}, 1\right)\) and has ecentricity \(\frac{1}{\sqrt{3}} .\) If a circle, centered at focus \(\mathrm{F}(\alpha, 0), \alpha>0\), of \(\mathrm{E}\) and radius \(\frac{2}{\sqrt{3}}\), intersects \(\mathrm{E}\) at two points \(\mathrm{P}\) and \(\mathrm{Q}\), then \(\mathrm{PQ}^{2}\) is equal to:

Let the product of the focal distances of the point \(\left(\sqrt{3}, \frac{1}{2}\right)\) on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(\mathrm{a}\gt\mathrm{b})\), be \(\frac{7}{4}\). Then the absolute difference of the eccentricities of two such ellipses is

A person throws two fair dice. He wins \(Rs.\, 15\) for throwing a doublet (same numbers on the two dice), wins \(Rs.\,12\) when the throw results in the sum of \(9\), and loses \(Rs.\, 6\) for any other outcome on the throw. Then the expected gain/loss (in \(Rs.\)) of the person is

The natural number \(m\), for which the coefficient of \(x\) in the binomial expansion of \(\left( x ^{ m }+\frac{1}{ x ^{2}}\right)^{22}\) is \(1540,\) is

If the domain of the function \(\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)\) is \((\alpha, \beta]\), then \(3 \alpha+10 \beta\) 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 ...........