Let the line x+y=1 meet the circle $x^2+y^2=4$ at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

Define a relation R on the interval $ \left[0, \frac{\pi}{2}\right) $ by $ x $ R $ y $ if and only if $ \sec^2x - \tan^2y = 1 $. Then R is :

Let M and m respectively be the maximum and the minimum values of

$f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 4 x\end{array}\right|, x \in R$

Then $ M^4 - m^4 $ is equal to :

The integral $80 \int\limits_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$ is equal to :