A tangent having slope $-\dfrac{1}{2}$ to the ellipse $3x^2 + 4y^2 = 12$ intersects the X-axis and Y-axis at the points A and B respectively. if O is the origin, then the area of $\triangle AOB$ is ...

If the function $f(x) = \dfrac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x}$ is continuous at $x = \dfrac{\pi}{4}$, then the value of $f\left(\dfrac{\pi}{4}\right)$ is ...

The value of $\lim\limits_{x \to 0}\left(\dfrac{8}{x^8}\right)\left[1 - \cos\dfrac{x^2}{2} - \cos\dfrac{x^2}{4} + \cos\dfrac{x^2}{2}\cdot\cos\dfrac{x^2}{4}\right]$ is equal to ...

The simplified switching circuit for the following circuit is