If the function $f(x) = \dfrac{4\sqrt{2}(\sin 3x + \sin x)}{2\sin 2x\sin\dfrac{3x}{2} + \cos\dfrac{5x}{2} - \cos\dfrac{3x}{2}}$ for $x \neq \dfrac{\pi}{2}$ is continuous at $x = \dfrac{\pi}{2}$, then the value of $f\left(\dfrac{\pi}{2}\right)$ is equal to

If $\lim\limits_{x \to 1} \dfrac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b} = -2$, then the quadratic equation having roots $a$ and $b$ is

Which of the following function is discontinuous at $x = 0$ ?

If $\lim\limits_{x \to 0}\dfrac{45^x - 9^x - 5^x + 1}{(k^x - 1)(3^x - 1)} = 2$, then the value of $k$ is ...