If $x = t^2$ and $y = t\sqrt{t}$, then $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{1}{4}$ is :

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

Match List-I with List-II

Choose the correct answer from the options given below:

The value of '$\lambda$' for which the function $f(x) = \begin{cases}\frac{1-\cos 4x}{4x^2}, & \text{if } x \neq 0 \\ \lambda, & \text{if } x = 0\end{cases}$ is continuous at $x=0$, is:

The function $f: \mathbb{R} \to \mathbb{R}$ defined by

$f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$

is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to