If $f$ is a function defined by $f(x) = \begin{cases} 5x^2 - x + 3, & x < 1 \\ 3x + 4, & x \geq 1 \end{cases}$, then, at $x = 1$, $f$ is

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

For the function $f(x)=|\cos x|$, which of the following statements are correct?

(A) $f(x)$ is continuous for all $x \in \mathbb{R}$

(B) $f(x)$ is differentiable for all $x \in \mathbb{R}$

(C) $f(x)$ is not differentiable at $x=(2n+1)\dfrac{\pi}{2}, n \in \mathbb{Z}$

(D) $f(x)$ is not differentiable at $x=n\pi, n \in \mathbb{Z}$

Choose the correct answer from the options given below:

If $x = at^2$ and $y = 4at$, then the value of $\frac{d^2y}{dx^2}$ at $t = 1/a$, is

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is: