Continuity and Differentiability — NDA Mathematics PYQs

If $f(x)$ is differentiable at $x=a$, then consider the following statements :
I. $f(x)$ is continuous at $x=a$
II. ${\lim }_{x\to a}f(x)=f(a)$
Which of the statements given above is/are correct ?

Directions : For the next two (02) items that follow :
Let $f(x)=\{\begin{array}{ll}ax(x-1),& x<1\\ x-1,& 1⩽x⩽3\\ p{x}^2+qx+2,& x>3\end{array}$
Given that $f(x)$ is continuous for all $x$ but not differentiable at $x=1$. Further $f^{\prime }(x)$ is continuous at $x=3$.

What is the value of $p$ ?

Directions : For the next two (02) items that follow :
Let $f(x)=\{\begin{array}{ll}ax(x-1),& x<1\\ x-1,& 1⩽x⩽3\\ p{x}^2+qx+2,& x>3\end{array}$
Given that $f(x)$ is continuous for all $x$ but not differentiable at $x=1$. Further $f^{\prime }(x)$ is continuous at $x=3$.

What is the value of $q$ ?

Directions : For the next two (02) items that follow :
Let $f(x)=\tan (x^2)$ and $g(x)=x|x|$ for $|x|<\sqrt{\frac{\pi }{2}}$.

If $p(x)=f(x)g(x)$, then which of the following statements is/are correct ?
I. $p(x)$ is continuous at $x=0$.
II. $p(x)$ is differentiable at $x=0$.
Select the answer using the code given below :

Directions : For the next two (02) items that follow :
Let $f(x)=\tan (x^2)$ and $g(x)=x|x|$ for $|x|<\sqrt{\frac{\pi }{2}}$.

If $q(x)=f\circ g(x)$, then which of the following statements is/are correct ?
I. $q(x)$ is continuous at $x=0$.
II. $q(x)$ is differentiable at $x=0$.
Select the answer using the code given below :

Consider the following for the two (02) items that follow :
Let $f(x)=\{\begin{array}{ll}{x}^3,& x^2<1\\ x^2,& x^2\ge 1\end{array}$

Consider the following statements :
I. The function is continuous at $x=-1$.
II. The function is differentiable at $x=1$.
Which of the statements given above is/are correct?

Directions : For the following three (03) items :
Consider the function $f(x)=x|x|$.

Consider the following statements :
I. The function is increasing in the interval $(-\infty ,\infty )$.
II. The function is differentiable at $x=0$.
Which of the statements given above is/are correct?

Directions : For the following two (02) items :
Consider the function
$$f(x)=\{\begin{array}{ll}4(5^x),& x<0\\ 8k+x,& x\ge 0\end{array}$$

If the function is continuous, then what is the value of $k$?

Directions : For the following two (02) items :
Let the function $f(x)=|x-3|+|x-4|$ be defined on the interval $[0,5]$.

Consider the following statements :
I. The function is differentiable at $x=3$.
II. The function is differentiable at $x=4$.
Which of the statements given above is/are correct?

*Directions : Consider the following for the next two (02) items that follow :

Let $f(x)=|x|+1$ and $g(x)=[x]-1$, where $[.]$ is the greatest integer function.

Let $h(x)=\frac{f(x)}{g(x)}$*

Consider the following statements :

1. $f(x)$ is differentiable for all $x<0$
2. $g(x)$ is continuous at $x=0.0001$
3. The derivative of $g(x)$ at $x=2.5$ is 1

Which of the statements given above are correct ?