Question 1:
If f(x)=∣x∣+∣x−1∣, then which of the following is correct?
Question 2:
If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?
Question 3:
If \(f(x)=\begin{cases}3x-2,&0 \lt x\le 1\\ 2x^{2}+ax,&1\lt x\lt 2\end{cases}\) is continuous for \(x\in(0,2)\), then a is equal to:
Question 4:
The function f defined by \(f(x)=\begin{cases}x,&if~x\le1\\ 5,&if~x>1\end{cases}\) is not continuous at:
Question 5:
If \(f(x)=\begin{cases}\frac{\sin^{2}ax}{x^{2}},&x\ne0\\ 1,&x=0\end{cases}\) is continuous at \(x=0\), then the value of a is:
Question 6:
If \(f(x)=\{[x],x\in R\}\) is the greatest integer function, then the correct statement is:
Question 7:
If \(f(x)=\begin{cases}\frac{\log(1+ax)+\log(1-bx)}{x},&for~x\ne0\\ k&,for~x=0\end{cases}\) is continuous at \(x=0\), then the value of k is:
Question 8:
If \( f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } x \geq 5 \end{cases} \) is continuous for all real numbers, then find the values of \(a\) and \(b\):
Question 9:
The number of points of discontinuity of \(f(x)=\begin{cases}|x|+3,& \text {if}~ x\le-3\\ -2x,& \text {if}~-3< x < 3\\ 6x+2,&\text {if}~x\ge 3\end{cases}\)
Question 10:
A function \(f(x)=|1-x+|x||\) is:
Question 11:
For what value of k, the function given below is continuous at \(x=0\) ? \(f(x)=\begin{cases}\frac{\sqrt{4+x}-2}{x},&x\ne0\\ k,&x=0\end{cases}\)
Question 12:
Assertion (A) :\(f(x) = \begin{cases} 3x-8, & x \le 5 \\ 2k, & x > 5 \end{cases}\) is continuous at \(x = 5\) for \(k = \dfrac{5}{2}\).
Reason (R) : For a function \(f\) to be continuous at \(x=a\), \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)\).
Question 13:
If \(f(x) = \begin{cases} 2x-3 & , -3 \leq x \leq -2 \\ x+1 & , -2 < x \leq 0 \end{cases}\)
Check the differentiability of \(f(x)\) at \(x=-2\).
Question 14:
Check whether the function \(f(x)=x^{2}|x|\) is differentiable at \(x=0\) or not.
Question 15:
Verify whether the function f defined by \(f(x)=\begin{cases}x~\sin(\frac{1}{x}),&x\ne0\\ 0,&x=0\end{cases}\) is continuous at \(x=0\) or not.
Question 16:
Check for differentiability of the function f defined by \(f(x)=|x-5|\), at the point \(x=5\).
Question 17:
Check the differentiability of \(f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}\) at \(x=1.\)
Question 18:
Find k so that \(f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}\) is continuous at \(x=-1.\)
Question 19:
Check the differentiability of function \(f(x)=x|x|\) at \(x=0\).
Question 20:
Find the value of a and b so that function f defined as: \(f(x)=\begin{cases}\frac{x-2}{|x-2|}+a,&if~x<2\\ a+b,&if~x=2\\ \frac{x-2}{|x-2|}+b,&if~x>2\end{cases}\) is a continuous function.