List of top Mathematics Questions on Differentiability asked in CBSE CLASS XII

If \( f(x) = |x| + |x - 1| \), then which of the following is correct?
Check the differentiability of the function $f(x) = |x|$ at $x = 0$.
Check the differentiability of \( f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1 \\ 3 - x, & 1 \leq x \leq 2 \end{cases} \) at \( x = 1 \).

A bacteria sample of a certain number of bacteria is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth is:

\[ \frac{dP}{dt} = kP, \] where \( P \) is the bacterial population.

Based on this, answer the following:

Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx \]
Check the differentiability of \( f(x) \) at \( x = 1 \), where: \[ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1, \\ 3 - x, & 1 \leq x \leq 2. \end{cases} \]