QJEEM23Maf940: JEE Main 2023 Mathematics - Class 12 JEEM23Maf940 Solution: Let and be the greatest integer . Then the number of points,…

Given that $f(x) = [a + 13\sin(x)]$ , where $[t]$ is the greatest integer function and $x \in (0, \pi)$ .

The function $[t]$ is not differentiable wherever $t$ is an integer because at these points, the function has a jump discontinuity.

For $f(x)$ to have a point of non-differentiability, the value inside the greatest integer function, i.e., $a + 13\sin(x)$ , should be an integer.

So, we need to find the values of $x$ in the interval $(0, \pi)$ for which $a + 13\sin(x)$ is an integer.

Now, $\sin(x)$ varies from 0 to 1 in the interval $(0, \pi)$ , so the maximum value of $13\sin(x)$ in this interval is 13.

For each integer value of $13\sin(x)$ between 0 and 13, we'll have a corresponding value of $x$ . There will be two such values of $x$ for each value of $13\sin(x)$ (because of the periodic and symmetric nature of sine function over the interval $(0, \pi)$ ), except for the maximum value, 13, which will have only one corresponding value of $x$ (namely $x = \frac{\pi}{2}$ ).

Thus, the total number of integer values of $13\sin(x)$ between 0 and 13 is 13. Excluding the maximum value, we have 12 integer values, each giving rise to two values of $x$ . Including the maximum value, which gives one value of $x$ , we have :

$12 \times 2 + 1 = 25$

Thus, $f(x)$ is not differentiable at 25 points in the interval $(0, \pi)$ .