NIMCET 2023 — Mathematics PYQ

To find the number of solutions ($n_1$ and $n_2$), we analyze the intersection points of the functions graphically.

Step 1: Determine $n_1$ for $x=|{\sin }^{-1}x|$

We look for the intersection of the line $y=x$ and the curve $y=|{\sin }^{-1}x|$.

• The domain of ${\sin }^{-1}x$ is $[-1,1]$.

• For $x\in [0,1]$, the equation becomes $x={\sin }^{-1}x$, which implies $\sin x=x$. The only solution is $x=0$.

• For $x\in [-1,0)$, $|{\sin }^{-1}x|$ is positive, while $x$ is negative, so there are no intersections.

• Thus, there is only one intersection point at $(0,0)$.

• Therefore, $n_1=1$.

Step 2: Determine $n_2$ for $x=\sin x$

We look for the intersection of the line $y=x$ and the curve $y=\sin x$.

• At $x=0$, both functions equal $0$ (the origin).

• For x > 0, \sin x < x.

• For x < 0, \sin x > x.

• Since the functions only intersect at the origin, there is only one solution.

• Therefore, $n_2=1$.

Step 3: Calculate $n_2-n_1$

Using the values found:

$$n_2-n_1=1-1=0$$

The correct option is B.