NIMCET 2022 — Mathematics PYQ

1. Condition for the Numerator (${\cos }^{-1}x$):

The inverse cosine function, ${\cos }^{-1}x$, is defined only when its input $x$ lies in the interval:

So, from the numerator, we have $x\in [-1,1]$.

2. Condition for the Denominator ($[x]$):

The function $f(x)$ is defined only when the denominator is non-zero. Here, $[x]$ represents the Greatest Integer Function.

We know that $[x]=0$ for all $x$ in the interval $[0,1)$. Therefore, to satisfy the condition $[x]\ne 0$, we must exclude the interval $[0,1)$ from our domain.

3. Combining both conditions:

We need the intersection of the results from Step 1 and Step 2:

• From Numerator: $x\in [-1,1]$

• From Denominator: x < 0 or $x\ge 1$

Intersection:

Since the only value in the second part of the union is exactly $1$, we can write it as:

Conclusion:

The domain of the function is the set of all $x$ such that -1 \leq x < 0 or $x=1$.

Correct Option:

A) $[-1,0)\cup \{1\}$