NIMCET 2023 — Mathematics PYQ

1. Factoring the Expressions

First, let's factor the quadratic expressions inside the absolute values:

• $x^2-4x=x(x-4)$

• $x^2-5x+6=(x-2)(x-3)$

The given equation is:

2. Identifying the Relationship between Terms

Let's define:

$a=x^2-5x+6$

$b=x-6$

If we add $a$ and $b$:

Now, substitute these back into the original equation:

Rewriting this:

3. Applying the Absolute Value Property

The property $|a+b|=|a|+|b|$ holds true if and only if $a$ and $b$ have the same sign (both non-negative or both non-positive). Mathematically:

Substitute the values of $a$ and $b$:

4. Solving the Inequality (Wavy Curve Method)

To solve $(x-2)(x-3)(x-6)\ge 0$, we find the critical points: $x=2,3,6$.

• For $x\ge 6$: The expression is $(+)(+)(+)\ge 0$ (True)

• For $3\le x\le 6$: The expression is $(+)(+)(-)\ge 0$ (False)

• For $2\le x\le 3$: The expression is $(+)(-)(-)\ge 0$ (True)

• For $x\le 2$: The expression is $(-)(-)(-)\ge 0$ (False)

Thus, the solution set is $x\in [2,3]\cup [6,\infty )$.

Final Answer

The value of $x$ lies in the interval $[2,3]\cup [6,\infty )$.

Correct Option: (B)