NIMCET 2008 — Mathematics PYQ

First, let's factorize the quadratic expression inside the absolute value:

So the equation becomes:

Step 2: Establish the constraint

Since the left side is an absolute value (which is always $\ge 0$), the right side must also be non-negative:

Step 3: Solve by cases

An equation $|A|=B$ is solved by considering $A=B$ and $A=-B$.

Case 1: $(x^2-x-6)=x+2$

Roots: $x=4$ and $x=-2$.

Check against constraint ($x\ge -2$): Both $4$ and $-2$ are valid.

Case 2: $(x^2-x-6)=-(x+2)$

Roots: $x=2$ and $x=-2$.

Check against constraint ($x\ge -2$): Both $2$ and $-2$ are valid.

Step 4: List the unique roots

Collecting all the values of $x$ found:

The distinct (unique) roots are $x=-2,2,$ and $4$.

Conclusion:

There are exactly 3 distinct roots satisfying the equation.

Correct Option: (b)