NIMCET 2020 — Mathematics PYQ

Detailed Step-by-Step Explanation:

We need to find the value of the given expression:

$$\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}$$

Step 1: Take the Least Common Multiple (LCM)

The LCM of the denominators $bc$, $ca$, and $ab$ is $abc$. Combining the fractions over a common denominator gives:

$$\frac{a^3+b^3+c^3}{abc}$$

Step 2: Use the Algebraic Identity

We know the standard algebraic identity:

$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

Step 3: Substitute the Given Condition

It is given that $a+b+c=0$. Substituting this value into the identity:

$$a^3+b^3+c^3-3abc=(0)(a^2+b^2+c^2-ab-bc-ca)$$

$$a^3+b^3+c^3-3abc=0$$

$$a^3+b^3+c^3=3abc$$

Step 4: Calculate the Final Value

Now substitute $a^3+b^3+c^3=3abc$ back into our simplified expression:

$$\frac{3abc}{abc}=3$$

Conclusion:

Therefore, the value of the expression is 3, matching option C.