Question:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion):
\[ \frac{ (x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3 }{ (x-y)^3+(y-z)^3+(z-x)^3 } = (x+y)(y+z)(z+x) \]
Statement-2 (Reason):
\[ \text{If } a+b+c=0, \text{ then } a^3+b^3+c^3=3abc \]
Solution:
Let \[ a=x^2-y^2, \quad b=y^2-z^2, \quad c=z^2-x^2 \]
Then \[ a+b+c=0 \]
Using identity:
\[ a^3+b^3+c^3=3abc \]
So,
\[ (x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3 \]
\[ = 3(x^2-y^2)(y^2-z^2)(z^2-x^2) \]
\[ = 3(x-y)(x+y)(y-z)(y+z)(z-x)(z+x) \]
Also, \[ (x-y)+(y-z)+(z-x)=0 \]
Again using \[ a^3+b^3+c^3=3abc \]
\[ (x-y)^3+(y-z)^3+(z-x)^3 \]
\[ = 3(x-y)(y-z)(z-x) \]
Therefore,
\[ \frac{ 3(x-y)(x+y)(y-z)(y+z)(z-x)(z+x) }{ 3(x-y)(y-z)(z-x) } \]
\[ = (x+y)(y+z)(z+x) \]
Thus, Statement-1 is true.
Statement-2 is also true and correctly explains Statement-1.
Hence, the correct answer is
\[ \boxed{(a)} \]