Assertion and Reason Question on Algebraic Identities
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):
\[ (a-b)^3+(b-c)^3+(c-a)^3 = 3(a-b)(b-c)(c-a) \]
Statement-2 (Reason):
\[ \text{If } a+b+c=0, \text{ then } a^3+b^3+c^3=3abc \]
Solution
Let
\[ x=a-b,\quad y=b-c,\quad z=c-a \]
Then,
\[ x+y+z=(a-b)+(b-c)+(c-a)=0 \]
Using the identity:
\[ \text{If } x+y+z=0, \text{ then } x^3+y^3+z^3=3xyz \]
Therefore,
\[ (a-b)^3+(b-c)^3+(c-a)^3 = 3(a-b)(b-c)(c-a) \]
Hence, both Statement-1 and Statement-2 are true, and Statement-2 correctly explains Statement-1.
\[ \boxed{(a)} \]