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):
If \[ 3x=a+b+c, \] then \[ (x-a)^3+(x-b)^3+(x-c)^3 = 3(x-a)(x-b)(x-c) \]
Statement-2 (Reason):
If \[ a+b+c=0, \] then \[ a^3+b^3+c^3=3abc \]
Solution
Let
\[ p=x-a,\quad q=x-b,\quad r=x-c \]
Then,
\[ p+q+r = (x-a)+(x-b)+(x-c) \]
\[ =3x-(a+b+c) \]
Since \[ 3x=a+b+c, \]
\[ p+q+r=0 \]
Using identity:
\[ \text{If } p+q+r=0, \text{ then } p^3+q^3+r^3=3pqr \]
Therefore,
\[ (x-a)^3+(x-b)^3+(x-c)^3 = 3(x-a)(x-b)(x-c) \]
Hence, both Statement-1 and Statement-2 are true, and Statement-2 correctly explains Statement-1.
\[ \boxed{(a)} \]