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 \[ a+b+c=5 \] and \[ ab+bc+ca=10, \] then \[ a^3+b^3+c^3-3abc=-25 \]
Statement-2 (Reason):
\[ a^3+b^3+c^3-3abc \]
\[ =(a+b+c)\{(a+b+c)^2-3(ab+bc+ca)\} \]
Solution
Using the identity:
\[ a^3+b^3+c^3-3abc \]
\[ =(a+b+c)\{(a+b+c)^2-3(ab+bc+ca)\} \]
Substituting the values:
\[ =5\{5^2-3(10)\} \]
\[ =5(25-30) \]
\[ =5(-5) \]
\[ =-25 \]
Hence, both Statement-1 and Statement-2 are true, and Statement-2 correctly explains Statement-1.
\[ \boxed{(a)} \]