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):
\[ a^3+\frac{3}{8}ax+\frac{1}{64}x^3-\frac{1}{8} \]
\[ = \left(a+\frac{x}{4}-\frac{1}{2}\right) \left( a^2+\frac{x^2}{16}+\frac{1}{4} -\frac{ax}{4} +\frac{x}{8} +\frac{a}{2} \right) \]
Statement-2 (Reason):
\[ a^3+b^3+c^3+3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
Solution:
Using identity:
\[ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
Let \[ b=\frac{x}{4}, \quad c=-\frac{1}{2} \]
Then,
\[ a^3+\left(\frac{x}{4}\right)^3+\left(-\frac{1}{2}\right)^3 -3\left(a\right)\left(\frac{x}{4}\right)\left(-\frac{1}{2}\right) \]
\[ = a^3+\frac{x^3}{64}-\frac{1}{8}+\frac{3ax}{8} \]
which matches Statement-1.
Hence Statement-1 is true.
Statement-2 is false because the correct identity is
\[ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
not with \(+3abc\).
Hence, the correct answer is
\[ \boxed{(c)} \]