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The question consists of two statements- assertion (A) and reason (R).

Assertion (A): The value of $(25) 3 + (− 16) 3 + (− 9) 3$ is 10800.

Reason (R): If a + b + c = 0, then $a 3 + b 3 + c 3 = 3 a b c$ .

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Both assertion (A) and reason (R) are true and reason (R) is a correct explanation of assertion (A).

Both assertion (A) and reason (R) are true but reason (R) is not a correct explanation of the assertion(A).

Assertion (A) is true and reason (R) is false.

Assertion (A) is false and reason (R) is true.

answer is A.

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Detailed Solution

Given data,
If a + b + c = 0, then

.
Considering,

(25) 3 + − 16 3 + − 9 3

As we know, f (a) = 0 if (x-a) is a factor of the quadratic equation f(x).
Let a =25, b=(-16) and c=(-9)
Using the formula,

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ca)

Now, using the assertion given,

25^{3} + {(- 16)}^{3} + {(- 9)}^{3} = 10800

Substituting the value of a, b, and c in the formula, we get,
(25)3+(-16)3+(-9)3-3(25)(-16)(-9)=(25-16-9)[252+162+92-25×(-16)-(-16)×(-9)-(-9)×25× (-16)-(-16)×(-9)-(-9)×25
Substituting the value again,

a + b + c = 25 - 16 - 9 = 0

{(25)}^{3} + {(- 16)}^{3} + {(- 9)}^{3} - 3 (25) (- 16) (- 9) = 0

{(25)}^{3} + {(- 16)}^{3} + {(- 9)}^{3} = 3 (25) (- 16) (- 9)

15625 - 4096 - 729 = 3 \times 3600

10800 = 10800

Therefore, both the assertion (A) and reason (R) are true, and reason (R) is the correct explanation of assertion (A).
Hence, option 1 is correct.

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