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UPSC CDS 2 2022 Math Paper

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Question : 3 of 100

Marks: +1, -0

If

(a^{3} + b^{3})

is proportional to

(a^{2} - b^{2})

(a^{2} - ab + b^{2})

is proportional to

(a - b)
(a + b)
(a + ab + b)
(a3 – b3)

Solution:

Concept:

If two expressions are proportional, then after factoring out common factors, the remaining parts are also proportional.

Explanation:

Given:

(a^3 + b^3) \propto (a^2 - b^2)

.

Use the identity:

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

.

Use the identity:

a^2 - b^2 = (a+b)(a-b)

.

Thus,

(a+b)(a^2 - ab + b^2) \propto (a+b)(a-b)

.

Cancel the common factor

(a+b)

, assuming

a+b \neq 0

.

Hence,

(a^2 - ab + b^2) \propto (a-b)

.

Answer:

Option A:

(a-b)

.

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