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UPSC CDS 2 2021 Math Paper

Question : 40 of 100

Marks: +1, -0

2x - 3y - 7 = 0

8x^{3} - 36x^{2}y + 54xy^{2} - 27y^{3} - 340

Solution:

Concept:

The expression

8x^{3} - 36x^{2}y + 54xy^{2} - 27y^{3}

is the expansion of

(2x - 3y)^{3}

The given equation

2x - 3y - 7 = 0

gives

2x - 3y = 7

, which can be cubed to relate the two expressions.

Explanation:

Start with the given equation:

2x - 3y - 7 = 0

Rearrange to

2x - 3y = 7

Cube both sides:

(2x - 3y)^{3} = 7^{3}

Expand the left side using the identity

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

(2x)^{3} - 3(2x)^{2}(3y) + 3(2x)(3y)^{2} - (3y)^{3} = 343

This simplifies to

8x^{3} - 36x^{2}y + 54xy^{2} - 27y^{3} = 343

Now subtract 340 from both sides:

8x^{3} - 36x^{2}y + 54xy^{2} - 27y^{3} - 340 = 343 - 340

Hence the required value is

3

Answer:

3

(Option D)

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