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SSC CPO 2022 PYQ — Question 127

SSC CPO - 9 Nov 2022 - Shift 2 • Quantitative Aptitude • Q127

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The value of (x−y)3+(y−z)3+(z−x)3(x2−y2)3+(y2−z2)3+(z2−x2)3\rm \frac{(x-y)^3+(y-z)^3+(z-x)^3}{(x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3} where x ≠ y ≠ z, is:

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Correct answer: C

Explanation

The correct answer is Option 3 i.e. 1(x+y)(y+z)(z+x)\rm \frac{1}{(x+y)(y+z)(z+x)}.

From the given expression, consider the denominator

⇒ (x2 - y2)3 + (y2 - z2)3 + (z2 - x2)3

(x2 - y2) = A, (y2 - z2) = B, (z2 - x2) = C

Let, A + B + C = x2 - y2 + y2 - z2 + z2 - x2 = 0

If A + B + C = 0, then (A3 + B3 + C3) = 3ABC

Therefore,

Denominator = 3(x2 - y2)(y2 - z2)(z2 - x2) and 

Numerator (x - y)3 + (y - z)3 + (z - x)3 = 3(x - y)(y - z)(z - x)

Therefore,

Numerator/Denominatorr = (3(x - y)(y - z)(z - x))/(3(x2 - y2)(y2 - z2)(z2 - x2))

= (x - y)(y - z)(z - x)/(x2 - y2)(y2 - z2)(z2 - x2)........(1)

We know that,

⇒ (a2 - b2) = (a + b)(a - b)

Now, eq 1 can be written as

⇒ ((x - y)(y - z)(z - x))/((x2 - y2)(y2 - z2)(z2 - x2))

⇒ ((x - y)(y - z)(z - x))/((x + y)(x - y)(y - z)(y + z)(z - x)(z + x))

⇒ 1(x+y)(y+z)(z+x)\rm \frac{1}{(x+y)(y+z)(z+x)}