Question : The value of $\frac{(p - q)^{3} + (q - r)^{3} + (r - p)^{3}}{12 (p - q) (q - r) (r - p)}$ , where $p \neq q \neq r$ , is equal to:
Option 1: $\frac{1}{9}$
Option 2: $\frac{1}{3}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{1}{2}$

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Question : The value of $\frac{(x - y)^{3} + (y - z)^{3} + (z - x)^{3}}{6 (x - y) (y - z) (z - x)}$ , where $x \neq y \neq z$ , is equal to:

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Answer 1

Correct Answer: $\frac{1}{4}$

Solution : $\frac{(p - q)^{3} + (q - r)^{3} + (r - p)^{3}}{12 (p - q) (q - r) (r - p)}$
We know that $a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
Let $a = p - q$ , $b = q - r$ and $c = r - p$
We observe that $a + b + c = p - q + q - r + - p = 0$
So, $a^{3} + b^{3} + c^{3} - 3 a b c = 0$
or, $a^{3} + b^{3} + c^{3} = 3 a b c$
or, $(p - q)^{3} + (q - r)^{3} + (r - p)^{3} = 3 (p - q) (q - r) (r - p)$
So, $\frac{(p - q)^{3} + (q - r)^{3} + (r - p)^{3}}{12 (p - q) (q - r) (r - p)} = \frac{3 (p - q) (q - r) (r - p)}{12 (p - q) (q - r) (r - p)}$
= $\frac{1}{4}$
Hence, the correct answer is $\frac{1}{4}$ .

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Question : The value of (p−q)3+(q−r)3+(r−p)312(p−q)(q−r)(r−p), where p≠q≠r, is equal to:Option 1: 19Option 2: 13Option 3: 14Option 4: 12

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Question : The value of $\frac{(p - q)^{3} + (q - r)^{3} + (r - p)^{3}}{12 (p - q) (q - r) (r - p)}$ , where $p \neq q \neq r$ , is equal to:
Option 1: $\frac{1}{9}$
Option 2: $\frac{1}{3}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{1}{2}$