Question : What is the simplified value of $\frac{(x + y + z) (x y + y z + z x) - x y z}{(x + y) (y + z) (z + x)}$ ?
Option 1: $y$
Option 2: $z$
Option 3: $1$
Option 4: $x$

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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: $1$

Solution : Given: The expression is $\frac{(x + y + z) (x y + y z + z x) - x y z}{(x + y) (y + z) (z + x)}$ .
$\frac{(x + y + z) (x y + y z + z x) - x y z}{(x + y) (y + z) (z + x)}$
Substitute the value of $x = y = z = 1$ , in the given expression, we get,
= $\frac{(1 + 1 + 1) (1 \times 1 + 1 \times 1 + 1 \times 1) - 1 \times 1 \times 1}{(1 + 1) (1 + 1) (1 + 1)}$
= $\frac{3 \times 3 - 1}{2 \times 2 \times 2} = \frac{9 - 1}{8}$
= $\frac{8}{8}$ = $1$
Hence, the correct answer is 1.

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Question : What is the simplified value of (x+y+z)(xy+yz+zx)–xyz(x+y)(y+z)(z+x)?Option 1: yOption 2: zOption 3: 1Option 4: x

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Question : What is the simplified value of $\frac{(x + y + z) (x y + y z + z x) - x y z}{(x + y) (y + z) (z + x)}$ ?
Option 1: $y$
Option 2: $z$
Option 3: $1$
Option 4: $x$