Question : If $x+y=2z$, then the value of $\frac{x}{x-z}+\frac{z}{y-z}$ is:
Option 1: $1$
Option 2: $3$
Option 3: $\frac{1}{2}$
Option 4: $2$
Correct Answer: $1$
Solution : Given: $x+y = 2z$ Now, $\frac{x}{x-z}+\frac{z}{y-z}$ $= \frac{x(y-z)+z(x-z)}{(x-z)(y-z)}$ $= \frac{xy−xz+zx−z^2}{xy−xz−zy+z^2}$ $= \frac{xy−z^2}{xy−z(x+y)+z^2}$ $= \frac{xy−z^2}{xy−z(2z)+z^2}$ $= \frac{xy−z^2}{xy−z^2}$ $= 1$ Hence, the correct answer is $1$.
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Question : If $x^2 = y+z$, $y^2=z+x$, $z^2=x+y$, then the value of $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}$ is:
Question : If $\frac{1}{x+2}=\frac{3}{y+3}=\frac{1331}{z+1331}=\frac{1}{3}$, then what is the value of $\frac{x}{x+1}+\frac{y}{y+6}+\frac{z}{z+2662}$?
Question : The value of $\frac{(x-y)^3+(y-z)^3+(z-x)^3}{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}$, where $x \neq y \neq z$, is:
Question : What is $\frac{\left (x^{2}-y^{2} \right)^{3}+\left (y^{2}-z^{2} \right )^{3}+\left (z^{2}-x^{2} \right )^{3}}{\left (x-y \right)^{3}+\left (y-z \right )^{3}+\left (z-x \right)^{3}}?$
Question : If $x+y+z=17, x y z=171$ and $x y+y z+z x=111$, then the value of $\sqrt[3]{\left(x^3+y^3+z^3+x y z\right)}$ is:
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