Find the value of $(a+b+c)^{3}-3(b+c)(c+a)(a+b)$ if $a=5, b=3$ and $c=2$.

Solve the given two equations and select the correct answer from the given options.
$I. x = \sqrt{1764}$
$II. y^2 = 1764$

If $x^{2}-3x+1=0$,then the value of $\frac{x^{6}+x^{4}+x^{2}+1}{x^{3}}$ is:

Suppose that $x$ and $y$ are positive numbers with $xy = \frac{1}{9}$,$x(y + 1) = \frac{7}{9}$,and $y(x + 1) = \frac{5}{18}$. The value of $(x + 1)(y + 1)$ equals:

Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax - 4 + a^2 = 0$ is smaller than $1$ and the larger root is greater than $6$ is

If the equations $x^{2}+2x-3=0$ and $x^{2}+3x-k=0$ have a common root,then the non-zero value of $k$ is: