If $x = a^{1/2} + a^{-1/2}$ and $y = a^{1/2} - a^{-1/2}$,then find the value of $(x^4 - x^2 y^2 - 1) + (y^4 - x^2 y^2 + 1)$.

If $5x + 9y = 5$ and $125x^3 + 729y^3 = 120$,then the value of the product of $x$ and $y$ is:

If $x$ is a rational number and $\frac{(x+1)^{3}-(x-1)^{3}}{(x+1)^{2}-(x-1)^{2}}=2$,then the sum of the numerator and denominator of $x$ is

If $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$ are the roots of the equations $ax^2 + bx + c = 0$ and $px^2 + qx + r = 0$ respectively,and the system of equations $\alpha_1 y + \alpha_2 z = 0$ and $\beta_1 y + \beta_2 z = 0$ has a non-zero solution,then:

If one of the roots of the equation ${x^2} + ax + 3 = 0$ is $3$ and one of the roots of the equation ${x^2} + ax + b = 0$ is three times the other root,then the value of $b$ is equal to

For what values of $k$ will the equation $x^2 - 2(1 + 3k)x + 7(3 + 2k) = 0$ have equal roots?