Solve the given two equations and select the correct answer from the given options.
$I.$ $(x+y)^{2} = 3136$
$II.$ $y+2513 = 2569$

If $x+\frac{1}{x}=2,$ then the value of $x^{12}-\frac{1}{x^{12}}$ is:

$Quantity \, 1$: Time taken by Manoj and Shubham to complete the work together. When Manoj works alone,he takes $4.8 \, \text{hrs}$ more than the time taken by Manoj and Shubham to complete the work together. When Shubham works alone,he takes $10.8 \, \text{hrs}$ more than the time taken by both of them to complete the work together.
$Quantity \, 2$: $7.4 \, \text{hrs}$.

Let $\alpha, \beta, \gamma, \delta$ be the roots of the equation $x^4 + x^2 + 1 = 0$. Then,the equation whose roots are $\alpha^2, \beta^2, \gamma^2, \delta^2$ is:

If one root of the equation $ax^2 + bx + c = 0$ is $n$ times the other root,then:

If the equation $x^2 + \lambda x + \mu = 0$ has equal roots and one root of the equation $x^2 + \lambda x - 12 = 0$ is $2$,then $(\lambda, \mu) = $