The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

If one root of $5x^2 + 13x + k = 0$ is the reciprocal of the other,then $k = $

If $10$ men and $15$ women complete a piece of work in $8$ days,while $12$ men and $8$ women can complete the same piece of work in $10$ days. $A$ boy,who is $50 \%$ less efficient than a man,can do the same work in $50$ days.
$Quantity \, I$: Time taken by $2$ men,$4$ women,and $18$ boys to complete the work.
$Quantity \, II$: Time taken by $9$ men,$3$ women,and $6$ boys to complete the same work.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt[3]{2197}$
$II.$ $2y^2 - 54y + 364 = 0$

Solve the given two equations and select the correct option.
$I.$ $7x + 3y = 26$
$II.$ $2x + 17y = -41$

Find the value of $k$ so that the sum of the roots of the equation $3 x^{2}+(2 k+1) x-k-5=0$ is equal to the product of the roots: