The number of values of $k$ for which the equation $x^2 - 3x + k = 0$ has two real and distinct roots lying in the interval $(0, 1)$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $7x + 3y = 77$
$II.$ $2x + 5y = (2601)^{1/2}$

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.

The values of $a$ for which one root of the equation $x^2 - (a + 1)x + a^2 + a - 8 = 0$ exceeds $2$ and the other is lesser than $2$,are given by

The complete solution set of the inequality $(\sec^{-1}x - 4)(\sec^{-1}x - 1)(\sec^{-1}x - 2) \ge 0$ is

If one root of $ax^2 + bx + c = 0$ is the square of the other,then the value of $b^3 + ac^2 + a^2c$ is