Let $\alpha $ and $\beta $ are roots of $5{x^2} – 3x – 1 = 0$ , then $\left[ {\left( {\alpha  + \beta } \right)x – \left( {\frac{{{\alpha ^2} + {\beta ^2}}}{2}} \right){x^2} + \left( {\frac{{{\alpha ^3} + {\beta ^3}}}{3}} \right){x^3} -……} \right]$ is

The sum of all non-integer roots of the equation $x^5-6 x^4+11 x^3-5 x^2-3 x+2=0$ is

Consider a three-digit number with the following properties:

$I$. If its digits in units place and tens place are interchanged, the number increases by $36$ ;

$II.$ If its digits in units place and hundreds place are interchanged, the number decreases by $198 .$

Now, suppose that the digits in tens place and hundreds place are interchanged. Then, the number

Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to

Exact set of values of $a$ for which ${x^3}(x + 1) = 2(x + a)(x + 2a)$ is having four real solutions is