The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

For what value of $\lambda$ the sum of the squares of the roots of ${x^2} + (2 + \lambda )\,x - \frac{1}{2}(1 + \lambda ) = 0$ is minimum

The maximum possible number of real roots of equation ${x^5} - 6{x^2} - 4x + 5 = 0$ is

For a real number $x$, let $[x]$ denote the largest integer less than or equal to $x$, and let $\{x\}=x-[x]$. The number of solutions $x$ to the equation $[x]\{x\}=5$ with $0 \leq x \leq 2015$ is

If $\alpha , \beta , \gamma$ are roots of equation $x^3 + qx -r = 0$ then the equation, whose roots are

$\left( {\beta \gamma  + \frac{1}{\alpha }} \right),\,\left( {\gamma \alpha  + \frac{1}{\beta }} \right),\,\left( {\alpha \beta  + \frac{1}{\gamma }} \right)$

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