The product of the roots of the equation $9 x^{2}-18|x|+5=0,$ is

If $a,b,c$ are real and ${x^3} - 3{b^2}x + 2{c^3}$ is divisible by $x - a$ and$x - b$, then

Leela and Madan pooled their music $CD's$ and sold them. They got as many rupees for each $CD$ as the total number of $CD's$ they sold. They share the money as follows: Leela first takes $10$ rupees, then Madan takes $10$ rupees and they continue taking $10$ rupees alternately till Madan is left out with less than $10$ rupees to take. Find the amount that is left out for Madan at the end, with justification.

Consider the equation ${x^2} + \alpha x + \beta  = 0$ having roots $\alpha ,\beta $ such that $\alpha  \ne \beta $ .Also consider the inequality $\left| {\left| {y - \beta } \right| - \alpha } \right| < \alpha $ ,then

If $|{x^2} - x - 6| = x + 2$, then the values of $x$ are

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1 .$ Let $p _{ n }=(\alpha)^{ n }+(\beta)^{ n },p _{ n -1}=11$ and $p _{ n +1}=29$ for some integer $n \geq 1 .$ Then, the value of $p _{ n }^{2}$ is .... .