If $\alpha ,\beta,\gamma$ are the roots of equation $x^3 + 2x -5 = 0$ and if equation $x^3 + bx^2 + cx + d = 0$ has roots $2 \alpha + 1, 2 \beta + 1, 2 \gamma + 1$ , then value of $|b + c + d|$ is (where $b,c,d$ are coprime)-

Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :

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.

If $x$ is real and $k = \frac{{{x^2} - x + 1}}{{{x^2} + x + 1}},$ then

One root of the following given equation $2{x^5} - 14{x^4} + 31{x^3} - 64{x^2} + 19x + 130 = 0$ is

Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in R$ has integral roots such that $P(6) = 3$, then $' a '$ cannot be equal to