Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6x^5+15x^4-20x^3+15x^2-6x+1=0$. Then $\frac{a^2+b^2}{a+b+1}$ is

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

The coefficient of $x^2$ in the expansion of the product $(2 - x^2)((1 + 2x + 3x^2)^6 + (1 - 4x^2)^6)$ is

The expression $(2 + \sqrt{2})^4$ has a value lying between

If $\frac{(1-px)^{-1}}{(1-qx)}=a_0+a_1x+a_2x^2+a_3x^3+\ldots$,then $a_n=$

Let $x = (8 \sqrt{3} + 13)^{13}$ and $y = (7 \sqrt{2} + 9)^9$. If $[t]$ denotes the greatest integer $\leq t$,then: