Let $r$ be a real number and $n \in N$ be such that the polynomial $2 x^2+2 x+1$ divides the polynomial $(x+1)^n-r$. Then, $(n, r)$ can be
$\left(4000,4^{1000}\right)$
$\left(4000, \frac{1}{4^{1000}}\right)$
$\left(4^{1000}, \frac{1}{4^{1000}}\right)$
$\left(4000, \frac{1}{4000}\right)$
Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$, then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$ is equal to ............
The number of roots of the equation $|x{|^2} - 7|x| + 12 = 0$ is
The sum of the roots of the equation, ${x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,$ is
Number of positive integral values of $'K'$ for which the equation $k = \left| {x + \left| {2x - 1} \right|} \right| - \left| {x - \left| {2x - 1} \right|} \right|$ has exactly three real solutions, is
If ${x^2} + 2ax + 10 - 3a > 0$ for all $x \in R$, then