Let $\alpha$ and $\beta$ be the roots of the equation $5 x^{2}+6 x-2=0 .$ If $S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots$ then :
$5 \mathrm{S}_{6}+6 \mathrm{S}_{5}=2 \mathrm{S}_{4}$
$5 \mathrm{S}_{6}+6 \mathrm{S}_{5}+2 \mathrm{S}_{4}=0$
$6 \mathrm{S}_{6}+5 \mathrm{S}_{5}+2 \mathrm{S}_{4}=0$
$6 \mathrm{S}_{6}+5 \mathrm{S}_{5}=2 \mathrm{S}_{4}$
The number of real solutions of the equation $|{x^2} + 4x + 3| + 2x + 5 = 0 $are
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
If the roots of ${x^2} + x + a = 0$exceed $a$, then
Let $\alpha, \beta(\alpha>\beta)$ be the roots of the quadratic equation $x ^{2}- x -4=0$. If $P _{ a }=\alpha^{ n }-\beta^{ n }, n \in N$, then $\frac{ P _{15} P _{16}- P _{14} P _{16}- P _{15}^{2}+ P _{14} P _{15}}{ P _{13} P _{14}}$ is equal to$......$
The product of all real roots of the equation ${x^2} - |x| - \,6 = 0$ is