If $x$ is real and $k = \frac{{{x^2} - x + 1}}{{{x^2} + x + 1}},$ then
$\frac{1}{3} \le k \le 3$
$k \ge 5$
$k \le 0$
None of these
The number of roots of the equation $\log ( - 2x)$ $ = 2\log (x + 1)$ are
Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :
The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is
The number of integers $n$ for which $3 x^3-25 x+n=0$ has three real roots is
The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 e^{-\sin x}=2$ is