$\alpha$, $\beta$ ,$\gamma$ are roots of equatiuon $x^3 -x -1 = 0$ then equation whose roots are $\frac{1}{{\beta + \gamma }},\frac{1}{{\gamma + \alpha }},\frac{1}{{\alpha + \beta }}$ is
$x^3 -x^2 + 1 = 0$
$x^3 + x^2 -1 = 0$
$x^3 + x -1 = 0$
$x^3 -x + 1 = 0$
If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0, \alpha\,>\,\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $\mathrm{n}$, then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$
The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is
The two roots of an equation ${x^3} - 9{x^2} + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots will be
lf $2 + 3i$ is one of the roots of the equation $2x^3 -9x^2 + kx- 13 = 0,$ $k \in R,$ then the real root of this equation
The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has: