The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is
$5$
$7$
$1$
$3$
If $\alpha , \beta , \gamma$ are roots of equation $x^3 + qx -r = 0$ then the equation, whose roots are
$\left( {\beta \gamma + \frac{1}{\alpha }} \right),\,\left( {\gamma \alpha + \frac{1}{\beta }} \right),\,\left( {\alpha \beta + \frac{1}{\gamma }} \right)$
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$......$
Let $x, y, z$ be positive reals. Which of the following implies $x=y=z$ ?
$I.$ $x^3+y^3+z^3=3 x y z$
$II.$ $x^3+y^2 z+y z^2=3 x y z$
$III.$ $x^3+y^2 z+z^2 x=3 x y z$
$IV.$ $(x+y+z)^3=27 x y z$
If $x$ is real and satisfies $x + 2 > \sqrt {x + 4} ,$ then
The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 e^{-\sin x}=2$ is