The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is
$5$
$7$
$1$
$3$
If the inequality $kx^2 -2x + k \geq 0$ holds good for atleast one real $'x'$ , then the complete set of values of $'k'$ is
The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$, is :
Number of integral values of '$m$' for which $\{x\}^2 + 5m\{x\} - 3m + 1 < 0 $ $\forall x \in R$, is (where $\{.\}$ denotes fractional part function)
If $x$ be real, then the minimum value of ${x^2} - 8x + 17$ is
Let $\alpha, \beta, \gamma$ be the three roots of the equation $x ^3+ bx + c =0$. If $\beta \gamma=1=-\alpha$, then $b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3$ is equal to $......$.