If $3 + 4i$ is a root of the equation ${x^2} + px + q = 0$ where $p$ and $q$ are real numbers,then:

The number of real solutions of the equation $e^{4x} + 4e^{3x} - 58e^{2x} + 4e^{x} + 1 = 0$ is:

The number of solutions of the equations $x+y+z=1$,$x^2+y^2+z^2=1$,and $x^3+y^3+z^3=1$ is:

Which among the following equations has roots that are negatives of the roots of the equation $x^3-x^2+x-4=0$?

If $m, n$ are the roots of the equation ${x^2} - x - 1 = 0$,then the value of $\frac{{\left( {1 + m{{\log }_e}3 + \frac{{{{(m{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)\left( {1 + n{{\log }_e}3 + \frac{{{{(n{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}{{\left( {1 + mn{{\log }_e}3 + \frac{{{{(mn{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}$ is:

Let $a, b$ and $c$ be the sides of a scalene triangle. If $\lambda$ is a real number such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,then the interval in which $\lambda$ lies is