The number of real roots of the equation, $\mathrm{e}^{4 \mathrm{x}}+\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}+1=0$ is

- [JEE MAIN 2020]

- A
$4$

- B
$2$

- C
$3$

- D
$1$

The number of solution$(s)$ of the equation $2^x = x^2$ is

Let $a, b, c$ be non-zero real roots of the equation $x^3+a x^2+b x+c=0$. Then,

- [KVPY 2020]

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

- [JEE MAIN 2015]

The population of cattle in a farm increases so that the difference between the population in year $n+2$ and that in year $n$ is proportional to the population in year $n+1$. If the populations in years $2010, 2011$ and $2013$ were $39,60$ and $123$,respectively, then the population in $2012$ was

- [KVPY 2014]

The number of real roots of the polynomial equation $x^4-x^2+2 x-1=0$ is

- [KVPY 2018]