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

    $1$

Similar Questions

Suppose $m, n$ are positive integers such that $6^m+2^{m+n} \cdot 3^w+2^n=332$. The value of the expression $m^2+m n+n^2$ is

  • [KVPY 2010]

Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to

  • [KVPY 2015]

Number of natural solutions of the equation $x_1 + x_2 = 100$ , such that $x_1$ and $x_2$ are not multiple of $5$

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$, with $\alpha>\beta$. For all positive integers $n$, define

$a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$

$b_1=1 \text { and } b_n=a_{n-1}+a_{n+1}, n \geq 2.$

Then which of the following options is/are correct?

$(1)$ $a_1+a_2+a_3+\ldots . .+a_n=a_{n+2}-1$ for all $n \geq 1$

$(2)$ $\sum_{n=1}^{\infty} \frac{ a _{ n }}{10^{ n }}=\frac{10}{89}$

$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n}=\frac{8}{89}$

$(4)$ $b=\alpha^n+\beta^n$ for all $n>1$

  • [IIT 2019]

If $\alpha ,\beta ,\gamma$  are the roots of $x^3 - x - 2 = 0$, then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-