The equation $e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R$ has:

Consider the quadratic equation $n x^2+7 \sqrt{n x+n}=0$ where $n$ is a positive integer. Which of the following statements are necessarily correct?

$I$. For any $n$, the roots are distinct.

$II$. There are infinitely many values of $n$ for which both roots are real.

$III$. The product of the roots is necessarily an integer.

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is

The number of real roots of the equation ${e^{\sin x}} - {e^{ - \sin x}} - 4$ $ = 0$ are

The number of solutions of the equation $x ^2+ y ^2= a ^2+ b ^2+ c ^2$. where $x , y , a , b , c$ are all prime numbers, is

Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?