The value of $x$ in the given equation ${4^x} – {3^{x\,\; – \;\frac{1}{2}}} = {3^{x + \frac{1}{2}}} – {2^{2x – 1}}$is

Let $p(x)=x^2-5 x+a$ and $q(x)=x^2-3 x+b$, where $a$ and $b$ are positive integers. Suppose HCF $(p(x), q(x))=x-1$ and $k(x)=1 cm (p(x), q(x))$ If the coefficient of the highest degree term of $k(x)$ is 1 , then sum of the roots of $(x-1)+k(x)$ is

If the quadratic equation ${x^2} + \left( {2 – \tan \theta } \right)x – \left( {1 + \tan \theta } \right) = 0$ has $2$ integral roots, then sum of all possible values of $\theta $ in interval $(0, 2\pi )$ is $k\pi $, then $k$ equals 

Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. Which of the following statements are true?

$I$. $4$ divides $x$ or $4$ divides $y$.

$II$. $3$ divides $x+y$ or $3$ divides $x-y$.

$III$. $5$ divides $z\left(x^2-y^2\right)$.

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