The least positive integer $x$ satisfying $2^{2010} \equiv 3x \pmod{5}$ is

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2 + px + q = 0$ $(p \neq 0)$,then:

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).

For what condition will the expression $a^2x^2 + bx + 1$ be positive for all $x \in R$?

Consider the following statements: For any integer $n$,
$I.$ $n^2+3$ is never divisible by $17$.
$II.$ $n^2+4$ is never divisible by $17$.
Then,

Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to