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

If $p$ and $q$ are positive real numbers such that $p^2 + q^2 = 1$,what is the maximum value of $(p + q)$?

If $\sqrt{2} \sin^2 x + (3\sqrt{2} + 1) \sin x + 3 > 0$ and $x^2 - 7x + 10 < 0$,then $x$ lies in the interval

Consider the quadratic equation $nx^2 + 7\sqrt{n}x + n = 0$ 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.

If the cubic equation $x^3-ax^2+ax-1=0$ is identical with the cubic equation whose roots are the squares of the roots of the given cubic equation,then the non-zero real value of $a$ is

If $ax + by = 1$,where $a, b, x$ and $y$ are integers,then which one of the following is not true?