If $\frac{x^2+ax+3}{x^2+x+1}$ takes all real values for all real values of $x$,then $a$ lies in the interval

If the extreme value of $3x - 2x^2 + 1$ is $k$, then the set of all real values of $x$ for which $kx^2 + 2x + 1 > 0$ is

Let $f(x)=(x-a)(x-b)-\left(\frac{a+b}{2}\right)$. If $f(x)=0$ has both non-negative roots,then the minimum value of $f(x)$ is:

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

If $\cos^4 \theta + \alpha$ and $\sin^4 \theta + \alpha$ are the roots of the equation $x^2 + 2bx + b = 0$,and $\cos^2 \theta + \beta$ and $\sin^2 \theta + \beta$ are the roots of the equation $x^2 + 4x + 2 = 0$,then find the value of $b$.

If $a, b, c$ are distinct positive real numbers and $a^2+b^2+c^2=1$,then the value of $ab+bc+ca$ is