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

Let $2 \sin^2 x + 3 \sin x - 2 > 0$ and $x^2 - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

Let $a, b, c, d$ be distinct real numbers such that $a, b$ are roots of $x^2-5cx-6d=0$,and $c, d$ are roots of $x^2-5ax-6b=0$. Then $b+d$ is

Let $f(x)=a x^2+b x+c$ and the $GCD$ of $a, b, c$ be $1$. If $\frac{-7+\sqrt{11} i}{6}$ is a root of $f(x)=0$ and $f\left(\frac{x}{k}\right)-L=(x+4)(3 x-5)$,then $k$ and $L$ are respectively:

If $m$ is chosen in the quadratic equation $(m^2 + 1)x^2 - 3x + (m^2 + 1)^2 = 0$ such that the sum of its roots is greatest,then the absolute difference of the cubes of its roots is

Evaluate $\sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}}$ for $x \ge 1$.