If the equation $\frac{{{x^2} + 5}}{2} = x - 2\cos \left( {ax + b} \right)$ has atleast one solution, then $(b + a)$ can be equal to

If graph of $y = ax^2 -bx + c$ is following, then sign of $a$, $b$, $c$ are

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

The number of real roots of the equation $x | x |-5| x +2|+6=0$, is

Number of natural solutions of the equation $xyz = 2^5 \times 3^2 \times  5^2$ is equal to

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