Let $P(x)$ be a polynomial, which when divided by $(x-3)$ and $(x-5)$ leaves remainders $10$ and $6$, respectively. If the polynomial is divided by $(x-3)(x-5)$, then the remainder is

The graph of $y = ax^2 + bx + c$ is shown. Which of the following does $NOT$ hold good?

If $\alpha$ and $\beta$ $(\alpha > \beta)$ are the multiple roots of the equation $4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0$,then $2\alpha - \beta = $

The $x$-coordinates of the vertices of a square of unit area are the roots of the equation $x^2 - 3|x| + 2 = 0$ and the $y$-coordinates of the vertices are the roots of the equation $y^2 - 3y + 2 = 0$. Then the possible vertices of the square are:

Let $S$ be the set of all real roots of the equation $3^{x}(3^{x}-1)+2=|3^{x}-1|+|3^{x}-2|$. Then $S$

If $2 + i\sqrt{3}$ is a root of the equation $x^2 + px + q = 0$,where $p$ and $q$ are real,then $(p, q) = $