If $(x + 1)$ is a factor of ${x^4} - (p - 3){x^3} - (3p - 5){x^2} + (2p - 7)x + 6$,then $p = $

The number of real solutions of the equation $(\frac{3}{2})^x = -x^2 + 5x - 10$ is:

Let $\alpha, \beta$ be two roots of the quadratic equation $x^2+ax-b=0, b \neq 0$. If the straight line $x \cos \theta + y \sin \theta = c$ touches the curve $(\frac{x}{\alpha})^n + (\frac{y}{\beta})^n = 2$ at the point $(\alpha, \beta)$, then $(\frac{a}{b})^2 + \frac{2}{b} =$

For which value of $k$,are the roots of the equation $2x^2 + 5x + k = 0$ rational?

If the equation $x^2 + \lambda x + \mu = 0$ has equal roots and one root of the equation $x^2 + \lambda x - 12 = 0$ is $2$,then $(\lambda, \mu) = $

If $3 + 4i$ is a root of the equation ${x^2} + px + q = 0$ where $p$ and $q$ are real numbers,then: