Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. If $a, b, c$ are in $A.P.$ and $\alpha + \beta = 15$,then $\alpha \beta$ equals:

Let $x, y$ be real numbers such that $x \neq y$ and $xy \neq 1$. If $ax + b \sec(\tan^{-1} x) = c$ and $ay + b \sec(\tan^{-1} y) = c$,then $\frac{x+y}{1-xy} =$

Two roots of the cubic equation $x^3 + 3x^2 + kx + 12 = 0$ are real and unequal but have the same absolute value. Find the value of $k$.

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + \frac{a}{2} x + b = 0$ and $(\alpha-\beta)(\alpha-\gamma)$,$(\beta-\alpha)(\beta-\gamma)$,$(\gamma-\alpha)(\gamma-\beta)$ are the roots of the equation $(y+a)^3 + K(y+a)^2 + L = 0$,then $\frac{L}{K} =$

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $x^{2}-bx+c=0$, then which of the following statements is/are correct?

If $3$ distinct real numbers $a, b, c$ satisfy $a^2(a + p) = b^2(b + p) = c^2(c + p)$ where $p \in \mathbb{R}$,then the value of $bc + ca + ab$ is: