If the roots of the equation $ax^2 + bx + c = 0$ are $\sin \alpha$ and $\cos \alpha$,then:

Two real numbers $\alpha$ and $\beta$ are such that $\alpha + \beta = 3$ and $|\alpha - \beta| = 4$. Then $\alpha$ and $\beta$ are the roots of the quadratic equation:

Let $\alpha, \beta$ be the roots of $x^{2}-x-1=0$ and $S_{n}=\alpha^{n}+\beta^{n}$ for all integers $n \geq 1$. Then, for every integer $n \geq 2$, which of the following is true?

Let $\alpha, \beta, \gamma$ be the roots of $x^3+x+10=0$ and $\alpha_1=\frac{\alpha+\beta}{\gamma^2}, \beta_1=\frac{\beta+\gamma}{\alpha^2}, \gamma_1=\frac{\gamma+\alpha}{\beta^2}$. Then,the value of $(\alpha_1^3+\beta_1^3+\gamma_1^3)-\frac{1}{10}(\alpha_1^2+\beta_1^2+\gamma_1^2)$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $5x^3 - 3x^2 + 2x - 4 = 0$,then find the value of $\sum \alpha^2 \beta^2$.

If the roots of the equation $2x^2 - 3x + 5 = 0$ are the reciprocals of the roots of the equation $ax^2 + bx + 2 = 0$,then: