Andhra Pradesh Board Solutions for Exercise 1: Exercise 4(a)

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3-2x^2+3x-4=0$, then find $\Sigma \alpha \beta (\alpha +\beta )$.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+p{x}^2+qx+r=0$. If $\sum \frac{1}{{\alpha }^2{\beta }^2}=\frac{p^2-2q}{r^k}$, then find the value of $k$.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+p{x}^2+qx+r=0$ then find $\frac{{\beta }^2+{\gamma }^2}{\beta \gamma }+\frac{{\gamma }^2+{\alpha }^2}{\gamma \alpha }+\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+p{x}^2+qx+r=0$ and $(\beta +\gamma -3\alpha )(\gamma +\alpha -3\beta )(\alpha +\beta -3\gamma )=3p^k-16pq+64r$, then find the value of $k$.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+p{x}^2+qx+r=0$ and $\Sigma {\alpha }^3{\beta }^3=q^3-3pqr+3r^k$, then find the value of $k$.

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-6x^2+11x-6=0$, then find the equation whose roots are ${\alpha }^2+{\beta }^2,{\beta }^2+{\gamma }^2,{\gamma }^2+{\alpha }^2$.

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-7x+6=0$, then find the equation whose roots are $(\alpha -\beta {)}^2,(\beta -\gamma {)}^2,(\gamma -\alpha {)}^2$.

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-3ax+b=0$, then prove that$\Sigma (\alpha -\beta )(\alpha -\gamma )=9a$.