91Ó°ÊÓ

If \(\alpha\) and \(\beta\) are the roots of \(x^{2}-p(x+1)-c=0\), show that \((\alpha+1)(\beta+1)=1-c .\) Hence prove that \(\frac{\alpha^{2}+2 \alpha+1}{\alpha^{2}+2 \alpha+c}+\frac{\beta^{2}+2 \beta+1}{\beta^{2}+2 \beta+c}=1\)

If one root of the equation \(5 x^{2}+13 x+k=0\) is reciprocal of other, then find the value of \(k\).

Find the condition that the roots of the equation \(a x^{2}+b x+c=0\) be such that i. One root is \(n\) times the other. ii. One root is three times the other.

If the equation \(\left(k^{2}-5 k+6\right) x^{2}+\left(k^{2}-3 k+2\right) x+\left(k^{2}-4\right)=0\) is satisfied by more than two values of \(x\), then determine the value of \(k\).

If the sum of the roots of \(a x^{2}+b x+c=0\) be equal to sum of their squares, prove that \(2 a c=a b+b^{2}\).

Form an equation whose roots are cubes of the roots of the equation \(a x^{3}+b x^{2}+c x+d=0\).

Show that a polynomial of an odd degree has at least one real root.

Show that a polynomial of an even degree has at least two real roots if it attains at least one value opposite in sign to the coefficient of its highest degree term.

If \(\alpha, \beta\) are the roots of \(x^{2}+p x+q=0\) and \(\gamma, \delta\) are the roots of \(x^{2}+r x+s=0\), evaluate \((\alpha-\gamma)(\alpha-\delta)(\beta-\gamma)(\beta-\delta)\) in terms of \(p, q, r\) and \(s\). Hence deduce the condition that the equations have a common root.

If each pair of the three equations \(x^{2}+p_{1} x+q_{1}=0, x^{2}+p_{2} x+q_{2}=0\) and \(x^{2}+p_{3} x+q_{3}=0\) have a common root, then prove that \(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+4\left(q_{1}+q_{2}+q_{3}\right)=2\left(p_{1} p_{2}+p_{2} p_{3}+p_{3} p_{1}\right)\)