Chapter 3

If \(p x^{2}-q x+r=0\) has \(\alpha\) and \(\beta\) as its roots, evaluate \(\alpha^{3} \beta+\beta^{3} \alpha\)

If roots of equation \(\frac{a}{x-a}+\frac{b}{x-b}=1\) are equal in magnitude but opposite in sign, then prove that \(a+b=0\).

f the sum of the roots of the quadratic equation \(a x^{2}+b x+c=0\) is equal to the sum of the square of their reciprocals, then \(a / c, b / a, c / b\) are in (a) A.P. (b) G.P. (c) H.P. (d) none of these

If \(\alpha\) and \(\beta\) are the roots of the equation \(a x^{2}\) \(+b x+c=0\), find the equation whose roots are \(\frac{1}{a \alpha+b}, \frac{1}{a \beta+b}\).

If \(\alpha\) and \(\beta\) are the roots of \(2 x^{2}-5 x+7=0\), find out the equation whose roots are \(2 \alpha+3 \beta\) and \(3 \alpha+2 \beta\).

If \(\alpha, \beta\) are the roots of \(a x^{2}+b x+c=0\) and \(2 \alpha+\beta, \alpha^{2}+\beta^{2}, \alpha^{3}+\beta^{3}\) are in G.P., where \(\Delta\) \(=b^{2}-4 a c\), then: [IIT (Screening) - 2005] (a) \(\Delta \neq 0\) (b) \(b \Delta=0\) (c) \(c b \neq 0\) (d) \(c \Delta=0\)

If \(\alpha\) and \(\beta\) be the roots of the equation \(p x^{2}\) \(+q x+r=0 .\) Hence, obtain the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

If \(\alpha\) and \(\beta\) are roots of the quadratic equation \(x^{2}-p x+36=0\) and \(\alpha^{2}+\beta^{2}=9\), then find the value of \(p\).

In a triangle \(\mathrm{PQR}, \angle R=\frac{\pi}{2}\). If \(\tan \left(\frac{p}{2}\right)\) and \(\tan \left(\frac{Q}{2}\right)\) are the roots of \(a x^{2}+b x+c=0, a\) \(\neq 0\), then: [AIEEE - 2005] (a) \(b=a+c\) (b) \(b=c\) (c) \(c=a+b\) (d) \(a=b+c\)

Sum of roots of the quadratic equation is 2 and sum of cube of roots is 98 . Find the equation of roots.

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

If the roots of the equation \(x^{2}-5 x+16=\) 0 are \(\alpha\) and \(\beta\), and the roots of equation \(x^{2}+p x+q=0\) are \(\alpha^{2}+\beta^{2}, \alpha \beta / 2\), then: (a) \(p=1, q=-56\) (b) \(p=-1, q=-56\) (c) \(p=1, q=56\) (d) \(p=-1, q=56\)

If roots of equation \(2 x^{2}+3(k-2) x+4-k\) \(=15 x\) are same but of opposite sign, then fi nd the value of \(k\).

If 8 and 2 are roots of the quadratic equation \(x^{2}+\alpha x+\beta=0\) and 3,3 are roots of the quadratic equation \(x^{2}+\alpha x+b=0 .\) Then, find the roots of the quadratic equation \(x^{2}+a x+b=0\)

Difference between the corresponding roots of \(x^{2}+a x+b=0\) and \(x^{2}+b x+a=0\) is same and \(a \neq b\), then (a) \(a+b+4=0\) (b) \(a+b-4=0\) (c) \(a-b-4=0\) (d) \(a-b+4=0\)

If roots of equation \(\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}\) are equal in magnitude but opposite in sign, then show that \(p+q=2 r\) and prove that product of roots is \(-\frac{p^{2}+q^{2}}{2}\).

The equation whose roots are \(\frac{1}{3+\sqrt{2}}\) and \(\frac{1}{3-\sqrt{2}}\) is (a) \(7 x^{2}-6 x+1=0\) (b) \(6 x^{2}-7 x+1=0\) (c) \(x^{2}-6 x+7=0\) (d) \(x^{2}-7 x+6=0\)

If the root of \(x^{2}-b x+c=0\) are two consecutive integers, then \(b^{2}-4 c\) is (a) 1 (b) 2 (c) 3 (d) 4

Show that \(2^{3 n}-7 n-1\) is divisible by 49 , where \(n \in N\)

If \(\alpha\) and \(\beta\) are the roots of the equation \(a x^{2}\) \(+b x+c=0\), then \(\frac{\alpha}{a \beta+b}+\frac{\beta}{a \alpha+b}=\) (a) \(2 / a\) (b) \(2 / b\) (c) \(2 / a\) (d) \(-2 / a\)

If the roots of the quadratic equation \(\frac{x-m}{m x+1}\) \(=\frac{x+n}{n x+1}\) are reciprocal to each other, then [MPPET-2001] (a) \(n=0\) (b) \(m=n\) (c) \(m+n=1\) (d) \(m^{2}+n^{2}=1\)

The number of values of \(a\) for which \(\left(a^{2}-3 a+2\right) x^{2}+\left(a^{2}-5 a+6\right) x+a^{2}-4=0\) is an identity in \(x\), is (a) 0 (b) 2 (c) 1 (d) 3

Two students while solving a quadratic equation in \(x\), one copied the constant term incorrectly and got the roots 3 and 2 . The other copied the constant term and coefficient of \(x^{2}\) correctly as \(-6\) and 1 respectively. The correct roots are (a) \(3,-2\) (b) \(-3,2\) (c) \(-6,-1\) (d) \(6,-1\)

If the product of roots of the equation \(x^{2}-\) \(3 k x+2 e^{2 \log k}-1=0\) is 7, then its roots will be a real when [IIT - 1984] (a) \(k=1\) (b) \(k=2\) (c) \(k=3\) (d) none of these

Let \(N\) be the number of quadratic equations with coefficients \(\\{0,1,2, \ldots, 9\\}\) such that zero is a solution of each equation. [Kerala PET - 2003] Then the value of \(N\) is (a) Infinite (b) \(2^{9}\) (c) 90 (d) 900

If the roots of \(a x^{2}+b x+c=0\) are \(\alpha, \beta\) and the roots of \(A x^{2}+B x+C=0\) are \(\alpha-k, \beta-\) \(k\), then \(\frac{B^{2}-4 A C}{b^{2}-4 a c}\) is equal to (a) 0 (b) 1 (c) \(\left(\frac{A}{a}\right)^{2}\) (d) \(\left(\frac{a}{A}\right)^{2}\)

If the sum of the roots of the equation \(x^{2}+\) \(p x+q=0\) is three times their difference, (a) \(9 p^{2}=2 q\) (b) \(2 q^{2}=9 p\) (c) \(2 p^{2}=9 q\) (d) \(9 q^{2}=2 p\)

The value of ' \(c\) ' for which \(\left|\alpha^{2}-\beta^{2}\right|=7 / 4\), where \(\alpha\) and \(\beta\) are the roots of \(2 x^{2}+7 x+c\) \(=0\), is (a) 4 (b) 0 (c) 6 (d) 2

If \(\alpha\) and \(\beta\) are the roots of the equation \(a x^{2}\) \(+b x+c=0, \alpha \beta=3\) and \(a, b, c\) are in A.P., then \(a+b\) is equal to (a) \(-4\) (b) \(-1\) (c) 4 (d) \(-2\)

If \(\alpha\) and \(\beta\) are roots of the equation \(A x^{2}+B x\) \(+C=0\), then value of \(a^{3}+\beta^{3}\) is [RPET-1996; DCE - 2005] (a) \(\frac{3 A B C-B^{3}}{A^{3}}\) (b) \(\frac{3 A B C+B^{3}}{A^{3}}\) (c) \(\frac{B^{3}-3 A B C}{A^{3}}\) (d) \(\frac{B^{3}-3 A B C}{B^{3}}\)

If 3 is a root of \(x^{2}+k x-24=0\), it is also a root of (a) \(x^{2}+5 x+k=0\) (b) \(x^{2}-5 x+k=0\) (c) \(x^{2}-k x+6=0\) (d) \(x^{2}+k x+24=0\)