1. Determine the nature of roots of each of the following equations
1. 2. 3. 4. 5. 6. 2. For what values of p will the equation ?

3. if the equation has equal roots, find k.

4. For what value of a will the equation have equal roots?

5. If the roots of the equation are equal, then show that 6. Show that the roots of the equation will be equal, if either or 7. If are rational and , show that the roots are rational.

8. Prove that the roots of the equation are real for all values of k.

9. Show that the roots of the equation are imaginary.

10. If the roots of the quadratic equation are real and unequal, prove that the roots of the equation are imaginary

1. From the equation whose roots are
1. 3,-2

2. -5,4

3. 4. 5. -3+5i,-i-5i

6. a+ib,a-ib

1. Find a quadratic equation whose roots are twice the roots of 2. Find a quadratic equation whose roots are reciprocals of the roots of 3. Find a quadratic equation whose roots are greater by h than the roots of 4. Find a quadratic equation whose roots are the squares of the roots of 2. Find a quadratic equation with rational coefficients one of whose roots is
1. 4+3i

2. 3. 3. Find the value of k so that the equation
1. has one root 3

2. has roots whose sum is equal to 6

3. has roots equal but opposite in sign

4. has roots numerically equal but opposite in sign

5. has one root equal to zero

6. has the reciprocal roots

7. has roots whose difference is 4. Show that -1 is a root of the equation x^2+(2a-b-c)x+(c+a-2b)=0\). Find the other root.

5. Find the value of m for which the equation will have (a) reciprocal roots (b) one root zero.

6. If the roots of the equation differ by 1, prove that 7. If are the roots of the equation , find the equation whose roots are
1. and 2. and 8. If are the roots of the equation , find the equation whose roots are
1. and 2. and 3. and 4. the reciprocal of the roots of given equation

1. If the roots of the equation be in the ratio of 3:4, prove that 2. If one root of the equation be four times the other root, show that 3. For what values of m, the equation may have its root in the ratio 2:3

1. If are the roots of the equation , prove that 2. If roots of the equation be in the ratio of p:q, prove that 9. If one root of the equation be square of the other root, prove that 1. Show that each pair of following equations has a common root
1. and 2. and 2. Find the value of p so that each pair of the equations may have one root common
1. and 2. and 3. If the quadratic equations and have common roots show that it must be either or 4. If the quadratic equations and have common roots show that it must be either or 5. If the quadratic equations and have common roots show that it must be either or 6. Prove that if the equations and have a common root, their other root will satisfy 