Question

(1) If ‘m’ and ‘n’ are the roots of the equation x² − 6x + 2 = 0 find the value of

(i) (m + n) mn

(ii)

(2) If ‘a’ and ‘b’ are the roots of the equation 3m² = 6m + 5, find the value of

(i)

(ii) (a + 2b) (2a + b)

(3) If ‘p’ and ‘q’ are the roots of the equations 2a² − 4a + 1 = 0, find the value of

(i) (p + q)² + 4pq

(ii) p³ + q³

(4) From the quadratic equation whose roots are

(5) Find the value of ‘k’ so that the equation x² + 4x + (k + 2) = 0 has one root equal to zero.

(6) Find the value of ‘q’ so that the equation 2x² − 3qx = 5q = 0 has one root which is twice the other.

(7) Find the value of ‘p’ so that the equation 4x² − 8px + 9 = 0 has roots whose difference is 4.

(8) If one root of the equations x² + px + q = 0 is 3 times the other prove that 3p² = 16q

Answer 1

(1)

Here, m and n are the roots of the quadratic equation.

(i)

Hence,

(ii)

Hence,

(2)

The above equation can be rewritten as _.

Here, a and b are the roots of the quadratic equation.

(i)

Hence,

(ii)

Hence,

(3)

Comparing the above equation with _,

It is given that p and q are the roots of the equation.

(i)

Hence,

(ii)

Hence,

(4) Letandbe the roots of the equation, where

The quadratic equation whose roots areandis given by:

Hence, the quadratic equation is.

(5)

Letandbe the roots of the equation.

Now,

It is given that one of the roots is 0.

Let

Hence, the value of k is.

(6)

Letandbe the roots of the equation.

Now,

It is given that.

Putting in, we get:

Hence, the value of q is.

(7)

Letandbe the roots of the equation.

Now,

It is given that.

Hence, the value of p is.

(8)

Letandbe the roots of the equation.

Now,

It is given that.

Hence,