Q) Given that 2 is a root of the equation 3x² – p(x + 1) =0 and that the equation px² – qx + 9 = 0 has equal roots, find the value of p and q.

Ans: 

Step 1: Since 2 is a root of the equation 3 x² – p (x + 1) =0, therefore it should satisfy the given equation.

3 x² – p (x + 1) =0

∴ 3 (2)² – p (2 + 1) =0

∴ 12 – 3 p =0

∴ 12 = 3 p

∴ p =

∴ p = 4

Step 2: Since the equation p x² – q x + 9 = 0 has equal roots

Therefore its discriminant, D will be zero

∴ D = b² – 4ac = 0

Next, let’s compare the given equation with standard quadratic equation ax2 + bx + c = 0, we get:

a = p

b = – q

c = 9

Step 3: Now by substituting the values of a, b and c in the discriminant equation, we get:

b² – 4 a c = 0

∴ (- q)² – 4 (p) (9) = 0

∴ q² – 36 p = 0

∴ q² = 36 p

∴ q =

Since we calculated p = 4

∴ q = 6 = 6 x 2

∴ q = 12

Therefore the values of p and q are 4 and 12 respectively.

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