CBSE Class 10 Answered

Let a,b and c are the zeroes of the polynomial f(x)=x^3-3px^2+qx-r.

Since it is given that they are in A.P

Therefore,     a + c = 2b ................(1)

Sum of zeroes of a cubic polynomial, a + b + c = 3p    .................(2)

Sum of zeroes taken two at a time, ab + bc + ca = q    ..............(3)

Product of zeroes, abc = r .....................(4)

From (1) and (2), we get

3b = 3p  ⇒ b = p ..........(5)

From (1) and (5), we get

ac = r/ p ..........(6)

Now from (2)

b (a + c) + ac = q

On substitution, we get

b (2b) + r/p = q

2p² + (r/p) = q

2p³ + r = pq

2p³ = pq - r.

Hence option (a) is correct.