The sum of first q terms of an AP is (63q - 3q²). If its p^th term is - 60, find the value of p. Also, find the 11th term of its AP.

Let a be the first term and d be the common difference.

Given: Sum of first q terms of an AP is given by:

S_q = [2a + (q - 1)d] = 63q - 3q²

Now, p^th term is given by: a_p = S_p - S_{p - 1}

∴ a_p = (63p - 3p²) - [63(p - 1) - 3(p - 1)²]

= (63p - 3p²) - [63p - 63 - 3p² - 3 + 6p]

= 63p - 3p² - 63p + 63 + 3p² + 3 - 6p

= 66 - 6p …………………(1)

Now, given that a_p = - 60

⇒ 66 - 6p = - 60

⇒ 6p = 126

⇒ p = 21

For 11^th term of AP, put p = 11 in the value of the p^th term in equation (1), we get

a₁₁ = 66 - 6 × (11 )

⇒ a₁₁ = 66 - 66

= 0

∴ a₁₁ = 0