The sum of first q terms of an AP is `(63q -3q^(2))`. If its pth term is -60, find the value of p. Also, find the 11th term of its AP.

Explanation

To solve the problem step by step, we will follow the outlined process:

### Step 1: Understand the given information
The sum of the first \( q \) terms of an arithmetic progression (AP) is given by:
\[ S_q = 63q - 3q^2 \]
We need to find the \( p \)th term of the AP, which is given as \( -60 \), and also find the 11th term of the AP.

### Step 2: Find the \( q \)th term of the AP
The \( q \)th term \( T_q \) can be found using the formula:
\[ T_q = S_q - S_{q-1} \]
First, we need to calculate \( S_{q-1} \):
\[ S_{q-1} = 63(q-1) - 3(q-1)^2 \]
Expanding this:
\[ S_{q-1} = 63q - 63 - 3(q^2 - 2q + 1) = 63q - 63 - 3q^2 + 6q - 3 \]
Combining like terms:
\[ S_{q-1} = 63q + 6q - 3q^2 - 66 = (69q - 3q^2 - 66) \]

Now, substituting \( S_q \) and \( S_{q-1} \) into the equation for \( T_q \):
\[ T_q = (63q - 3q^2) - (69q - 3q^2 - 66) \]
Simplifying this:
\[ T_q = 63q - 3q^2 - 69q + 3q^2 + 66 \]
The \( 3q^2 \) terms cancel out:
\[ T_q = -6q + 66 \]

### Step 3: Set up the equation for the \( p \)th term
We know that the \( p \)th term \( T_p \) is given as \( -60 \):
\[ T_p = -6p + 66 = -60 \]

### Step 4: Solve for \( p \)
Rearranging the equation:
\[ -6p + 66 = -60 \]
Subtracting 66 from both sides:
\[ -6p = -60 - 66 \]
\[ -6p = -126 \]
Dividing by -6:
\[ p = 21 \]

### Step 5: Find the 11th term of the AP
Now, we need to find the 11th term \( T_{11} \):
\[ T_{11} = -6(11) + 66 \]
Calculating this:
\[ T_{11} = -66 + 66 = 0 \]

### Final Answers
The value of \( p \) is \( 21 \) and the 11th term of the AP is \( 0 \).