Question

The roots of the quadratic equation \( 3x^2 - px + q = 0 \) are the 10th and 11th terms of an arithmetic progression with common difference \( \frac{3}{2} \). If the sum of the first 11 terms of this arithmetic progression is 88, then \( q - 2q \) is equal to:

Hint:

In problems involving arithmetic progressions, use the sum and product of the roots formula for quadratic equations to find the required values.

Answer 1

Correct Answer: 474

Step 1: Given information

The sum of the first 11 terms of an arithmetic progression (A.P.) is 88, expressed as: \[ S_{11} = \frac{11}{2}(2a + 10d) = 88 \]

Simplifying this equation yields: \[ a + 5d = 8 \]

Step 2: Finding the first term \( a \)

Given the common difference \( d = \frac{3}{2} \), substitute this value into the simplified equation: \[ a + 5 \times \frac{3}{2} = 8 \] \[ a + \frac{15}{2} = 8 \] \[ a = 8 - \frac{15}{2} = \frac{1}{2} \]

Step 3: Finding the 10th and 11th terms

The formula for the \( n^{th} \) term of an A.P. is: \[ T_n = a + (n-1)d \]. Using this, we calculate:

\[ T_{10} = a + 9d = \frac{1}{2} + 9 \times \frac{3}{2} = \frac{1}{2} + \frac{27}{2} = 14 \]

\[ T_{11} = a + 10d = \frac{1}{2} + 10 \times \frac{3}{2} = \frac{1}{2} + 15 = \frac{31}{2} \]

Step 4: Finding the value of \( p \)

Given the relationship: \[ \frac{p}{3} = T_{10} + T_{11} \]. Substituting the calculated values of \( T_{10} \) and \( T_{11} \):

\[ \frac{p}{3} = 14 + \frac{31}{2} = \frac{28 + 31}{2} = \frac{59}{2} \]

Solving for \( p \): \[ p = \frac{3 \times 59}{2} = \frac{177}{2} \]

Step 5: Finding the value of \( q \)

Given the relationship: \[ \frac{q}{3} = T_{10} \times T_{11} \]. Substituting the calculated values:

\[ \frac{q}{3} = 14 \times \frac{31}{2} = 7 \times 31 = 217 \]

Solving for \( q \): \[ q = 3 \times 217 = 651 \]

Step 6: Finding the final expression \( q - 2p \)

Calculate \( q - 2p \) using the determined values of \( p \) and \( q \):

\[ q - 2p = 651 - 2 \times \frac{177}{2} \] \[ q - 2p = 651 - 177 = 474 \]

Final Answer:

\[ \boxed{q - 2p = 474} \]