Question

Consider two sets $ A $ and $ B $, each containing three numbers in A.P. Let the sum and the product of the elements of $ A $ be 36 and $ p $, respectively, and the sum and the product of the elements of $ B $ be 36 and $ q $, respectively. Let $ d $ and $ D $ be the common differences of A.P's in $ A $ and $ B $, respectively, such that $ D = d + 3 $, $ d>0 $. If $ \frac{p+q}{p-q} = \frac{19}{5} $, then $ p - q $ is equal to:

• A. 600
• B. 450
• C. 630
• D. 540

Hint:

For problems involving sums and products of terms in an A.P., use the standard formulas for sum and product in A.P. and substitute the given values accordingly.

Answer 1

The Correct Option is D

Let the elements of set \( A \) be \( a - d, a, a + d \) (since they are in A.P.) and the elements of set \( B \) be \( b - D, b, b + D \). 
The sum of the elements of set \( A \) is given by: \[ (a - d) + a + (a + d) = 3a = 36 \quad \Rightarrow \quad a = 12 \] The product of the elements of set \( A \) is: \[ (a - d) \cdot a \cdot (a + d) = a(a^2 - d^2) = p \] \[ 12 \cdot (12^2 - d^2) = p \quad \Rightarrow \quad 12(144 - d^2) = p \] Similarly, the sum of the elements of set \( B \) is: \[ (b - D) + b + (b + D) = 3b = 36 \quad \Rightarrow \quad b = 12 \] The product of the elements of set \( B \) is: \[ (b - D) \cdot b \cdot (b + D) = b(b^2 - D^2) = q \] \[ 12 \cdot (12^2 - D^2) = q \quad \Rightarrow \quad 12(144 - D^2) = q \] We are given that \( D = d + 3 \), so substitute \( D = d + 3 \) into the equation for \( q \): \[ q = 12(144 - (d + 3)^2) \] Now, we are given the relation: \[ \frac{p + q}{p - q} = \frac{19}{5} \] Substitute the expressions for \( p \) and \( q \) into this relation, and solve for \( p - q \). After solving, we get \( p - q = 540 \). 
Thus, the correct answer is \( 540 \).

Answer 2

Step 1: Given points: 

\[ A(a - d, a, a + d), \quad B(b - D, b, b + D) \] with \( a = 12 \) and \( b = 12 \).

Step 2: Expressions for \( p \) and \( q \):

\[ p = 12(144 - d^2) \] \[ q = 12(144 - D^2) \]

Step 3: Sum and Difference of \( p \) and \( q \):

\[ \frac{p + q}{p - q} = \frac{19}{5} \] Simplifying: \[ \frac{p}{q} = \frac{24}{14} = \frac{12}{7} \]

Step 4: Solving for \( d \):

From: \[ \frac{144 - d^2}{144 - (d^2 + 6d + 9)} = \frac{12}{7} \] Simplifying: \[ 1008 - 7d^2 = -12d^2 - 72d + 1620 \] Leading to: \[ 5d^2 + 72d - 612 = 0 \] Solving this gives: \[ d = 6. \]

Step 5: Finding \( D \):

\[ D = 9. \]

Step 6: Final Calculation for \( p - q \):

\[ p - q = 12(D^2 - d^3) \] Substituting \( D = 9 \) and \( d = 6 \): \[ p - q = 12(81 - 36) = 12(45) = 540. \]