text: Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
  
  # The Sum of an Infinite G.P. is 4 and the Sum of the Cubes of Its Terms is 92. the Common Ratio of the Original G.P. is - Mathematics
  
  MCQ
  
  The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
  
  • (a) 1/2
  
  • (b) 2/3
  
  • (c) 1/3
  
  • (d) −1/2
  
  #### Solution
  
  (a) 1/2
  
  $\text{ Let the G . P . be a, ar }, a r^2 , a r^3 , . . . , \infty .$
  $S_\infty = 4$
  $\Rightarrow \frac{a}{1 - r} = 4 (i)$
  $\text{ Also, sum of the cubes }, S_1 = 92$
  $\Rightarrow \frac{a^3}{\left( 1 - r^3 \right)} = 92 (ii)$
  $\text{ Putting the value of a from } (i) \text{ to } (ii):$
  $\Rightarrow \frac{\left( 4(1 - r) \right)^3}{\left( 1 - r^3 \right)} = 92$
  $\Rightarrow \frac{64(1 - r )^3}{\left( 1 - r^3 \right)} = 92$
  $\Rightarrow \frac{\left( 1 - r \right)^3}{\left( 1 - r \right)\left( 1 + r + r^2 \right)} = \frac{92}{64}$
  $\Rightarrow \frac{\left( 1 - r \right)^2}{\left( 1 + r + r^2 \right)} = \frac{23}{16}$
  $\Rightarrow 16\left( 1 - 2r + r^2 \right) = 23\left( 1 + r + r^2 \right)$
  $\Rightarrow 7 r^2 + 55r + 7 = 0$
  $\text{ Using the quadratic formula }:$
  $\Rightarrow r = \frac{- 55 + \sqrt{{55}^2 - 4 \times 7 \times 7}}{2 \times 7}$
  $\Rightarrow r = \frac{- 55 + \sqrt{{55}^2 - {14}^2}}{14}$
  $\Rightarrow r = \frac{- 55 + \sqrt{2829}}{14}$
  
  Is there an error in this question or solution?
  
  #### APPEARS IN
  
  RD Sharma Class 11 Mathematics Textbook
  Chapter 20 Geometric Progression
  Q 10 | Page 57
explanation: The document is a math question from a Karnataka PUC Class 11 Mathematics textbook. It presents a multiple-choice question (MCQ) about finding the common ratio of an infinite geometric progression (G.P.) given the sum of the G.P. and the sum of the cubes of its terms. The document also includes a detailed solution to the problem, leading to the correct answer, which is (a) 1/2.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
  
  # The Sum of an Infinite G.P. is 4 and the Sum of the Cubes of Its Terms is 92. the Common Ratio of the Original G.P. is - Mathematics
  
  MCQ
  
  The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
  
  • (a) 1/2
  
  • (b) 2/3
  
  • (c) 1/3
  
  • (d) −1/2
  
  #### Solution
  
  (a) 1/2
  
  $\text{ Let the G . P . be a, ar }, a r^2 , a r^3 , . . . , \infty .$
  $S_\infty = 4$
  $\Rightarrow \frac{a}{1 - r} = 4 (i)$
  $\text{ Also, sum of the cubes }, S_1 = 92$
  $\Rightarrow \frac{a^3}{\left( 1 - r^3 \right)} = 92 (ii)$
  $\text{ Putting the value of a from } (i) \text{ to } (ii):$
  $\Rightarrow \frac{\left( 4(1 - r) \right)^3}{\left( 1 - r^3 \right)} = 92$
  $\Rightarrow \frac{64(1 - r )^3}{\left( 1 - r^3 \right)} = 92$
  $\Rightarrow \frac{\left( 1 - r \right)^3}{\left( 1 - r \right)\left( 1 + r + r^2 \right)} = \frac{92}{64}$
  $\Rightarrow \frac{\left( 1 - r \right)^2}{\left( 1 + r + r^2 \right)} = \frac{23}{16}$
  $\Rightarrow 16\left( 1 - 2r + r^2 \right) = 23\left( 1 + r + r^2 \right)$
  $\Rightarrow 7 r^2 + 55r + 7 = 0$
  $\text{ Using the quadratic formula }:$
  $\Rightarrow r = \frac{- 55 + \sqrt{{55}^2 - 4 \times 7 \times 7}}{2 \times 7}$
  $\Rightarrow r = \frac{- 55 + \sqrt{{55}^2 - {14}^2}}{14}$
  $\Rightarrow r = \frac{- 55 + \sqrt{2829}}{14}$
  
  Is there an error in this question or solution?
  
  #### APPEARS IN
  
  RD Sharma Class 11 Mathematics Textbook
  Chapter 20 Geometric Progression
  Q 10 | Page 57
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a math question from a Karnataka PUC Class 11 Mathematics textbook. It presents a multiple-choice question (MCQ) about finding the common ratio of an infinite geometric progression (G.P.) given the sum of the G.P. and the sum of the cubes of its terms. The document also includes a detailed solution to the problem, leading to the correct answer, which is (a) 1/2.
  
  Label: question-and-answer