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NDA 1 2024 Maths, question 26

Topic: Algebra

Q26

If a,b,ca,b,c are in AP; b,c,db,c,d are in GP; c,d,ec,d,e are in HP, then which of the following is/are correct ?

  1. a,ca,c and ee are in GP

  2. 1a,1c,1e\frac{1}{a},\frac{1}{c},\frac{1}{e} are in GP

Select the correct answer using the code given below :

  1. (a) 1 only
  2. (b) 2 only
  3. (c) Both 1 and 2Correct answer
  4. (d) Neither 1 nor 2

Solution.

Answer: (c) Both 1 and 2

  1. The three conditions give 2b=a+c, c²=bd and 2/d=1/c+1/e. Combining the first two gives 2c²=d(a+c).
  2. The HP equation gives e=cd/(2c−d); substitution yields e=c²/a, hence ae=c².
  3. Thus a,c,e form a GP.
  4. Taking reciprocals gives (1/c)²=(1/a)(1/e), so statement 2 also holds.
  5. The HP conditions ensure the necessary denominators are nonzero.