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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Sequences and Series

NDA 2020 · Mathematics · Sequences and Series

If p^2, q^2 and r^2 (where .p, q, r 0) are in GP, then which of the...

If p2,q2andr2p^2, q^2 and r^2 (where p,q,r>0)\left.p, q, r \gt 0\right) are in GP, then which of the following is/are correct? 1. p,qandr\quad p, q and r are in GP. 2. ln⁡p,ln⁡qandln⁡r\ln p, \ln q and \ln r are in AP . Select the correct answer using the code given below :

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

(1) p2,q2andr2p^2, q^2 and r^2 in G.P. Then, (q2)2=p2⋅r2⇒q2=(p2⋅r2)12⇒q2=p⋅r\left(q^2\right)^2=p^2 \cdot r^2 \Rightarrow q^2=\left(p^2 \cdot r^2\right)^{\frac{1}{2}}\Rightarrow q^2=p \cdot r Hence, p,qandrp, q and r in G.P. (2) As, q2=p⋅rq^2=p \cdot r Taking log⁡\log on both sides, we have ln⁡(q2)=ln⁡(p⋅r)2ln⁡(q)=ln⁡p+ln⁡r\begin{aligned} & \ln \left(q^2\right)=\ln (p \cdot r) \\ & 2 \ln (q)=\ln p+\ln r \end{aligned} Hence, ln⁡p,ln⁡q\ln p, \ln q and ln⁡r\ln r are in A.P.
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