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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Sequences and Series

NDA 2022 · Mathematics · Sequences and Series

If a, b, c are in GP where a > 0, b > 0, c > 0 , then which of the...

If a,b,ca, b, c are in GP where a>0,b>0,c>0a > 0, b > 0, c > 0, then which of the following are correct? 1. a2,b2,c2a^2, b^2, c^2 are in GP 2. 1a,1b,1c\frac{1}{a}, \frac{1}{b}, \frac{1}{c} are in GP 3. a,b,c\sqrt{a}, \sqrt{b}, \sqrt{c} are in GP Select the correct answer using the code given below :

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1, 2 and 3

Correct answer

D. 1, 2 and 3

Step-by-step solution

∵a,b,care in G.P ∴b2=ac\begin{array}{ll}\because & \mathrm{a}, \mathrm{b}, \mathrm{c} \text{are in G.P } \\ \therefore \quad b^2=a c\end{array}SquaringbothsidesSquaring both sides(b2)2=a2c2∴a2,b2,c2 are in  G.P \begin{aligned} & \left(b^2\right)^2=a^2 c^2 \\ & \therefore \quad a^2, b^2, c^2 \text{ are\text{ in } G.P }\end{aligned}From(i)From(i)\left(\frac{1}{b}\right)^2=\frac{1}{a} \times \frac{1}{c}∴\therefore \quad\frac{1}{a}, \frac{1}{b},\frac{1}{c}areinG.PFrom(i)are in G.P From(i)b=\sqrt{a c} \Rightarrow(\sqrt{b})^2=\sqrt{a} \sqrt{b}[\because a \gt 0, b \gt 0 and c \gt 0]\therefore \quad \sqrt{a}, \sqrt{b}, \sqrt{c}$ are in G.P.
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