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SEQUENCES AND SERIES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSEQUENCES AND SERIES1 Mark
MCQ
If p, q be two A.M.'s and G be one G.M. between two numbers, then $G^2$ =
  • ✓
    $(2\text{p}-\text{q})(\text{p}-2\text{q})$
  • B
    $(2\text{p}-\text{q})(2\text{q}-\text{p})$
  • C
    $(2\text{p}-\text{q})(\text{p}+2\text{q})$
  • D
    None of these.
✓

Answer

Correct option: A.
$(2\text{p}-\text{q})(\text{p}-2\text{q})$
  1. $(2\text{p}-\text{q})(\text{p}-2\text{q})$
Solution:
Let the two numbers be a and b.
a, p, q and b are in A.P.
$\therefore\text{ p}-\text{a}=\text{q}-\text{q}=\text{b}-\text{q}$
$\Rightarrow\text{ p}-\text{a}=\text{q}-\text{p}\text{ and}\text{ q}-\text{p}=\text{b}-\text{q}$
$\Rightarrow\text{ a}=2\text{p}-\text{q}\text{ and}\text{ b}=2\text{q}-\text{p}\cdots(\text{i})$
Also, a, G and b are in G.P.
$\therefore\text{G}^2=\text{ab}$
$\Rightarrow\text{G}^2=(2\text{p}-\text{q})(2\text{q}-\text{p})$

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