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Model Paper 3 — Applied Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceApplied MathsModel Paper 33 Marks
Question
If $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in G.P, show that $\mathrm{a}^{2}+\mathrm{b}^{2}, \mathrm{~b}^{2}+\mathrm{c}^{2}, \mathrm{c}^{2}+\mathrm{d}^{2}$ are in G.P.
✓

Answer

From the question, it is given that $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in G.P
So, bc = ad
$b^{2}=\mathrm{ac}$
$c^{2}=b d$
We have to show that,
$a^{2}+b^{2}, b^{2}+c^{2}, c^{2}+d^{2}$ are in G.P.
Then, $\left(b^{2}+c^{2}\right)^{2}=\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$
Consider the LHS $=\left(b^{2}+c^{2}\right)^{2}$
$=b^{4}+c^{4}+2 b^{2} c^{2}$
From the equation (ii) and equation (iii)
$=a^{2} c^{2}+b^{2} d^{2}+a^{2} d^{2}+b^{2} c^{2}$
$=c^{2}\left(a^{2}+b^{2}\right)+d^{2}\left(a^{2}+b^{2}\right)$
$=\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$
Now consider the
RHS $=\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$
By comparing the LHS and RHS
LHS = RHS
Hence it is proved that, $a^{2}+b^{2}, b^{2}+c^{2}, c^{2}+d^{2}$ are in G.P.

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