Download App

Geometric Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions4 Marks
Question
If a, b, c, are in G.P., prove that: $\frac{1}{\text{a}^2+\text{b}^2},\frac{1}{\text{b}^2+\text{c}^2},\frac{1}{\text{c}^2+\text{d}^2}\text{ are in G.P.}$
✓

Answer

a, b, c, d are in G.P. $\therefore\text{b}^2=\text{ac}$ $\text{ad}=\text{bc}$ $\text{c}^2=\text{bd}\cdots(1)$ $\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big)^2=\Big(\frac{1}{\text{b}^2}\Big)^2+\frac{2}{\text{b}^2\text{c}^2}+\Big(\frac{1}{\text{c}^2}\Big)^2$ $\Rightarrow\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big)^2=\Big(\frac{1}{\text{ac}}\Big)^2+\frac{1}{\text{b}^2\text{c}^2}+\frac{1}{\text{b}^2\text{c}^2}+\Big(\frac{1}{\text{bd}}\Big)^2$ [Using (1)] $\Rightarrow\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big)^2=\frac{1}{\text{a}^2\text{c}^2}+\frac{1}{\text{a}^2\text{d}^2}+\frac{1}{\text{b}^2\text{c}^2}+\frac{1}{\text{b}^2\text{d}^2}$ [Using (1)]$\Rightarrow\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big)^2=\frac{1}{\text{a}^2}\Big(\frac{1}{\text{c}^2}+\frac{1}{\text{d}^2}\Big)+\frac{1}{\text{b}^2}\Big(\frac{1}{\text{c}^2}+\frac{1}{\text{d}^2}\Big)$
$\Rightarrow\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big)^2=\Big(\frac{1}{\text{a}^2+\text{b}^2}\Big)\Big(\frac{1}{\text{c}^2}+\frac{1}{\text{d}^2}\Big)$
$\therefore\Big(\frac{1}{\text{b}^2+\text{c}^2}\Big),\Big(\frac{1}{\text{c}^2+\text{d}^2}\Big)\text{ and are also in G.P.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Geometric Progressions→Geometric Progressions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\text{bx}}$
→
$\frac{\text{a}^2\sin(\text{B}-\text{C})}{\sin\text{A}}+\frac{\text{b}^2\sin(\text{C}-\text{A})}{\sin\text{B}}+\frac{\text{c}^2\sin(\text{A}-\text{B})}{\sin\text{C}}=0$
→
From 4 officers and 8 jawans in how many ways can 6 be chosen:
  1. To include exactly one officer.
  2. To include at least one officer?
→
Find the acute angle between the lines 2x - y + 3 = 0 and x + y + 2 = 0.
→
Show that the points A(1, 3, 4), B(-1, 6, 10), C(-7, 4, 7) and D(-5, 1, 1) are the vertices of a rhombus.
→
Prove the following by using the principle of mathematical induction for all n ∈ N:$1^3+2^3+3^3+....+\text{n}^3=\Big(\frac{\text{n}(\text{n+1)}}{2}\Big)^2.$
→
Solve the following systems of linear inequations graphically: $2\text{x}+3\text{y}\leq35,\text{y}\geq3,\text{x}\geq2,\text{x}\geq0,\text{y}\geq0$
→
Find the modulus and argument of the complex number $\frac{1+2\text{i}}{1-3\text{i}}.$
→
If $a, b, c$, are in G.P., prove that the following are also in G.P. $\text{a}^2+\text{b}^2,\text{ab}+\text{bc},\text{b}^2+\text{c}^2$
→
Find the equation to the ellipse whose foci are (-4, 0), and (-4, 0), eccentricity = $\frac{1}{3}.$
→