Download App

Geometric Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsGeometric Progressions5 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 "5 Marks Questions" from Geometric Progressions→Geometric Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$
→
The angle of a quadrilateral are in A.P. and the greatest angle is 120°. Express the angles in radians.
→
Find the equation to the ellipse whose centre is $(-2, 3)$ and semi-axis are $3$ and $2$ when major axis is parallel to $x-$axis
→
Find the mean deviation from the mean for the following data:
classes
0-10
10-20
20-30
30-40
40-50
50-60
Frequencies
6
8
14
16
4
2
→
Prove that the term independent of x in the expansion of $\Big(\text{x}+\frac{1}{\text{x}}\Big)^{2\text{n}}$ is $\frac{1,3,5.....(2\text{n}-1)}{\text{n}!}.2^{\text{n}}.$
→
Reduce each of the following expressions to the sine and cosin of a single expression:
$\sqrt{3}\sin\text{x}-\cos\text{x}$
→
In what direction should a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 is at a distance $\frac{\sqrt{6}}{3}$ from the given point.
→
show that the solution set of the following system of linear inequalities is an unbounded region $2\text{x}+\text{y}\geq8,\text{x}+2\text{y}\geq10,\text{x}\geq0,\text{y}\geq0.$
→
If $A (-2,2,3)$ and $B (13,-3,13)$ are two points. Find the locus of a point P which moves in such a way that $3 PA =2 PB$.
→
Use the Principle of Mathematical Induction in the following Exercis.
Show that $\frac{\text{n}^5}{5}+\frac{\text{n}^3}{3}+\frac{7\text{n}}{15}$ is a natural number for all $\text{n}\in\text{N}.$
→