Download App

Arithmetic Progressions — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions4 Marks
Question
If a, b, c are in A.P., prove that:
$\text{a}^2+\text{c}^2+4\text{ac}=2(\text{ab}+\text{bc}+\text{ca})$
✓

Answer

If $\text{a}^2+\text{c}^2+4\text{ac}=2(\text{ab}+\text{bc}+\text{ca})$
Then,
$\text{a}^2+\text{c}^2+2\text{ac}-2\text{ab}=2(\text{ab}+\text{bc}+\text{ca})$
or $(​​\text{a}+​​\text{b}+​-\text{c})^2-​​\text{b}^2=0$ $[\therefore(​​\text{a}+​​\text{b}+​​\text{c})^2=​​\text{a}^2+​​\text{b}^2+​​\text{c}^2+2​​\text{ab}+2​​\text{ac}+2​​\text{bc}]$
or $​​\text{b}=​​\text{a}+​​\text{c}-​​\text{b}$
or $2​​\text{b}=​​\text{a}+​​\text{c}$
$​​\text{b}=\frac{​​\text{a}+​​\text{b}}{2}$
and since,
$​​\text{a},\text{b},\text{c}$ are in A.P
$​​\text{b}=\frac{​​\text{a}+​​\text{c}}{2}$
Thus, $\text{a}^2+​​\text{c}^2+4​​\text{ac}=2(​​\text{ab}+​​\text{bc}+​​\text{ca})$
Hence proved.

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 "Solve the Following Question.(4 Marks)" from Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the minors and cofactors of elements of the determinants.

$\left|\begin{array}{ccc}-1 & 0 & 4 \\ -2 & 1 & 3 \\ 0 & -4 & 2\end{array}\right|$

→
In how many ways can a football team of 11 players be selected from 16 players? How many of these will.
  1. Include 2 particular players?
  2. Exclude 2 particular players?
→
In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?
→
How many permutations can be formed by the letters of the word, $'VOWELS',$ when.
  1. There is no restriction on letters$?$
  2. Each word begins with $E?$
  3. Each word begins with $O$ and ends with $L?$
  4. All vowels come together$?$
  5. All consonants come together$?$
→
If the $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ terms of a G.P. are $x, y$ and $z$ respectiveiy. Prove that $x, y, z$ are in G.P.
→
A man saved ₹ 66000 in 20 years. In each succeeding year after the first year he saved ₹ 200 more than what he saved in the previous year. How much did he save in the first year?
→
Prove that:
$\cos20^\circ\cos40^\circ\cos80^\circ=\frac{1}{8}$
→
Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION'.
→
In how many ways can one select a circket team of eleven from17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?
→
Prove the following identities:
$\cos\text{x}(\tan\text{x}+2)(2\tan\text{x}+1)=2\sec\text{x}+5\sin\text{x}$
→