Download App

Arithmetic Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions3 Marks
Question
If $\frac{1}{\text{a}},\ \frac{1}{\text{b}},\ \frac{1}{\text{c}}$ are in A.P., Prove that:
$​\text{a}(​\text{b}+​\text{c}),\ ​\text{b}(​\text{c}+​\text{a}),\ ​\text{c}(​\text{a}+​\text{b})$ are in A.P.
✓

Answer

$​\text{a}(​\text{b}+​\text{c}),\ ​\text{b}(​\text{c}+​\text{a}),\ ​\text{c}(​\text{a}+​\text{b})$ are in A.P if $​​\text{b}(​​\text{c}+​​\text{a})-​​\text{a}(​​\text{b}+​​\text{c})=​​\text{c}(​​\text{a}+​​\text{b})=​​\text{c}(​​\text{a}+​​\text{b})-​​\text{b}(​​\text{c}+​​\text{a})$
$​​\text{LHS}=​​\text{b}(​​\text{c}+​​\text{a})-​​\text{a}(​​\text{b}+​​\text{c})$
$=​​\text{bc}+​​\text{ab}-​​\text{ab}-​​\text{ac}$
$=​​\text{c}(​​\text{b}-​​\text{a})$
$​​\text{RHS}=​​\text{c}(​​\text{a}+​​\text{b})-​​\text{b}(​​\text{c+​​a})$
$=\text{ca}+\text{cd}-\text{bc}-\text{ba}$
$=\text{a}(\text{c}-\text{d})\ .....(2)$
and $\frac{1}{\text{a}},\ \frac{1}{\text{b}},\ \frac{1}{\text{c}}$ are in A.P
$\therefore\frac{1}{​\text{}a​}-\frac{1}{\text{b}}=\frac{1}{\text{b}}-\frac{1}{\text{c}}$
or $\text{c}(\text{b}-\text{a})=\text{a}(\text{c}-\text{b})\ .....(3)$
From (1), (2) and (3)
$\text{a}(\text{b}+\text{c}),\ \text{b}(\text{c}+\text{a}),\ \text{c}(\text{a}+\text{b})$ are in A.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 "3 Marks Question" from Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

In a $\triangle\text{ABC},$ Prove that:
$\cos\Big(\frac{\text{A+B}}{2}\Big)=\sin\frac{\text{C}}{2}$
→
Find the perpendicular distance of the line joining the points $\big(\cos\theta, \sin \theta\big) $ and $\big(\cos \phi, \sin \phi\big)$ from the origin.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}}{\text{x}-3}-\frac{4}{\text{x}^2-2\text{x}}\Big)$
→
Find the equation of the circle having $(1, -2)$ as its centre and passing through the intersection of the lines $3x + y = 14$ and $2x + 5y = 18.$
→
Write the following relations as sets of ordered pairs and find which of them are functions:
$\big\{(\text{x},\text{y})=\text{x}+\text{y}=3,\text{x},\text{y}\in\{0,1,2,3\}\big\}$
→
Evaluate the following limits:
$\lim\limits_{\text{x}\rightarrow\infty}\frac{3\text{x}^{-1}+4\text{x}^{-2}}{5\text{x}^{-1}+6\text{x}^{-2}}$
→
If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions:
(fg)(0)
→
Differentiate the following function with respect to x:$\text{x}^3\text{e}^\text{x}\cos\text{x}$
→
In a group of 950 person, 750 can speak Hindi and 460 can speak English. Find:
how many can speak English only.
→
Find the mean deviation about the mean for the data
Height in cm $95-105$ $105-115$ $115-125$ $125-135$ $135-145$ $145-155$
Number of boys $9$ $13$ $26$ $30$ $12$ $10$
→