If ,| |,| | , < 1, , ,1 - + ^2 - ^3 + ... = s_1 , 1 - + ^2 - ^3 +… - Vidyadip

MCQ

${\rm{If}}\,|\alpha |,|\beta |\, < 1,\,\,1 - \alpha + {\alpha ^2} - {\alpha ^3} + ...\infty = {s_1},$ $1 - \beta + {\beta ^2} - {\beta ^3} + ....\infty = {s_2},$ then $1 - \alpha \beta + {a^2}{\beta ^2} - {a^3}{\beta ^3} + ....\infty \,\,{\rm{equals}}$

A
${s_1}{s_2}$
B
$\frac{{{s_1}{s_2}}}{{1 + {s_1}{s_2}}}$
✓
$\frac{{{s_1}{s_2}}}{{1 - {s_1} - {s_2} + 2{s_1}{s_2}}}$
D
$\frac{1}{{1 + {s_1}{s_2}}}$

✓

Answer

Correct option: C.

$\frac{{{s_1}{s_2}}}{{1 - {s_1} - {s_2} + 2{s_1}{s_2}}}$

c
(c) ${s_1} = \frac{1}{{1 + \alpha }},{s_2} = \frac{1}{{1 + \beta }}$

Let $s = 1 - \alpha \beta + {\alpha ^2}{\beta ^2}.......$

==> $s = \frac{1}{{1 + \alpha \beta }}$
$\alpha = \frac{1}{{{s_1}}} - 1,\,\,\beta = \frac{1}{{{s_2}}} - 1$;

$\therefore s = \frac{1}{{1 + \left( {\frac{1}{{{s_1}}} - 1} \right)\,\left( {\frac{1}{{{s_2}}} - 1} \right)}}$.

$s = \frac{{{s_1}{s_2}}}{{2{s_1}{s_2} + 1 - {s_1} - {s_2}}}$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5 , 5) is :

Let $\vec{\alpha}=4 \hat{ i }+3 \hat{ j }+5 \hat{ k }$ and $\vec{\beta}=\hat{ i }+2 \hat{ j }-4 \hat{ k }$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, then the value of $5 \vec{\beta}_2 \cdot(\hat{ i }+\hat{ j }+\hat{ k })$ is

The co-ordinates of the foot of perpendicular drawn from the origin to the line joining the points $(-9, 4, 5)$ and $(10, 0, -1)$ will be

If $AB = C$, then matrices $A,B,C$are

If the set $R=\{(a, b) ; a+5 b=42, a, b \in N\}$ has $m$ elements and $\sum_{n=1}^m\left(1+i^{n!}\right)=x+i y, $ where $I=\sqrt{-1},$ then the value of $m+x+y$ is :

The solution of linear programming problem, maximize $\mathrm{Z}=3 x_{1}+5 x_{2}$ subject to $3 x_{1}+$ $2 x_{2} \leq 18, x_{1} \leq 4, x_{2} \leq 6, x_{1} \geq 0, x_{2} \geq 0$ is $.....$

If$f(x) = \left\{ \begin{array}{l}\frac{{|x - a|}}{{x - a}},{\rm{when\,\,}}\,x \ne a\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,1,{\rm{when\,\,}}\,x = a\end{array} \right.,$ then

An $A.P.$, a $G.P.$ and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

If $x, y, z \in R^+$ are such that $z > y > x > 1$ , ${\log _y}x + {\log _x}y = \frac{5}{2}$ and ${\log _z}y + {\log _y}z = \frac{{10}}{3}$ then ${\log _x}z$ is equal to

$\int_0^{1/\sqrt 2 } {\frac{{{{\sin }^{ - 1}}x}}{{{{(1 - {x^2})}^{3/2}}}}dx = } $