$(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + ...... (\alpha + q)^{m - 1}$
વિસ્તરણમાં $\alpha ^t$ નો સહગુણક મેળવો.
જ્યાં $\alpha \ne - q$ અને $p \ne q$
$(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + ...... (\alpha + q)^{m - 1}$
વિસ્તરણમાં $\alpha ^t$ નો સહગુણક મેળવો.
જ્યાં $\alpha \ne - q$ અને $p \ne q$
- A
$\frac{{^m{C_t}\,\,\left( {{p^t}\, - \,{q^t}} \right)}}{{p\, - \,q}}$
- B
$\frac{{^m{C_t}\,\,\left( {{p^{m\, - \,t}}\, - \,{q^{m\, - \,t}}} \right)}}{{p\, - \,q}}$
- C
$\frac{{^m{C_t}\,\,\left( {{p^t}\, + \,{q^t}} \right)}}{{p\, - \,q}}$
- D
$\frac{{^m{C_t}\,\,\left( {{p^{m\, - \,t}}\, + \,{q^{m\, - \,t}}} \right)}}{{p\, - \,q}}$
Similar Questions
જો ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ ના સહગુણકનો સરવાળો શૂન્ય હોય તો $\alpha $ મેળવો.
જો ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ ના સહગુણકનો સરવાળો શૂન્ય હોય તો $\alpha $ મેળવો.
- [IIT 1991]
$\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = $
$\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = $
- [JEE MAIN 2017]
$(1+ x )^{ n +2}$ ના દ્રીપદી વિસ્તરણમાં $1:3:5$ ગુણોત્તરમાં હોય તેવા ત્રણ ક્રમિક પદોના સહગુણકોનો સરવાળો $........$ થાય.
$(1+ x )^{ n +2}$ ના દ્રીપદી વિસ્તરણમાં $1:3:5$ ગુણોત્તરમાં હોય તેવા ત્રણ ક્રમિક પદોના સહગુણકોનો સરવાળો $........$ થાય.
- [JEE MAIN 2023]
જો ${\left( {1 + x} \right)^{10}} = \sum\limits_{r = 0}^{10} {{C_r}{x^r}} $ ,${\left( {1 + x} \right)^7} = \sum\limits_{r = 0}^7 {{d_r}{x^r}} $ અને $P = \sum\limits_{r = 0}^5 {{C_{2r}}} $ તથા $Q = \sum\limits_{r = 0}^3 {{d_{2r + 1}}} $ ,હોય તો $\frac{P}{{2Q}}$ ની કિમત મેળવો
જો ${\left( {1 + x} \right)^{10}} = \sum\limits_{r = 0}^{10} {{C_r}{x^r}} $ ,${\left( {1 + x} \right)^7} = \sum\limits_{r = 0}^7 {{d_r}{x^r}} $ અને $P = \sum\limits_{r = 0}^5 {{C_{2r}}} $ તથા $Q = \sum\limits_{r = 0}^3 {{d_{2r + 1}}} $ ,હોય તો $\frac{P}{{2Q}}$ ની કિમત મેળવો
$\left( \begin{array}{l}30\\0\end{array} \right)\,\left( \begin{array}{l}30\\10\end{array} \right) - \left( \begin{array}{l}30\\1\end{array} \right)\,\left( \begin{array}{l}30\\11\end{array} \right)$ + $\left( \begin{array}{l}30\\2\end{array} \right)\,\left( \begin{array}{l}30\\12\end{array} \right) + ....... + \left( \begin{array}{l}30\\20\end{array} \right)\,\left( \begin{array}{l}30\\30\end{array} \right) = .$ . ..
$\left( \begin{array}{l}30\\0\end{array} \right)\,\left( \begin{array}{l}30\\10\end{array} \right) - \left( \begin{array}{l}30\\1\end{array} \right)\,\left( \begin{array}{l}30\\11\end{array} \right)$ + $\left( \begin{array}{l}30\\2\end{array} \right)\,\left( \begin{array}{l}30\\12\end{array} \right) + ....... + \left( \begin{array}{l}30\\20\end{array} \right)\,\left( \begin{array}{l}30\\30\end{array} \right) = .$ . ..
- [IIT 2005]