If sum of coefficients in expansion of {({α ^2}{x^2} - 2α {rm{ }}x + 1)^{51}} vanishes, then value

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7.Binomial Theorem

medium

If the sum of the coefficients in the expansion of ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ vanishes, then the value of $\alpha $ is

$2$

$-1$

$1$

$-2$

(IIT-1991)

Solution

(c) The sum of the coefficients of the polynomial ${({\alpha ^2}{x^2} – 2\alpha \,x + 1)^{51}}$is obtained by putting $x = 1$ in ${({\alpha ^2}{x^2} – 2\alpha \,x + 1)^{51}}$.

Therefore by hypothesis ${({\alpha ^2} – 2\alpha + 1)^{51}} = 0 \Rightarrow \alpha = 1$

Standard 11

Mathematics

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Co-efficient of $\alpha ^t$ in the expansion of,

$(\alpha + p)^{m – 1} + (\alpha + p)^{m – 2} (\alpha + q) + (\alpha + p)^{m – 3} (\alpha + q)^2 + …… (\alpha + q)^{m – 1}$

where $\alpha \ne – q$ and $p \ne q$ is :

normal

If the sum of the coefficients in the expansion of ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ vanishes, then the value of $\alpha $ is

Solution

Similar Questions

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If ${(1 – x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + …. + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + …. + {a_{2n}} = $

If ${a_r}$ is the coefficient of ${x^r}$, in the expansion of ${(1 + x + {x^2})^n}$, then ${a_1} – 2{a_2} + 3{a_3} – …. – 2n\,{a_{2n}} = $

Co-efficient of $\alpha ^t$ in the expansion of, $(\alpha + p)^{m – 1} + (\alpha + p)^{m – 2} (\alpha + q) + (\alpha + p)^{m – 3} (\alpha + q)^2 + …… (\alpha + q)^{m – 1}$ where $\alpha \ne – q$ and $p \ne q$ is :

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Co-efficient of $\alpha ^t$ in the expansion of,

$(\alpha + p)^{m – 1} + (\alpha + p)^{m – 2} (\alpha + q) + (\alpha + p)^{m – 3} (\alpha + q)^2 + …… (\alpha + q)^{m – 1}$

where $\alpha \ne – q$ and $p \ne q$ is :