Sum of co-efficients of all odd degree terms in expansion of {left( {x + √ {{x^3}

English

Gujarati

Hindi

7.Binomial Theorem

hard

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5},\left( {x > 1} \right)$

$0$

$1$

$2$

$-1$

(JEE MAIN-2018)

Solution

$(C)$ Since we know that,

${(x+a)^{5}+(x-a)^{5}}$

${=2\left[^{5} C_{0} x^{5}+^{5} C_{2} x^{3} \cdot a^{2}+^{5} C_{4} x \cdot a^{4}\right]}$

$\therefore \quad(x+\sqrt{x^{3}-1})^{5}+(x-\sqrt{x^{3}-1})^{5}$

$=2\left[^{5} \mathrm{C}_{0} \mathrm{x}^{5}+^{5} \mathrm{C}_{2} \mathrm{x}^{3}\left(\mathrm{x}^{3}-1\right)+^{5} \mathrm{C}_{4} \mathrm{x}\left(\mathrm{x}^{3}-1\right)^{2}\right]$

$\Rightarrow 2\left[x^{5}+10 x^{6}-10 x^{3}+5 x^{7}-10 x^{4}+5 x\right]$

$\therefore $ Sum of coefficients of odd degree terms $=2$

Standard 11

Mathematics

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

hard

(JEE MAIN-2024)

The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + ….. + (a + nb)C_n$ is where $Cr's$ denotes combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$

normal

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then

medium

If $f(y) = 1 – (y – 1) + {(y – 1)^2} – {(y – 1)^{^3}} + … – {(y – 1)^{17}},$ then the coefficient of $y^2$ in it is

hard

(AIEEE-2012)

If $C_r= ^{100}{C_r}$ , then $1.C^2_0 – 2.C^2_1 + 3.C^2_3 – 4.C^2_0 + 5.C^2_4 – …. + 101.C^2_{100}$ is equal to

normal

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5},\left( {x > 1} \right)$

Solution

Similar Questions

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + ….. + (a + nb)C_n$ is where $Cr's$ denotes combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then

If $f(y) = 1 – (y – 1) + {(y – 1)^2} – {(y – 1)^{^3}} + … – {(y – 1)^{17}},$ then the coefficient of $y^2$ in it is

If $C_r= ^{100}{C_r}$ , then $1.C^2_0 – 2.C^2_1 + 3.C^2_3 – 4.C^2_0 + 5.C^2_4 – …. + 101.C^2_{100}$ is equal to

Start a Free Trial Now