Let alpha, beta, gamma and delta be the coeff | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $y = \sqrt{x^3-1}$. The expression is $(x+y)^5 + (x-y)^5$. Using the binomial expansion,$(x+y)^5 + (x-y)^5 = 2[\binom{5}{0}x^5 + \binom{5}{2}x

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(A) Let $y = \sqrt{x^3-1}$. The expression is $(x+y)^5 + (x-y)^5$. Using the binomial expansion,$(x+y)^5 + (x-y)^5 = 2[\binom{5}{0}x^5 + \binom{5}{2}x^3y^2 + \binom{5}{4}xy^4]$. Substituting $y^2 = x^3-1$ and $y^4 = (x^3-1)^2 = x^6 - 2x^3 + 1$: $= 2[1 \cdot x^5 + 10 \cdot x^3(x^3-1) + 5 \cdot x(x^6 - 2x^3 + 1)]$. $= 2[x^5 + 10x^6 - 10x^3 + 5x^7 - 10x^4 + 5x]$. $= 10x^7 + 2x^5 - 20x^4 - 20x^3 + 10x$. Comparing coefficients: $\alpha = 10, \beta = 2, \gamma = -20, \delta = 10$. Given equations: $10u + 2v = 18$ and $-20u + 10v = 20$. Divide the first by $2$: $5u + v = 9$. Divide the second by $10$: $-2u + v = 2$. Subtracting the equations: $(5u + v) - (-2u + v) = 9 - 2$ $\Rightarrow 7u = 7$ $\Rightarrow u = 1$. Substituting $u=1$ into $5u+v=9$ gives $v=4$. Therefore,$u+v = 1+4 = 5$.

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

Similar Questions

The coefficient of $x^{101}$ in the expression $(5+x)^{500} + x(5+x)^{499} + x^{2}(5+x)^{498} + \ldots + x^{500}$ for $x > 0$ is:

The coefficient of $x^6$ in the power series expansion of $\frac{x^4-12x^2+7}{(x^2+1)^3}$ is

If $(1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + .... + a_{12}x^{12}$,then the expression $a_2 + a_4 + a_6 + .... + a_{12}$ has the value

The sum of the coefficients of all odd degree terms in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5$,where $x > 1$,is:

The coefficient of $x^n$ in the expansion of $(1 + x + x^2 + ....)^{-n}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module