Let a be the sum of all coefficients in the expansion of (1-2x+2x^2)^2023(3-4x^2+2x^3)^202…

Question

Let $a$ be the sum of all coefficients in the expansion of ${(1-2x+2{x}^{2})}^{2023}{(3-4{x}^{2}+2{x}^{3})}^{2024}$ and $b=\underset{x	o 0}{lim}(\frac{{\int }_{x}^{0}\frac{\log (1+t)}{{t}^{2024}+1}dt}{{x}^{2}})$. If the equation $c{x}^{2}+dx+e=0$ and $2b{x}^{2}+ax+4=0$ have a common root, where $c,d,e\in R$, then $d:c:$ e equals

Accepted Answer

Put $x=1$

$	herefore a=1$

$b=\underset{x	o 0}{lim}\frac{\int _{0}^{x}\frac{\ln (1+t)}{1+{t}^{2024}}dt}{{x}^{2}}$

Using L' HOPITAL Rule

$b=\underset{x	o 0}{lim}\frac{\ln (1+x)}{(1+{x}^{2024})}	imes \frac{1}{2x}=\frac{1}{2}$

Now, ${cx}^{2}+dx+e=0,{x}^{2}+x+4=0$

$(D$

Solutions · 1