Question

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-3 x+3=0$. If $S$ is the algebraic sum of the coefficients and $T$ is the term independent of $x$ in $\left((\alpha+1) x +\frac{(\beta+1)}{ x }\right)^{20}$, then

• A. $S=5^{20}$
• B. $S =3^{20}$
• C. $T ={ }^{20} C _{10} \cdot 5^{10}$
• D. $T ={ }^{20} C _{10} \cdot 7^{10}$

Answer 1

Answer: (A), (D)

$\left((\alpha+1) x+\frac{(\beta+1)}{x}\right)^{20}$
$S=(\alpha+1+\beta+1)^{20}=5^{20}$
For the term independent of $x$
$ T ={ }^{20} C _{ r } \cdot(\alpha+1)^{20- r } \cdot x ^{20- r } \cdot \frac{(\beta+1)^{ r }}{ x ^{ r }} $
$ 20-2 r =0 \Rightarrow r =10 $
$\therefore T ={ }^{20} C _{10} \cdot(\alpha+1)^{10} \cdot(\beta+1)^{10}={ }^{20} C _{10} \cdot(\alpha \beta+\alpha+\beta+1)^{10}={ }^{20} C _{10} \cdot 7^{10}$