Question

If ${}^7{C}_r+3^7{C}_{r+1}+3^7{C}_{r+2}{+}^7{C}_{r+3}{>}^{10}{C}_4$, then the quadratic equation whose roots are $\alpha ,\beta$ and ${\alpha }^{r-1},{\beta }^{r-1}$ have

• A. no common roots
• B. only one common root
• C. two common root
• D. none of these

Answer 1

The correct option is C two common root

${}^7{C}_r+3^7{C}_{r+1}+3^7{C}_{r+2}{+}^7{C}_{r+3}{>}^{10}{C}_4$
$\Rightarrow {}^7{C}_r+{}^7{C}_{r+1}+2({}^7{C}_{r+1}+{}^7{C}_{r+2})+{}^7{C}_{r+2}+{}^7{C}_{r+3}>{}^{10}{C}_4$ $\Rightarrow {}^8{C}_{r+1}+2{}^8{C}_{r+2}+{}^8{C}_{r+3}>{}^{10}{C}_4$
$\Rightarrow {}^{10}{C}_{r+3}>{}^{10}{C}_4$
$\Rightarrow r+3=5\Rightarrow r=2$
Now roots of two different equation are $\alpha ,\beta$ and ${\alpha }^{r-1},{\beta }^{r-1}$
i.e. ${\alpha }^1,{\beta }^1$ and ${\alpha }^{2-1},{\beta }^{2-1}$, i.e. ${\alpha }^1,{\beta }^1$ and ${\alpha }^1,{\beta }^1$ $\Rightarrow$ both roots are same
So Number of common roots are $2$.