Question

If ${}^6{C}_n+2^6{C}_{n+1}{+}^6{C}_{n+2}{>}^8{C}_3$,then the quadratic equations whose roots are $\alpha ,\beta$ and ${\alpha }^{n-1},{\beta }^{n-1}$ have

• A. 2 common root
• B. 1 common root
• C. no common roots
• D. imaginary roots

Answer 1

Answer: (A)

2 common root

${}^6{C}_n+2^6{C}_{n+1}{+}^6{C}_{n+2}{>}^8{C}_3$

${}^6{C}_n{+}^6{C}_{n+1}{+}^6{C}_{n+1}{+}^6{C}_{n+2}$

${\Rightarrow }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$

${\Rightarrow }^7{C}_{n+1}{+}^7{C}_{n+2}{=}^8{C}_{n+2}$

${\Rightarrow }^8{C}_{n+2}{>}^8{C}_3$

$\Rightarrow n+2>3$

$\Rightarrow n>1$

$\therefore n=2$

So, roots are $\alpha ,\beta$;${\alpha }^{n-1},{\beta }^{n-1}$

$\alpha ,\beta$; ${\alpha }^{{}^{\prime }}{\beta }^{{}^{\prime }}$

$\therefore$ Both roots are equal. So, 2 common roots.