Question

Assertion :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 ${\alpha }^{n-1},{\beta }^{n-1}$ (for least value of n) have two common roots. Reason: Equation $a{x}^2+bc+c=0$ and $a^1{x}^2+b^{1}x+c^1=0$ have two roots common then $(b{c}^1-b^{1}c)(a{b}^1-a^{1}b)=(c{a}^1-c^{1}a{)}^2$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Statement-1 is true but statement-2 is false. ${}^6{C}_n+{2.}^6{C}_{n+1}{+}^6{C}_{n+2}{>}^8{C}_3$
$\Rightarrow {(}^6{C}_n{+}^6{C}_{n+1})+{(}^6{C}_{n+1}{+}^6{C}_{n+2}){>}^8{C}_3$.
${\Rightarrow }^7{C}_{n+1}{+}^7{C}_{n+2}{>}^8{C}_3$
${\Rightarrow }^8{C}_{n+2}{>}^8{C}_3$
$\Rightarrow n+2>3\Rightarrow n+2=4\Rightarrow n=2$
Hence, quadratic equation having roots $\alpha ,\beta$ and ${\alpha }^{n-1},{\beta }^{n-1}$ are identical.
Statement-2 is true if one root common.