Question

Assertion :If $a_i,b_iϵN$, for i=1, 2, 3 and
$\Delta \left(x\right)=\left|\begin{array}{ccc}{(1+x)}^{a_1{b}_1}& {(1+x)}^{a_1{b}_2}& {(1+x)}^{a_1{b}_3}\\ {(1+x)}^{a_2{b}_1}& {(1+x)}^{a_2{b}_2}& {(1+x)}^{a_2{b}_3}\\ {(1+x)}^{a_3{b}_1}& {(1+x)}^{a_3{b}_2}& {(1+x)}^{a_3{b}_3}\end{array}\right|$
then coefficient of x in expansion of $\Delta \left(x\right)$ is 0. Reason: If $P\left(x\right)={(1+x)}^n$, $nϵN$ then coefficient of x in the expansion of P(x) is $P^{\prime }\left(0\right)$.

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

We have
$P\left(x\right)={(1+x)}^n=1+\left(\begin{array}{c}n\\ 1\end{array}\right)x+\left(\begin{array}{c}n\\ 2\end{array}\right)x^2+...+\left(\begin{array}{c}n\\ n\end{array}\right)x^n\Rightarrow P^{\prime }\left(x\right)=\left(\begin{array}{c}n\\ 1\end{array}\right)+\left(\begin{array}{c}n\\ 2\end{array}\right)x+...+\left(\begin{array}{c}n\\ n\end{array}\right)x^{n-1}$
$\Rightarrow P^{\prime }\left(0\right)=\left(\begin{array}{c}n\\ 1\end{array}\right)=$ coefficient of x in the expansion of $P\left(x\right)$
Therefore, reason is true
Note $\Delta \left(x\right)$ consists of terms of the form ${(1+x)}^n$
Thus, coefficient of $x$ in $\Delta \left(x\right)={\Delta }^{\prime }\left(0\right)$
But
${\Delta }^{\prime }\left(0\right)=\left|\begin{array}{ccc}{a}_1{b}_1& a_1{b}_2& a_1{b}_3\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|+\left|\begin{array}{ccc}1& 1& 1\\ a_2{b}_1& a_2{b}_2& a_2{b}_3\\ 1& 1& 1\end{array}\right|+\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ a_3{b}_1& a_3{b}_2& a_3{b}_3\end{array}\right|=0$