Question

If ${\Delta }_r=\left|\begin{array}{ccc}2r-1& {}^m{C}_r& 1\\ m^2-1& 2^m& m+1\\ si{n}^2\left(m^2\right)& si{n}^2\left(m\right)& si{n}^2\left(m+1\right)\end{array}\right|$ then value of $\sum _{r=0}^m{\Delta }_r$ is

• A. 0
• B. 4
• C. 3
• D. 1

Answer 1

Answer: (A)

${\Delta }_r=\left|\begin{array}{ccc}2r-1& {}^m{C}_r& 1\\ m^2-1& 2^m& m+1\\ si{n}^2\left(m^2\right)& si{n}^2\left(m\right)& si{n}^2\left(m+1\right)\end{array}\right|$
$\Rightarrow \sum _{r=0}^m{\Delta }_r=\left|\begin{array}{ccc}\sum _{r=0}^m\left(2r-1\right)& \sum _{r=0}^m{}^m{C}_r& \sum _{r=0}^{m}1\\ m^2-1& 2^m& m+1\\ si{n}^2\left(m^2\right)& si{n}^2\left(m\right)& si{n}^2\left(m+1\right)\end{array}\right|$
$=\left|\begin{array}{ccc}{m}^2-1& 2^m& m+1\\ m^2-1& 2^m& m+1\\ si{n}^2\left(m^2\right)& si{n}^2\left(m\right)& si{n}^2\left(m+1\right)\end{array}\right|=0$