Question

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

• A. 1
• B. 0
• C. 2
• D. None of these

Answer 1

Answer: (B)

${\Delta }_r=\left|\begin{array}{ccc}2r-1& {}^m{C}_r& 1\\ m^2-1& 2^m& m+1\\ {\sin }^2\left(m^2\right)& {\sin }^2(m)& {\sin }^2(m+1)\end{array}\right|$
$\therefore \sum _{r=0}^m{\Delta }_r=\left|\begin{array}{ccc}\sum _{r=0}^m(2r-1)& \sum _{r=0}^m{}^m{C}_r& \sum _{r=0}^{m}1\\ m^2-1& 2^m& m+1\\ {\sin }^2\left(m^2\right)& {\sin }^2(m)& {\sin }^2(m+1)\end{array}\right|$
$=\left|\begin{array}{ccc}{m}^2-1& 2^m& m+1\\ m^2-1& 2^m& m+1\\ {\sin }^2\left(m^2\right)& {\sin }^2(m)& {\sin }^2(m+1)\end{array}\right|$
$=0(\because$ two rows are identical)