Question

I. If A,B,C are angles of angle and $\left|\begin{array}{ccc}1& 1& 1\\ 1+sinA& 1+sinB& 1+sinC\\ sinA+si{n}^{2}A& sinB+si{n}^{2}B& sinC+si{n}^{2}C\end{array}\right|$ =0 then triangle is isosceles
II. lf $a=1+2+4+---$ upto $n$ terms $b=1+3+9+---$ up to $n$ terms $c=1+5+25+----$up to $n$ terms then $\Delta \left|\begin{array}{ccc}a& 2b& 4c\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$ =0

• A. I, II both are true
• B. only I is true
• C. only II is true
• D. neither of them are true

Answer 1

The correct option is A I, II both are true

$\left|\begin{array}{ccc}1& 1& 1\\ 1+sinA& 1+sinB& 1+sinC\\ sinA+si{n}^{2}A& sinB+si{n}^{2}B& sinC+si{n}^{2}C\end{array}\right|$=0
$\Rightarrow \sin B{\sin }^{2}C-\sin C{\sin }^{2}B+\sin C{\sin }^{2}A-\sin A{\sin }^{2}C+\sin A{\sin }^{2}B-\sin B{\sin }^{2}A=0$
$\Rightarrow (\sin A-\sin B)(\sin B-\sin C)(\sin C-\sin A)=0$
$\Rightarrow (\sin A-\sin B)=0$ or $(\sin B-\sin C)=0$ or $(\sin C-\sin A)=0$
$\Rightarrow A=B$ or $B=C$ or $C=A$
$a=1+2+4+---$ upto $n$ terms
$\Rightarrow a=2^n-1$
$b=1+3+9+---$ upto $n$ terms
$\Rightarrow b=\frac{3^n-1}{2}$
$c=1+5+25+----$ upto $n$ terms
$\Rightarrow c=b=\frac{5^n-1}{4}$
$\Delta =\left|\begin{array}{ccc}a& 2b& 4c\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=\left|\begin{array}{ccc}{2}^n-1& 3^n-1& 5^n-1\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=\left|\begin{array}{ccc}{2}^n& 3^n& 5^n\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|+\left|\begin{array}{ccc}-1& -1& -1\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=0-2\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 2^n& 3^n& 5^n\end{array}\right|=0$