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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Matrices
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If $a=1+2+4+.........$upto n terms,$b=1+3+9+.....$upto n terms,$c=1+5+25+.....$upto n terms then $\begin{vmatrix}a&2b&4c\\2&2&2\\2^n&3^n&5^n\end{vmatrix}=$

$\begin{array}{1 1}(A)\;(30)^n&(B)\;(10)^n\\(C)\;0&(D)\;2^n+3^n+5^n\end{array}$

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1 Answer

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We have $a=\large\frac{1(2^n-1)}{2-1}$
$b=\large\frac{1(3^n-1)}{3-1}$
$c=\large\frac{1(5^n-1)}{5-1}$
$\begin{vmatrix}a&2b&4c\\2&2&2\\2^n&3^n&5^n\end{vmatrix}=\begin{vmatrix}2^n-1&3^n-1&5^n-1\\2&2&2\\2^n&3^n&5^n\end{vmatrix}$
Applying $R_3-R_1=R_3$ and $R_2=\large\frac{1}{2}$$R_2$ we get
$\Rightarrow 2.\begin{vmatrix}2^n-1&3^n-1&5^n-1\\1&1&1\\1&1&1\end{vmatrix}$
On expanding along $R_1$ we get
$2^n-1(1-1)+3^n-1(1-1)+5^n-1(1-1)$
$\Rightarrow 0$
Hence (C) is the correct answer.
answered Apr 23, 2014 by sreemathi.v
 

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