The remainder when the determinant | lll 2014^ 2014 & 2015^ 2015 … - Vidyadip

MCQ

The remainder when the determinant $\left|\begin{array}{lll} 2014^{2014} & 2015^{2015} & 2016^{2016} \\ 2017^{2017} & 2018^{2018} & 2019^{2019} \\ 2020^{2020} & 2021^{2021} & 2022^{2022} \end{array}\right|$ is divided by $5$ is

A
$1$
B
$2$
C
$3$
✓
$4$

✓

Answer

Correct option: D.

$4$

d
(d)

Let

$D=\left|\begin{array}{lll}2014^{2014} & 2015^{2015} & 2016^{2016} \\ 2017^{2017} & 2018^{2018} & 2019^{2019} \\ 2020^{2020} & 2021^{2021} & 2022^{2022}\end{array}\right|$

$D=\left|\begin{array}{ccc}(2015-1)^{2014} & (2015)^{2015} & (2015+1)^{2016} \\ (2015+2)^{2017} & (2020-2)^{2018} & (2020-1)^{2019} \\ (2020)^{2020} & (2020+1)^{2021} & (2020+2)^{2022}\end{array}\right|$

Remainder when divided by $5$ , is

$D=\left|\begin{array}{ccc}1 & 0 & 1 \\2^{2017} & 2^{2018} & -1 \\0 & 1 & 2^{2022}\end{array}\right|$

$=1\left(2^{4040}+1\right)+2^{2017}$

$=(5-1)^{2020}+1+2(5-1)^{1008}$

$=1+1+2=4$

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