KVPY SX SB Exam 2015 Question Paper

Question : 90 of 120

Marks: +1, -0

The remainder when the determinant Determinant

|\table 2014^2014, 2015^2015 ,2016^2016;2017^2017, 2018^2018, 2019^2019;2020^2020, 2021^ 2021,2022^2022|

is divided by 5 is -

Solution:

Determinant =

|\table (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|

Determinant of remainder =

|\table ( 1)^2014, 0 ,1; 2 ^2017, 2 ^2018, ( -1)^2019;0, 1 ^ 2021, 2 ^2022|

= 1\{2^4040 + 1\} + 1 \{2^2017\}

{\{(4)2020 + 1\}} + 2 . 2^2016

⇒ (5 -1)^2020 + 1 + 2.4^1008

= (5- 1)^2020 + 1 + 2. (5-1)^1008

remainder,

(-1)^2020 + 1 + 2. (- 1)^1008

= 1 + 1 + 2 = 4

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