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Determinants

Question
CBSEENMA12034482
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If a, b, c are in A.P, then the determinant
open vertical bar table row cell straight x plus 2 end cell cell space space space straight x plus 3 end cell cell space space space straight x plus 2 straight a end cell row cell straight x plus 3 end cell cell space space straight x plus 4 end cell cell space space straight x plus 2 straight b end cell row cell straight x plus 4 end cell cell space space straight x plus 5 end cell cell space space space straight x plus 2 straight c end cell end table close vertical bar

  • 0

  • 1

  • v

  • 2x

Solution

A.

0

Since a, b, c are in A.P.
therefore space space space space space space space 2 straight b space equals space straight a plus straight c space                                                     ...(1)
Let space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space space space space straight x plus 3 end cell cell space space space straight x plus 2 straight a end cell row cell straight x plus 3 end cell cell space space space straight x plus 4 end cell cell space space straight x plus 2 straight b end cell row cell straight x plus 4 end cell cell space space straight x plus 5 end cell cell space space straight x plus 2 straight c end cell end table close vertical bar space space space space space space space space space space equals space 1 half open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 3 end cell cell space space straight x plus 2 straight a end cell row cell 2 straight x plus 6 end cell cell space space space 2 straight x plus 8 end cell cell space space space 2 straight x plus 4 straight b end cell row cell straight x plus 4 end cell cell space straight x plus 5 end cell cell space space straight x plus 2 straight c end cell end table close vertical bar comma space by space straight R subscript 2 space rightwards arrow space 2 straight R subscript 2 space space space space space space space space space space equals space 1 half open vertical bar table row cell straight x plus 2 end cell cell space space space space space straight x plus 3 end cell cell space space space straight x plus 2 straight a end cell row 0 cell space 0 end cell cell space space space space space 4 straight b minus 2 straight a minus 2 straight c end cell row cell straight x plus 4 end cell cell space space space straight x plus 5 end cell cell straight x plus 2 straight c end cell end table close vertical bar comma space space by space straight R subscript 2 rightwards arrow straight R subscript 2 space minus straight R subscript 1 space minus straight R subscript 3 space space space space space space space space space space equals space 1 half open vertical bar table row cell straight x plus 2 end cell cell space space space space space straight x plus 3 end cell cell space space space straight x plus 2 straight a end cell row 0 cell space space 0 end cell 0 row cell straight x plus 4 end cell cell space space straight x plus 5 end cell cell straight x plus 2 straight c end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets space space space space space space space space space space equals space 1 half cross times 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space each space element space of space second space row space is space zero close square brackets space space space space space space space space space space space equals space 0 therefore space space space space space left parenthesis straight A right parenthesis space is space correct space answer. space space space space space space

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