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Determinants

Question
CBSEENMA12034450
Wired Faculty App

Using the properties of determinants, show that:
open vertical bar table row 1 cell space space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 1 cell space straight y plus straight z end cell cell space space straight y squared plus straight z squared end cell row 1 cell space straight z plus straight x end cell cell space space straight z squared plus straight x squared end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis

Solution

Let space space increment space equals space open vertical bar table row 1 cell space space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 1 cell space space straight y plus straight z end cell cell space space space straight y squared plus straight z squared end cell row 1 cell space straight z plus straight x end cell cell space space space straight z squared plus straight x squared end cell end table close vertical bar space space space space space space space space space space
           equals space open vertical bar table row 1 cell space space straight x plus straight y end cell cell space space space straight x squared plus straight y squared end cell row 0 cell space space straight z minus straight x end cell cell space space space straight z squared minus straight x squared end cell row 0 cell space space straight z minus straight y end cell cell space space space straight z squared minus straight y squared end cell end table close vertical bar space by space straight R subscript 2 minus straight R subscript 1 comma space space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight z minus straight x end cell cell space space space space straight z squared minus straight x squared end cell row cell straight z minus straight y end cell cell space space space space straight z squared minus straight y squared end cell end table close vertical bar comma space by space space expanding space with space straight C subscript 1 equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space open vertical bar table row 1 cell space space space straight z plus straight x end cell row 1 cell space space space space straight z plus straight y end cell end table close vertical bar space equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y minus straight z minus straight x right parenthesis equals space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight y minus straight x right parenthesis equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis

Question
CBSEENMA12034458
Wired Faculty App

Using properties of determinants, prove that
open vertical bar table row straight x cell space straight x squared end cell cell space yz end cell row straight y cell space straight y squared end cell cell space zx end cell row straight z cell space straight z squared end cell cell space xy end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis xy plus yz plus zx right parenthesis

Solution

L.H.S. = open vertical bar table row straight x cell space straight x squared end cell cell space yz end cell row straight y cell space straight y squared end cell cell space zx end cell row straight z cell space straight z squared end cell cell space xy end cell end table close vertical bar
        equals space 1 over xyz open vertical bar table row cell straight x squared end cell cell space space straight x cubed end cell cell space space xyz end cell row cell straight y squared end cell cell space space straight y cubed end cell cell space space xyz end cell row cell straight z squared end cell cell space space straight z cubed end cell cell space xyz end cell end table close vertical bar space equals space xyz over xyz space open vertical bar table row cell straight x squared end cell cell space straight x cubed end cell cell space space 1 end cell row cell straight y squared end cell cell space straight y cubed end cell cell space 1 end cell row cell straight z squared end cell cell space straight z cubed end cell cell space 1 end cell end table close vertical bar space equals space open vertical bar table row 1 cell space straight x squared end cell cell space straight x cubed end cell row 1 cell space straight y squared end cell cell space straight y cubed end cell row 1 cell space straight z squared end cell cell space straight z cubed end cell end table close vertical bar equals space open vertical bar table row 1 cell space space space straight x squared end cell cell space space space straight x cubed end cell row 0 cell space space space space straight y squared minus straight x squared end cell cell space space space space space straight y cubed minus straight x cubed end cell row 0 cell space space space straight z squared minus straight x squared end cell cell space space straight z cubed minus straight x cubed end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 2 comma space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight y squared minus straight x squared end cell cell space space space space straight y cubed minus straight x cubed end cell row cell straight z squared minus straight x squared end cell cell space space space space straight z cubed minus straight x cubed end cell end table close vertical bar space equals space space open vertical bar table row cell left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight y plus straight x right parenthesis end cell cell left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight y squared plus straight x squared plus yx right parenthesis end cell row cell left parenthesis straight z minus straight x right parenthesis space left parenthesis straight z plus straight x right parenthesis end cell cell left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z squared plus straight x squared plus zx right parenthesis end cell end table close vertical bar equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis space open vertical bar table row cell straight y plus straight x end cell cell space space space space space space space space straight y squared plus straight x squared plus yx end cell row cell straight z plus straight x end cell cell space space space space space space space straight z squared plus straight x squared plus zx end cell end table close vertical bar equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open vertical bar table row cell straight y plus straight x end cell cell space straight y squared plus straight x squared plus yx end cell row cell straight z minus straight y end cell cell space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y right parenthesis space plus straight x space left parenthesis straight z minus straight y right parenthesis end cell end table close vertical bar space equals space left parenthesis straight y minus straight x right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight z minus straight y right parenthesis space open vertical bar table row cell straight y plus straight x end cell cell space space space space space space space space straight y squared plus straight x squared plus yx end cell row 1 cell space space space space space space straight z plus straight y plus straight x end cell end table close vertical bar equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space open square brackets yz plus straight y squared plus xy plus zx plus xy plus straight x squared minus straight y squared minus straight x squared minus xy close square brackets equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis xy plus yz plus zx right parenthesis equals space straight R. straight H. straight S.

Question
CBSEENMA12034459
Wired Faculty App

Prove that
open vertical bar table row cell straight a cubed plus 1 end cell cell space space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space straight b squared end cell cell space space straight b end cell row cell straight c cubed plus 1 end cell cell straight c squared end cell cell space straight c end cell end table close vertical bar space equals space minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a space straight b space straight c space plus space 1 right parenthesis.

Solution

Let space increment space equals space open vertical bar table row cell straight a cubed plus 1 end cell cell space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space space straight b squared end cell cell space straight b end cell row cell straight c cubed plus 1 end cell cell space straight c squared end cell cell space space straight c end cell end table to the power of blank close vertical bar space equals space open vertical bar table row cell straight a cubed end cell cell space space straight a squared end cell cell space straight a end cell row cell straight b cubed end cell cell space straight b squared end cell straight b row cell straight c cubed end cell cell space straight c squared end cell straight c end table close vertical bar space plus space open vertical bar table row 1 cell space space straight a squared end cell cell space space straight a end cell row 1 cell space straight b squared end cell cell space space straight b end cell row 1 cell space straight c squared end cell cell space space straight c end cell end table close vertical bar space space space space space equals negative open vertical bar table row straight a cell space space straight a squared end cell cell space straight a cubed end cell row straight b cell space straight b squared end cell cell space straight b cubed end cell row straight c cell space straight c squared end cell cell space straight c cubed end cell end table close vertical bar minus 1 open vertical bar table row 1 cell space space straight a end cell cell space space straight a squared end cell row 1 cell space straight b end cell cell space space straight b squared end cell row 1 cell space space straight c end cell cell space space straight c squared end cell end table close vertical bar space space space space equals negative abc space open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar minus open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar space equals space minus left parenthesis abc plus 1 right parenthesis space open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar equals negative left parenthesis abc plus 1 right parenthesis space open vertical bar table row 1 cell space space straight a end cell cell space space space straight a squared end cell row 0 cell space space straight b minus straight a end cell cell space space space space straight b squared minus straight a squared end cell row 0 cell space space straight c minus straight a end cell cell space space space straight c squared minus straight a squared end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1 equals space minus left parenthesis abc plus 1 right parenthesis thin space open vertical bar table row cell straight b minus straight a end cell cell space space space space space straight b squared minus straight a squared end cell row cell straight c minus straight a end cell cell straight c squared minus straight a squared end cell end table close vertical bar equals space minus left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space open vertical bar table row 1 cell space space space space space space straight b plus straight a end cell row 1 cell space space space space space space straight c plus straight a end cell end table close vertical bar equals space minus left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b minus straight a right parenthesis equals negative left parenthesis abc plus 1 right parenthesis thin space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b right parenthesis equals negative left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis abc plus 1 right parenthesis therefore space space space space space space open vertical bar table row cell straight a cubed plus 1 end cell cell space space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space straight b squared end cell cell space straight b end cell row cell straight c cubed plus 1 end cell cell space straight c squared end cell cell space straight c end cell end table close vertical bar space equals space minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis abc plus 1 right parenthesis

Question
CBSEENMA12034460
Wired Faculty App

If x = – 9 is a root of
open vertical bar table row straight x cell space space 3 end cell cell space space 7 end cell row 2 cell space space straight x end cell cell space space 2 end cell row 7 cell space 6 end cell cell space straight x end cell end table close vertical bar space equals 0 comma
find the other roots.

Solution

The given equation is
open vertical bar table row straight x cell space 3 end cell cell space 7 end cell row 2 cell space straight x end cell cell space 2 end cell row 7 cell space 6 end cell cell space straight x end cell end table close vertical bar space equals space 0
or          open vertical bar table row cell straight x plus 9 end cell cell straight x plus 9 end cell cell straight x plus 9 end cell row 2 straight x 2 row 7 6 straight x end table close vertical bar space equals 0 comma space space by space space space straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3
or space left parenthesis straight x plus 9 right parenthesis space open vertical bar table row 1 cell space 1 end cell cell space 1 end cell row 2 cell space straight x end cell cell space 2 end cell row 7 cell space 6 end cell cell space straight x end cell end table close vertical bar space equals space 0 or space left parenthesis straight x plus 9 right parenthesis thin space open vertical bar table row 1 0 cell space space space 0 end cell row cell 2 space end cell cell space space space straight x minus 2 end cell cell space space space 0 end cell row 7 cell negative 1 end cell cell straight x minus 7 end cell end table close vertical bar space equals 0 comma space by space straight C subscript 2 minus straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 space or space left parenthesis straight x plus 9 right parenthesis space open square brackets left parenthesis 1 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x minus 7 right parenthesis close square brackets space equals space 0 space space space space space space space space space space space space space space space left square bracket Product space of space diagonal space elements right square bracket or space space left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x minus 7 right parenthesis thin space left parenthesis straight x plus 9 right parenthesis space equals space 0 therefore space space space space straight x space equals space 2 comma space space 7 comma space space minus 9 because space space space straight x space equals space minus 9 space is space given space root therefore space space other space roots space are space 2 comma space 7.

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