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Show that:
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight q plus straight r end cell cell space straight r plus straight p end cell cell space straight p plus straight q end cell row cell straight y plus straight z end cell cell space straight z plus straight x end cell cell space straight x plus straight y end cell end table close vertical bar space equals space 2 open vertical bar table row straight a cell space space space space straight b end cell cell space space straight c end cell row straight p cell space space space straight q end cell cell space space straight r end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar$

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102.

Show that:
$open vertical bar table row straight a cell space space straight b end cell cell space space straight c end cell row cell straight a plus 2 straight x end cell cell space straight b plus 2 straight y end cell cell space space space straight c plus 2 space straight z end cell row straight x straight y cell space straight z end cell end table close vertical bar space equals space 0$

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103. Factorise the determinant:

open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight a cell space space straight b end cell cell space straight c end cell row cell straight a squared end cell cell space space straight b squared end cell cell space space straight c squared end cell end table close vertical bar.

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Long Answer Type

104.

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

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Short Answer Type

105.

By using properties of determinants, show that:
$open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space straight b end cell cell space space space straight b squared end cell row 1 cell space space straight c end cell cell space space space straight c squared end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis space left parenthesis straight c minus straight a right parenthesis$

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106.

Prove the following identities:
$open vertical bar table row 1 cell space space space 1 end cell cell space space space 1 end cell row straight a cell space space straight b end cell cell space space space straight c end cell row cell straight a cubed end cell cell space space space straight b cubed end cell cell space space space straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a plus straight b plus straight c right parenthesis$

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107.

Prove the following identities:
$open vertical bar table row 1 cell space space space straight a end cell cell space space space straight a cubed end cell row 1 cell space space straight b end cell cell space space straight b cubed end cell row 1 cell space space straight c end cell cell space space straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a plus straight b plus straight c right parenthesis$

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108.
Prove the following identities:
$open vertical bar table row 1 cell space space space straight x end cell cell space space space straight x cubed end cell row 1 cell space space straight y end cell cell space space straight y cubed end cell row 1 cell space straight z end cell cell space space straight z cubed end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space left parenthesis straight x plus straight y plus straight z right parenthesis$

Let $increment space equals space open vertical bar table row 1 cell space space space space straight x end cell cell space space space straight x cubed end cell row 1 cell space space space straight y end cell cell space space space straight y cubed end cell row 1 cell space space straight z end cell cell space space space straight z cubed end cell end table close vertical bar$

$equals space open vertical bar table row 1 cell space space space space space space straight x end cell cell space space space space straight x cubed end cell row 0 cell space space space space space straight y minus straight x end cell cell space space space space space space straight y cubed minus straight x cubed end cell row 0 cell space space space straight z minus straight x end cell cell space 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 1 comma space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight y minus straight x end cell cell space space space space space straight y cubed minus straight x cubed end cell row cell straight z minus straight x end cell cell space space space space space space straight z cubed minus straight x cubed end cell end table close vertical bar space equals space open vertical bar table row cell straight y minus straight x end cell cell space space space space space 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 straight z minus straight x end cell cell space space space space space space 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 thin space space open vertical bar table row 1 cell space space space space space straight y squared plus straight x squared plus yx end cell row 1 cell space space space space straight z squared plus straight x squared plus zx 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 square bracket straight z squared plus straight x squared plus zx minus straight y squared minus straight x squared minus yx right square bracket 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 square brackets left parenthesis straight z squared minus straight y squared right parenthesis space plus straight x left parenthesis straight z minus straight y right parenthesis close square brackets 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 open square brackets left parenthesis straight z minus straight y right parenthesis thin space left parenthesis straight z plus straight y right parenthesis plus straight x space left parenthesis straight z minus straight y right parenthesis close square brackets 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 thin space left square bracket straight z plus straight y plus straight x right square bracket 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 straight x plus straight y plus straight z right parenthesis$

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109.

Using properties of determinants, prove that:
$open vertical bar table row straight x cell space space space straight y end cell cell space space space straight z end cell row cell straight x squared end cell cell space space straight y squared end cell cell space space space straight z squared end cell row cell straight x cubed end cell cell space space straight y cubed end cell cell space space straight z cubed end cell end table close vertical bar space equals space space straight x space straight y space straight z space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$

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Long Answer Type

110.

Prove that:
$open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight alpha cell space space straight beta end cell cell space straight gamma end cell row βγ cell space space γα end cell cell space space αβ end cell end table close vertical bar space equals space left parenthesis straight beta minus straight gamma right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis$

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108. Prove the following identities:

Long Answer Type