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Prove that:
$open vertical bar table row 1 cell space straight a end cell cell space space bc end cell row 1 cell space straight b end cell cell space space ca end cell row 1 cell space straight c end cell cell space ab end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis$

70 Views

112.
Prove that:
$open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space space straight y end cell cell space space zx end cell row 1 cell space straight z end cell cell space space xy end cell end table close vertical bar space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis$

Let $increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space yz end cell row 1 cell space straight y end cell cell space zx end cell row 1 cell space straight z end cell cell space xy end cell end table close vertical bar space equals space 1 over xyz space open vertical bar table row straight x cell space space straight x squared end cell cell space space xyz end cell row straight y cell space straight y squared end cell cell space space xyz end cell row straight z cell space straight z squared end cell cell space xyz end cell end table close vertical bar$ ,
by multiplying $straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3 space and space with space straight a comma space straight b comma space straight c space respectively.$
$equals xyz over xyz open vertical bar table row straight x cell space space straight x squared end cell cell space 1 end cell row straight y cell space straight y squared end cell cell space 1 end cell row straight z cell space straight z squared end cell cell space 1 end cell end table close vertical bar equals space open vertical bar table row 1 cell space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space space straight z end cell cell space space straight z squared end cell end table close vertical bar comma space space by space passing space straight R subscript 3 space over space two space rows. equals space open vertical bar table row 1 cell space space space straight x end cell cell space space space space space space straight x squared end cell row 0 cell space space space straight y minus straight x end cell cell space space space space space space straight y squared minus straight x squared end cell row 0 cell space space straight z minus straight x end cell cell space space space space space straight z squared minus straight x squared end cell end table close vertical bar comma space 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 open vertical bar table row cell straight y minus straight x end cell cell space space space space space space space straight y squared minus straight x squared end cell row cell straight z minus straight x end cell cell space space space space space space straight z squared minus straight x squared end cell end table close vertical bar comma space by space expanding space with space straight C subscript 1 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 1 cell space space space straight y plus straight x end cell row 1 cell space space space straight z plus straight x 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 plus straight x minus straight y minus straight x right parenthesis 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 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.$

93 Views

113.

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

73 Views

Short Answer Type

114.

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

69 Views

Long Answer Type

115.

Prove that:
$open vertical bar table row straight alpha cell space space straight beta end cell cell space space space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space space straight gamma squared end cell row cell straight beta plus straight gamma end cell cell space space space straight gamma plus straight alpha end cell cell space space space space straight alpha plus straight beta end cell end table close vertical bar space equals space left parenthesis straight beta minus straight gamma right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis thin space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis$

62 Views

Short Answer Type

116.

Show that:
$open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space straight y end cell cell space space zx end cell row 1 cell space straight z end cell cell space space xy end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space space straight x end cell cell space space straight x squared end cell row 1 cell space space straight y end cell cell space space straight y squared end cell row 1 cell space straight z end cell cell space space straight z squared end cell end table close vertical bar$
and hence factorise.

77 Views

117.

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

89 Views

Long Answer Type

118. 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

139 Views

119.

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