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

93 Views

113.
Prove that:
$open vertical bar table row straight a straight b straight c row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row bc ca ab end table close vertical bar space equals space left parenthesis ab plus bc plus ca right parenthesis thin space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis$

Let $increment space equals space open vertical bar table row straight a cell space straight b end cell cell space straight c end cell row cell straight a squared end cell cell space straight b squared end cell cell space straight c squared end cell row bc ca ab end table close vertical bar$

$equals space 1 over abc open vertical bar table 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 row abc abc abc end table close vertical bar space equals space abc over abc space open vertical bar table row cell straight a squared end cell cell space space straight b squared end cell cell space space straight c squared end cell row cell straight a cubed end cell cell space straight b cubed end cell cell space space straight c cubed end cell row 1 1 1 end table close vertical bar equals space 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 equals space open vertical bar table row 1 0 cell space space 0 end cell row cell straight a squared end cell cell space space straight b squared minus straight a squared end cell cell space space space straight c squared minus straight a squared end cell row cell straight a cubed end cell cell straight b cubed minus straight a cubed end cell cell space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space space by space straight C subscript 2 minus straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space open vertical bar table row cell straight b squared minus straight a squared end cell cell space space space space space space straight c squared minus straight a squared end cell row cell straight b cubed minus straight a cubed end cell cell space space space space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space expanding space with space first space row comma equals space open vertical bar table row cell left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight b plus straight a right parenthesis end cell cell space space space space space space space space space space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a right parenthesis end cell row cell left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight b squared plus straight a squared plus ba right parenthesis end cell cell space space space space space space space space space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c squared plus straight a squared plus ca right parenthesis end cell end table close vertical bar equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space straight c plus straight a end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space space space space space space space space straight c squared plus straight a squared plus ca end cell end table close vertical bar equals space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space space straight c minus straight b end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space straight c squared minus straight b squared plus ca minus ba end cell end table close vertical bar comma space by space straight C subscript 2 minus straight C subscript 1 equals left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space left parenthesis straight c minus straight b right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space space space space space space space space space space 1 end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space space space space space space space straight c plus straight b plus straight a end cell end table close vertical bar equals 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 space open square brackets bc plus straight b squared plus ab plus ca plus ab plus straight a squared minus straight b squared minus straight a squared minus ab close square brackets equals 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 ab plus bc plus ca 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
$open vertical bar table row cell straight a cubed plus 1 end cell cell space space space straight a squared end cell cell space space straight a end cell row cell straight b cubed plus 1 end cell cell space straight b squared end cell cell space space straight b end cell row cell straight c cubed plus 1 end cell cell straight c squared end cell cell space straight c end cell end table close vertical bar space equals space minus left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a space straight b space straight c space plus space 1 right parenthesis.$