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Prove that:
$open vertical bar table row cell straight a squared end cell bc cell ac plus straight c squared end cell row cell straight a squared plus ab end cell cell straight b squared end cell ac row ab cell straight b squared plus bc end cell cell straight c squared end cell end table close vertical bar space equals space 4 straight a squared straight b squared straight c squared.$

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92.

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

73 Views

93.

Using properties of determinats, prove the following:
$open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space straight a cubed plus straight b cubed plus straight c cubed minus 3 abc.$

74 Views

94.
Prove that
$open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar space equals space straight a space straight b space straight c space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space space or space space straight a space straight b plus straight b space straight c plus straight c space straight a plus straight a space straight b space straight c$

Let $increment space equals space open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar$

$equals space abc space open vertical bar table row cell 1 over straight a plus 1 end cell cell 1 over straight a end cell cell 1 over straight a end cell row cell 1 over straight b end cell cell 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space space 1 over straight c plus 1 end cell end table close vertical bar$
by taking a, b, c common from $straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3$ respectively
$equals space abc space open vertical bar table row cell 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell space space space space space 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell cell space space space space space 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c end cell row cell 1 over straight b end cell cell 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space space space space 1 over straight c plus 1 end cell end table close vertical bar$

by $straight R subscript 1 plus straight R subscript 2 plus straight R subscript 3$
$equals space abc space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 1 1 row cell 1 over straight b end cell cell space space space 1 over straight b plus 1 end cell cell 1 over straight b end cell row cell 1 over straight c end cell cell 1 over straight c end cell cell space 1 over straight c plus 1 end cell end table close vertical bar equals space abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space open vertical bar table row 1 cell space space 0 end cell cell space space space 0 end cell row cell 1 over straight b end cell cell space 1 end cell cell space space space 0 end cell row cell 1 over straight c end cell 0 cell space space space 1 end cell end table close vertical bar comma space by space straight C subscript 2 space minus space straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space abc open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space left square bracket 1. space 1. space 1 space right square bracket space space space space space space space space left square bracket Product space of space diagonal space elements right square bracket equals space abc space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses.$

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95.

Show that:
$increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x space straight y end cell cell straight z space straight x end cell row cell straight x space straight y end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y space straight z end cell row cell straight x space straight z end cell cell straight y space straight z end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar space equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis squared$

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96.

If a, b and c are real numbers, and
$increment space equals space open vertical bar table row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 0 comma$ show that either $straight a plus straight b plus straight c space equals space 0 space or space space straight a space equals space straight b space equals space straight c.$

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Short Answer Type

97.

Using properties of determinants, prove that
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis$

72 Views

Long Answer Type

98.

Using properties of determinants, show that:
$open vertical bar table row cell straight b squared straight c end cell bc cell straight b plus straight c end cell row cell straight c squared straight a squared end cell ca cell straight c plus straight a end cell row cell straight a squared straight b squared end cell ab cell straight a plus straight b end cell end table close vertical bar space equals space 0$

77 Views

99.

Prove that:
$open vertical bar table row cell straight a plus straight b end cell cell space space straight b plus straight c end cell cell space space space straight c plus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell row cell straight c plus straight a end cell cell space space straight a plus straight b end cell cell space space space straight b plus straight c end cell end table close vertical bar space equals space 2 space open vertical bar table row straight a cell space space straight b end cell cell space straight c end cell row straight b cell space space straight c end cell cell space straight a end cell row straight c cell space straight a end cell cell space straight b end cell end table close vertical bar.$