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

72 Views

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:
$open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space straight a cubed plus straight b cubed plus straight c cubed minus 3 abc.$

L.H.S. = $open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space open vertical bar table row cell straight a plus straight b plus straight c end cell cell space space space space straight a plus straight b plus straight c end cell cell space space straight a plus straight b plus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell space straight c minus straight a end cell row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell end table close vertical bar comma space by space straight R subscript 1 plus straight R subscript 3$

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

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94.

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

106 Views

95.

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

78 Views

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 $straight a plus straight b plus straight c space equals space 0 space or space space straight a space equals space straight b space equals space straight c.$

71 Views

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