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

74 Views

94.

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

106 Views

95.
Show that:
$increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x space straight y end cell cell straight z space straight x end cell row cell straight x space straight y end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y space straight z end cell row cell straight x space straight z end cell cell straight y space straight z end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar space equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis squared$

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

Applying

straight R subscript 1 space rightwards arrow space xR subscript 1 comma space straight R subscript 2 rightwards arrow straight y space straight R subscript 2 comma space straight R subscript 3 space rightwards arrow space zR subscript 3

increment

and dividing by x y z , we get

increment space equals space 1 over xyz open vertical bar table row cell straight x left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x squared straight y end cell cell straight x squared straight z end cell row cell straight x space straight y end cell cell straight y left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y squared straight z end cell row cell straight x space straight z squared end cell cell straight y space straight z squared end cell cell straight z left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar

Taking common factors, x y, z, from

straight C subscript 1 comma space straight C subscript 2 space and space straight C subscript 3 comma space

respectively, we get

increment space equals space xyz over xyz space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x squared end cell cell straight x squared end cell row cell straight y squared end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y squared end cell row cell straight z squared end cell cell straight z squared end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar

Again

straight C subscript 2 space rightwards arrow 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 comma

we get

increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared space space space space end cell cell straight x squared minus left parenthesis straight y plus straight z right parenthesis squared end cell cell space space space space space straight x squared minus left parenthesis straight y plus straight z right parenthesis squared end cell row cell straight y squared end cell cell left parenthesis straight x plus straight z right parenthesis squared minus straight y squared end cell 0 row cell straight z squared end cell 0 cell left parenthesis straight x plus straight y right parenthesis squared minus straight z squared end cell end table close vertical bar

Taking common factor (x + y + z) from

straight C subscript 2 space and space straight C subscript 3

, we have

increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell space space space straight x minus left parenthesis straight y plus straight z right parenthesis end cell cell space space space space straight x minus left parenthesis straight y plus straight z right parenthesis end cell row cell straight y squared end cell cell space left parenthesis straight x plus straight z right parenthesis minus straight y end cell cell space space 0 end cell row cell straight z squared end cell cell space 0 end cell cell space space space space left parenthesis straight x plus straight y right parenthesis minus straight z end cell end table close vertical bar

Applying

straight R subscript 1 space rightwards arrow space straight R subscript 1 space minus space left parenthesis straight R subscript 2 space plus space straight R subscript 3 right parenthesis comma space we space have

increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell 2 space straight y space straight z end cell cell negative 2 space straight z end cell cell negative 2 straight y end cell row cell straight y squared end cell cell straight x minus straight y plus straight z end cell 0 row cell straight z squared end cell 0 cell straight x plus straight y minus straight z end cell end table close vertical bar

Applying

straight C subscript 2 space rightwards arrow space open parentheses straight C subscript 2 plus 1 over straight y straight C subscript 1 close parentheses space and space straight C subscript 3 space rightwards arrow space open parentheses straight C subscript 3 plus 1 over straight z straight C subscript 1 close parentheses comma

we get

increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell 2 yz end cell 0 0 row cell straight y squared end cell cell straight x plus straight z end cell cell straight y squared over straight z end cell row cell straight z squared end cell cell straight z squared over straight y end cell cell straight x plus straight y end cell end table close vertical bar

increment space equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space left parenthesis 2 yz right parenthesis thin space left square bracket left parenthesis straight x plus straight z right parenthesis thin space left parenthesis straight x plus straight y right parenthesis space minus space yz right square bracket

[Expanding by first row]

equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space left parenthesis 2 space space straight y space straight z right parenthesis thin space left parenthesis straight x squared plus straight x space straight y plus straight x space straight z right parenthesis equals space left parenthesis straight x plus straight y plus straight z right parenthesis cubed space left parenthesis 2 space straight x space straight y space straight z right parenthesis equals space 2 space straight x space straight y space straight z space left parenthesis straight x plus straight y plus straight z right parenthesis cubed

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

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