Important Questions of Determinants Mathematics | Zigya

Important Questions of Determinants Mathematics | Zigya

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Mathematics : Determinants

Short Answer Type

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81.

show that
$open vertical bar table row cell straight x minus straight y minus straight z end cell cell 2 straight x end cell cell 2 straight x end cell row cell 2 straight y end cell cell space space space straight y minus straight z minus straight x end cell cell 3 straight x end cell row cell 2 straight z end cell cell 2 space straight z end cell cell space space straight z minus straight x minus straight y end cell end table close vertical bar space equals space left parenthesis x plus y plus z right parenthesis cubed$

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82.

show space that open vertical bar table row cell straight a plus straight b plus nc end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis space straight c end cell cell straight b plus straight c plus na end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis straight c end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell straight c plus straight a plus nb end cell end table close vertical bar space equals straight n left parenthesis straight a plus straight b plus straight c right parenthesis cubed

68 Views

83.

show space that open vertical bar table row 1 cell space space space space straight b plus straight c space end cell cell space space space space space straight b squared plus straight c squared end cell row 1 cell space space straight c plus straight a end cell cell space space space straight c squared plus straight a squared end cell row 1 cell space space straight a plus straight b end cell cell space space space space straight a squared plus straight b squared end cell end table close vertical bar

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84.

prove space that open vertical bar table row cell 1 plus straight a squared plus straight b squared end cell cell 2 ab end cell cell negative 2 straight b end cell row cell 2 ab end cell cell 1 minus straight a squared space plus straight b squared end cell cell 2 straight a end cell row cell 2 straight b end cell cell negative 2 straight a end cell cell 1 minus straight a squared minus straight b squared end cell end table close vertical bar space equals left parenthesis 1 plus straight a squared space plus straight b squared right parenthesis cubed

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85.

By space using space properties space of space determinants comma space show space that open vertical bar table row cell straight x plus 4 end cell cell 2 space straight x end cell cell 2 space straight x end cell row cell 2 space straight x end cell cell space space straight x plus 4 end cell cell 2 space straight x end cell row cell 2 space straight x end cell cell 2 space straight x end cell cell space space straight x plus 4 end cell end table close vertical bar space equals space left parenthesis 5 space straight x plus 4 right parenthesis space left parenthesis 4 minus straight x right parenthesis squared

66 Views

86.

space By space using space properties space of space determinants comma space show space that space space space space space space space space open vertical bar table row cell straight y plus straight k end cell straight y straight y row straight y cell space space space straight y plus straight k end cell straight y row straight y straight y cell space space straight y plus straight k end cell end table close vertical bar space equals straight k squared space left parenthesis 3 straight y plus straight k right parenthesis

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87. Using the properties of determinants, prove that

open vertical bar table row cell straight a plus straight b plus straight c end cell cell negative straight c end cell cell negative straight b end cell row cell negative straight c end cell cell space space space straight a plus straight b plus straight c end cell cell negative straight a end cell row cell negative straight b end cell cell negative straight a end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar space equals 2 left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis

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Long Answer Type

88.

show space that space open vertical bar table row cell negative straight a left parenthesis straight b squared plus straight c squared minus straight a squared right parenthesis end cell cell 2 straight b cubed end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell space space space minus straight b left parenthesis straight c squared plus straight a squared plus straight b squared right parenthesis end cell cell 2 straight c cubed end cell row cell 2 straight a cubed end cell cell 2 straight b cubed end cell cell space space space minus straight c left parenthesis straight a squared plus straight b squared space minus straight c squared right parenthesis end cell end table close vertical bar space equals abc space left parenthesis straight a squared plus straight b squared plus straight c squared right parenthesis

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89.

prove space that space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space equals left parenthesis straight b squared minus ac right parenthesis left parenthesis ax squared plus 2 space straight b space straight x space straight y space plus straight c space straight y cubed right parenthesis

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Short Answer Type

90.

prove space that space space increment space equals space open vertical bar table row cell straight a plus straight b space straight x end cell cell straight c plus straight d space straight x end cell cell space space space straight p plus straight q space straight x end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x plus straight d end cell cell space space straight p space straight x plus straight q end cell row straight upsilon straight nu straight omega end table close vertical bar space equals space left parenthesis 1 minus straight x squared right parenthesis space space open vertical bar table row straight a cell space straight c end cell cell space space straight p end cell row straight b cell space space space straight d space space end cell cell space straight q end cell row straight u cell space straight v end cell cell space space straight w end cell end table close vertical bar

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