Important Questions of Determinants Mathematics | Zigya
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    Mathematics : Determinants

    Multiple Choice QuestionsShort Answer Type

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    51.  Without expanding, prove that the following determinants vanish :
     
    open vertical bar table row cell straight a minus straight b end cell cell space space straight b minus straight c space end cell cell straight c minus straight a end cell row cell straight b minus straight c end cell cell space straight c minus straight a end cell cell space straight a minus straight b end cell row cell straight c minus straight a end cell cell space straight a minus straight b end cell cell space straight b minus straight c end cell end table close vertical bar
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    52. Without space expanding comma space prove space that space the space following space determinants space vanish space open vertical bar table row 1 cell space space space space space space bc end cell cell space space space space space straight a left parenthesis straight b plus straight c right parenthesis end cell row 1 cell space space space space space ca end cell cell space space space space space straight b left parenthesis straight c plus straight a right parenthesis end cell row 1 cell space space space space space space ab end cell cell space space space space space straight c left parenthesis straight a plus straight b right parenthesis end cell end table close vertical bar
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    53.

    Without expansion, prove that the following determinants vanish 
    open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space space space space straight a squared end cell cell space space bc end cell row cell begin inline style 1 over straight b end style end cell cell space space space straight b squared end cell cell space space ca end cell row cell begin inline style 1 over straight c end style end cell cell space space straight c squared end cell cell space space ab end cell end table close vertical bar

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    54.  Without expansion, prove that the following determinants vanish
     
    open vertical bar table row cell begin inline style 1 over straight a end style end cell cell space straight a space end cell bc row cell begin inline style 1 over straight b end style end cell cell space straight b end cell cell space ca end cell row cell begin inline style fraction numerator 1 over denominator space straight c end fraction end style end cell cell space straight c end cell cell space ab end cell end table close vertical bar
    107 Views
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    55. Without expanding, prove that 
    open vertical bar table row 1 cell space straight omega end cell cell space space straight omega squared end cell row straight omega cell space space straight omega squared end cell 1 row cell straight omega squared end cell cell space 1 end cell cell space straight omega end cell end table close vertical bar

    vanishes, where ω, ω2 are imaginary cube roots of unity.
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    56. Without expansion, show that

    open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space equals space 0
    69 Views
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    57. If A + B + C = π. Find value of

    open vertical bar table row cell sin space left parenthesis straight A plus straight B plus straight C right parenthesis end cell cell space sin space straight B end cell cell space space cos space straight C end cell row cell negative sin space straight B end cell 0 cell space tan space straight A end cell row cell cos left parenthesis straight A plus straight B right parenthesis end cell cell space minus tan space straight A end cell cell space 0 end cell end table close vertical bar
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    58. open vertical bar table row cell 2 straight y plus 4 end cell cell space space space space 5 straight y plus 7 end cell cell space 8 straight y plus straight a end cell row cell 3 straight y plus 5 end cell cell space space space 6 straight y plus 8 end cell cell space space 9 straight y plus straight b end cell row cell 4 straight y plus 6 end cell cell space space space 7 straight y plus 9 end cell cell space space 10 straight y plus straight c end cell end table close vertical barIf a, b, c are in A.P. find value of
    71 Views
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    59.  Using properties of determinants, prove that
     
     space open square brackets table row cell straight x minus 3 end cell cell space space straight x minus 4 space end cell cell space space straight x minus straight alpha end cell row cell straight x minus 2 end cell cell space space straight x minus 3 end cell cell space space straight x minus straight beta end cell row cell straight x minus 1 end cell cell space space straight x minus 2 end cell cell space space straight x minus straight gamma end cell end table close square brackets space equals 0
     
    where α, β, γ, are in A.P.
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    60. open vertical bar table row 1 cell space space space straight a space end cell cell space bc end cell row 1 cell space space straight b end cell cell space space ca end cell row 1 cell space space straight c end cell cell space space ab end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space space straight a space end cell cell space space space straight a squared end cell row 1 cell space space straight b end cell cell space space straight b squared end cell row 1 cell space space straight c end cell cell space space straight c squared end cell end table close vertical bar spaceWithout expanding, prove that
    73 Views
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