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61. Without expanding, prove that

open vertical bar table row cell straight x plus straight y end cell cell space space straight x end cell cell space space space straight x end cell row cell 5 straight x plus 4 straight y end cell cell space space 4 straight x end cell cell space space space space 2 straight x end cell row cell 10 straight x plus 8 straight y end cell cell space space 8 straight x end cell cell space space space space 3 straight x end cell end table close vertical bar space equals space x cubed

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62. Without expanding, prove that

open vertical bar table row cell straight a plus straight l end cell straight m straight n row straight I cell straight a plus straight m end cell straight n row straight I straight m cell straight a plus straight n end cell end table close vertical bar space equals a squared space left parenthesis a plus l plus m plus n right parenthesis

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63. Using properties of determinants, prove that

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

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64. Prove that

open square brackets table row cell straight x plus straight a end cell straight b straight c row straight a cell space space space straight x plus straight b end cell straight c row straight a straight b cell space space space straight x plus straight c end cell end table close square brackets space equals space x squared space left parenthesis x plus a plus b plus c right parenthesis

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65. Evaluate

open vertical bar table row cell 1 space end cell cell space straight x end cell straight y row 1 cell space space straight x plus straight y end cell straight y row 1 straight x cell space straight x plus straight y end cell end table close vertical bar

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66.

Evaluate open vertical bar table row straight x straight y cell space space space straight x plus straight y end cell row straight y cell space space space straight x plus straight y end cell straight x row cell straight x plus straight y end cell straight x straight y end table close vertical bar

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67. Show that

open vertical bar table row cell straight x plus 1 end cell cell space straight x plus 2 end cell cell space space straight x plus straight a end cell row cell straight x plus 2 end cell cell space space space straight x plus 3 space space end cell cell straight x plus straight b end cell row cell straight x plus 3 end cell cell space straight x plus 4 end cell cell space straight x plus straight c end cell end table close vertical bar space equals 0

,where a,b,c and in A.P

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68.

show space that space open vertical bar table row straight a cell space space space space space space space space space space space space straight a plus straight b end cell cell space straight a plus straight b plus straight c end cell row cell 2 straight a end cell cell space space space space space space space space straight a end cell cell space space space space 4 straight a plus 3 straight b plus 2 straight c end cell row cell 3 straight a end cell cell space space space space space space space space 3 straight a end cell cell space space space space 10 straight a plus 6 straight b plus 3 straight c end cell end table close vertical bar space equals straight a cubed

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69. Prove that

open vertical bar table row cell 3 straight a end cell cell space space minus straight a plus straight b end cell cell space space minus straight a plus straight c end cell row cell negative straight b plus straight a end cell cell 3 straight b end cell cell space minus straight b plus straight c end cell row cell negative straight c plus straight a end cell cell space minus straight c plus straight b end cell cell 3 straight c end cell end table close vertical bar space equals 3 left parenthesis straight a plus straight b plus straight c right parenthesis left parenthesis ab plus bc plus ca right parenthesis

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70. Prove that

open vertical bar table row 1 cell space space space space straight a end cell cell space space space space space straight a squared minus bc end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared minus ca end cell row 1 cell space space space straight c end cell cell space space space space straight c squared minus ab end cell end table close vertical bar space equals 0

71 Views