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AP EAMCET202123 Aug 2021Morning ShiftMathematicsDeterminantsActual

If is a root of the equation x+ 1 x +1=0 , then | array ccc 1 & 1+ & 1+ + ^2 3 & 4+3 & 5+4 +3 ^2 6 & 9+6 & 11+9 +6 ^2 array | is equal to

Options

  1. A1
  2. B-1
  3. C0
  4. D1+

Correct answer

B. -1

Step-by-step solution

is root of x+ 1 x +1=0 , i.e. x^2+x+1=0 This gives roots -1 1-4 2 = -1 i 3 2 = -1+ -3 2 and ^2= -1- -3 2 are roots of x^2+x+1=0 Also, ^3= ^2=1 and 1+ + ^2=0 Then, A= | array ccc 1 & 1+ & 1+ + ^2 3 & 4+3 & 5+4 +3 ^2 6 & 9+6 & 11+9 +6 ^2 array | is written as, aligned & |A|= | array ccc 1 & 1+ & 1+ + ^2 3 & 4+3 & 3 (1+ + ^2 )+2+ 6 & 9+6 & 6 (1+ + ^2 )+5+3 array | & |A|= | array ccc 1 & 1+ & 0 3 & 4+3 & 2+ 6 & 9+6 & 5+3 array | aligned Expand along row 1, aligned & |A|=(4+3 )(5+3 )-(2+ )(9+6 ) & -(1+ ) 15+9 -12-6 & =2

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