If and are the roots of the equation 2 z^2-3 z-2 i =0 , where i = -1 , then 16 Re ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ ) lm ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ ) is equal to

Step-by-step solution

2 z^2-3 z-2 i=0 ...(i) 2 (z- i z )=3 As , are roots of (i) aligned & - i = 3 2 & ^2- 1 ^2 -2 i= 9 4 & ^2- 1 ^2 = 9 4 +2 i aligned Squaring both sides aligned & ^4+ 1 ^4 -2= 81 16 -4+9 i & ^4+ 1 ^4 = 49 16 +9 i aligned Similarly, ^4+ 1 ^4 = 49 16 +9 i aligned & ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ & = ¹⁵ ( ^4+ 1 ^4 )+ ¹⁵ ( ^4+ 1 ^4 ) ¹⁵+ ¹⁵ & = 49 16 +9 i & Re ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ )= 49 16 & Im ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ )=9 & 16 Re ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ ) lm ( ¹⁹+ ¹⁹+ ¹¹+ ¹¹ ¹⁵+ ¹⁵ ) & =16 49 16 9 & =441 aligned