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JEE MainMathematicsComplex Number

Let S = Re ( z^2 - 2z + 9 z^2 - 5z + 9 ) : z C , |z| = 3, z^2 - 5z + 9 0 . Which of the following intervals correctly represents the set S ?

Options

  1. A[ 8 11 , 4 ]
  2. B[- 1 2 , 5 8 ]
  3. C(- , - 1 2 ] [ 5 8 , )
  4. D(- , 8 11 ] [4, )

Correct answer

D. (- , 8 11 ] [4, )

Step-by-step solution

Given |z| = 3 , we have |z|^2 = 9 , which implies z z = 9 , or 9 z = z . The given expression is z^2 - 2z + 9 z^2 - 5z + 9 . Dividing the numerator and the denominator by z , we get: z - 2 + 9 z z - 5 + 9 z = z + z - 2 z + z - 5 Let z = x + iy . Then z + z = 2x , where x = Re (z) . Since |z| = 3 , the real part x must satisfy -3 x 3 . Substitute z + z = 2x into the expression: g(x) = 2x - 2 2x - 5 Notice that g(x) is purely real, so Re (g(x)) = g(x) . We need to find the range of g(x) for x [-3, 3] 2.5 . Rewrite g(

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