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JEE Main 2016

JEE Main 2016 Mathematics question on Complex Numbers and Quadratic Equations

The sum of all real values of xx satisfying the equation (x2−5x+5)x2+4x−60=1(x^2 - 5x + 5)^{x^2 + 4x - 60} = 1 is
  1. A.
    3
  2. B.
    -4
  3. C.
    6
  4. D.
    5
Show answer and solution

Correct option: A

Solution

For ab = 1, we need either: (i) b = 0 with a ≠ 0, (ii) a = 1, or (iii) a = -1 with b even.

Case (i): x² + 4x - 60 = 0 → (x+10)(x-6) = 0 → x = -10 or x = 6. Check base ≠ 0: both valid.

Case (ii): x² - 5x + 5 = 1 → x² - 5x + 4 = 0 → (x-1)(x-4) = 0 → x = 1 or x = 4.

Case (iii): x² - 5x + 5 = -1 → x² - 5x + 6 = 0 → (x-2)(x-3) = 0 → x = 2 or x = 3. Check exponent is even: at x=2, exponent = 4+8-60 = -48 (even ✓); at x=3, exponent = 9+12-60 = -39 (odd ✗). So only x = 2.

Valid solutions: x = -10, 6, 1, 4, 2. Sum = -10 + 6 + 1 + 4 + 2 = 3.

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