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AP EAMCET201921 Apr 2019Evening ShiftMathematicsQuadratic EquationActual

Let θ be an acute angle such that the equation x 3 + 4 x 2 cos θ + x cot θ = 0 has multiple roots. Then the value of θ (in radians) is

Options

  1. Aπ 3
  2. Bπ 8
  3. Cπ 12 or 5 π 12
  4. Dπ 6 or 5 π 12

Correct answer

C. π 12 or 5 π 12

Step-by-step solution

The given equation is, x 3 + 4 x 2 cos θ + x cot θ = 0 x x 2 + 4 x cos θ + cot θ = 0 One root of the equation is x = 0 and x 2 + 4 x cos θ + cot θ = 0 has multiple solutions. So, discriminant, ( 4 cos θ ) 2 - 4 ( 1 ) cot θ = 0 ⇒ 16 cos 2 θ = 4 cot θ ⇒ 4 cos θ = 1 sin θ or cos θ = 0 ⇒ θ = π 2 For 4 cos θ = 1 sin θ ⇒ 2 sin θ cos θ = 1 2 ⇒ sin 2 θ = 1 2 2 θ = π 6 ,   5 π

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