Consider P ( x )= x 2+( sin φ-1) x -(1/2) cos 2 φ. Let α and β be unequal real roots of the equation P(x)=0 such that (α2+β2) has the maximum value, then non-negative difference of the roots of P ( x )=0, is

Question

Consider P ( x )= x 2+( sin φ-1) x -(1/2) cos 2 φ. Let α and β be unequal real roots of the equation P(x)=0 such that (α2+β2) has the maximum value, then non-negative difference of the roots of P ( x )=0, is (A) 1 (B) 2 (C) 0 (D) 4

Tardigrade Admin · Accepted Answer

Θ Equation has unequal real roots ∴ D >0 ⇒( sin φ-1)2-4 ⋅ 1((-1/2) cos 2 φ)>0 ⇒ sin 2 φ-2 sin φ+1+2(1- sin 2 φ)>0 ⇒- sin 2 φ-2 sin φ+3>0 ⇒( sin φ+3)( sin φ-1)<0 ⇒ sin φ ≠ 1 Θ α2+β2=(α+β)2-2 α β=( sin φ-1)2-2 ⋅ 1 ⋅((-1/2) cos 2 φ)=2-2 sin φ ∴ maximum value =4 when sin φ=-1 ∴ cos φ=0 ∴ P ( x )= x 2-2 x = x ( x -2) ∴ Difference of roots =2

Q. Consider P(x)=x2+(sinϕ−1)x−21​cos2ϕ. Let α and β be unequal real roots of the equation P(x)=0 such that (α2+β2) has the maximum value, then non-negative difference of the roots of P(x)=0, is

1

2

0

4

Q. Consider $P (x) = x^{2} + (sin ϕ - 1) x - \frac{1}{2} cos^{2} ϕ$ . Let $α$ and $β$ be unequal real roots of the equation $P (x) = 0$ such that $(α^{2} + β^{2})$ has the maximum value, then non-negative difference of the roots of $P (x) = 0$ , is