JEE Advanced2010MathematicsQuadratic EquationActual
Let p and q be real numbers such that p 0, p^3 q and p^3 -q . If and are non-zero complex numbers satisfying + =-p and ^3+ ^3=q , then a quadratic equation having and as its roots is
Options
- A(p^3+q ) x^2- (p^3+2 q ) x + (p^3+q )=0
- B(p^3+q ) x^2- (p^3-2 q ) x + (p^3+q )=0
- C(p^3-q ) x^2- (5 p^3-2 q ) x + (p^3-q )=0
- D(p^3-q ) x^2- (5 p^3+2 q ) x + (p^3-q )=0
Correct answer
B. (p^3+q ) x^2- (p^3-2 q ) x + (p^3+q )=0
Step-by-step solution
Sum of roots = ^2+ ^2 and product =1 Given, + =-p and ^3+ ^3=q aligned & ( + ) ( ^2- + ^2 )=q & ^2+ ^2- = -q p aligned and ( + )^2=p^2 ^2+ ^2+2 =p^2 From Eqs. (i) and (ii), we get ^2+ ^2= p^3-2 q 3 p and = p^3+q 3 p Required equation is gathered x^2- (p^3-2 q ) x (p^3+q ) +1=0 (p^3+q ) x^2- (p^3-2 q ) x+ (p^3+q )=0 gathered