99 Percentile Qs Bank for JEE MainMathematicsQuadratic Equation
If , are the roots of x^2+p x+q=0 , then the values of ^3+ ^3 and ^4+ ^2 ^2+ ^4 are respectively ...... and ......
Options
- A(3 p q-p^3 ) and (p^4-3 p^2 q+3 q^2 )
- B-p (3 q-p^2 ) and (p^2-q ) (p^2+3 q )
- C(p q-4) and (p^4-q^4 )
- D(3 p q-p^3 ) and (p^2-q ) (p^2-3 q )
Correct answer
D. (3 p q-p^3 ) and (p^2-q ) (p^2-3 q )
Step-by-step solution
Since and are roots of equation aligned & x^2+p x+q=0, so & + =-p and =q aligned As we know that, ^3+ ^3=( + ) ( ^2+ ^2- ) aligned & =( + ) [( + )^2-3 ] & =(-p) [p^2-3 q ]=3 p q-p^3 and & ^4+ ^2 ^2+ ^4 & = ( ^2+ ^2 )^2- ^2 ^2= [( + )^2-2 ]^2-( )^2 & = ((-p)^2-2 q )^2-q^2 & = (p^2-2 q )^2-q^2= (p^2-2 q-q ) (p^2-2 q+q ) & = (p^2-q ) (p^2-3 q ) aligned