A QUADRATIC EQUATION IS CHOSEN FROM THE SET OF ALL QUADRATIC EQUATION WHICH ARE UNCHANGED BY SQUARING THEIR ROOTS. THE CHANCE THAT THE CHOSEN EQUATION HAS REAL ROOTS IS

Dear Student,
Please find below the solution to the asked query:

First we need to find how may quadratic equation exists for whichremain unchange after squaring this roots.If roots of original equation are α and βThen equation should not change when roots are α2 and β2HenceProduct of roots=Product of rootsα.β=α2.β2α.β2-α.β=0α.βα.β-1=0Hence α.β=0 or α.β=1α=0 or β=0 or α=1βand Sum of roots=Sum of rootsα+β=α2+β2α+β=α+β2-2α.β....iWhen α.β=0α+β=α+β2-0α+β2-α+β=0α+βα+β-1=0α+β=1 or α+β=0 and α.β=0Equationx2-α+βx+α.β=0Equation isx2-x=0 or x2=0By iα+β=α+β2-2α.βWhen α.β=1α+β=α+β2-2α+β2-α+β-2=0α+β2-2α+β+α+β-2=0α+βα+β-2+1α+β-2=0α+β-2α+β+1=0α+β=2 or α+β=-1and αβ=1Equationx2-α+βx+α.β=0x2-2x+1=0 and x2-x+1=0Hence we have four equationsx2-x=0x2=0x2-2x+1=0x2-x+1=0Out of which only x2-x+1=0 has imaginary roots and 3 have real roots.HenceRequired Probabilty=34 Answer  

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards

  • -1
What are you looking for?