Akshay_Budhkar 4 years ago let w=e^{(i*pie)/3} and a,b,c,x,y,z be non zero complex numbers such that a+b+c=x a+bw+cw^2=y a+bw^2+cw=z thn the value of ([x]^2+[y]^2+[z]^2)/([a]^2+[b]^2+[c]^2) is??? where [q] is mod q...

1. Akshay_Budhkar

question from iit 2011

2. Ishaan94

ohkay

3. Akshay_Budhkar

w is cube root of unity i believe.. the complex cube root of unity

4. estudier

What's iit?

5. Ishaan94

yeah it is

6. Akshay_Budhkar

IIT-indian institute of technology

7. Ishaan94

it is the best institute in india for engineering

8. estudier

Ok, thks, I didn't know...

9. estudier

Not really an engineering question:-)

10. Ishaan94

yeah but after 12 th we have to give entrance test jee( i don't know the full form) it's from that

11. Akshay_Budhkar

i had solved the ques using a short cut to save time

12. Ishaan94

jee to get into iit

13. Akshay_Budhkar

jee-joint entrance exam

14. estudier

Mind u, maybe u have to be an engineer to solve it..

15. Ishaan94

oh well let me try it well i get confused jee e for entrance or engineering lol

16. Akshay_Budhkar

just take a=b=c=1 and get x,y,z and ur done!!!!! lol!!!! but now i want the technical answer

17. Ishaan94

well 3a = x+y+z

18. Akshay_Budhkar

that did get me 4 marks in the paper..substitution

19. Akshay_Budhkar

right add all the three. 1+w+wsquare is 0

20. Akshay_Budhkar

any suggestions estudier??? ishaan i cant find a way ahead of it..

21. Ishaan94

me too but wait a minute

22. estudier

I don't like this kind of problems..:-

23. Ishaan94

well |3a| = |x + y+z|

24. Akshay_Budhkar

dude do u remeber (a+b+c)(a+bw+cw^2)(a+bw^2+cw)=?????? its a formula.. i am not remembering

25. Ishaan94

but x^2 makes the possible minus vanish and hence |x^2 + y^2+x^2| = x^2 + y^2 + z^2 \ just wait like 20 sec

26. Ishaan94

a^2 + b^2 +c^2 -ab +ac -cb

27. Ishaan94

not including (a+b+c)

28. Akshay_Budhkar

-ac dude.. all signs were same no?

29. Ishaan94

well including a+b+c it becomes x^3 + y^3 +z^3 -3xyz x=a y=b z=c

30. Akshay_Budhkar

ya right.. that was d formula..

31. Ishaan94

lets do it x^2 + y^2 +z^2 you find y while i find x^2 and z^2

32. Akshay_Budhkar

ok.. but i believe it is not so lenthy dude.. we get jus 3 minutes for a problem.. there must be some trick

33. Ishaan94

34. Ishaan94

hey (x + y+z) ^2 = x2 +y2 + z2 +2xy + 2xz+2yz

35. Ishaan94

x + y +z = 3a

36. estudier

hey (x + y+z) ^2 = x2 +y2 + z2 +2xy + 2xz+2yz I saw that in your puzzle from yesterday...

37. Ishaan94

9a^2 = x^2 + y^2 +z^2 +2xy + 2zy +2xz

38. Ishaan94

yeah

39. Akshay_Budhkar

right

40. Ishaan94

so ^2 make minus sign vanish

41. Ishaan94

it implies x^2 = |x|^2

42. Akshay_Budhkar

ofcourse.. its given just to confuse mod is not needed

43. Ishaan94

-ve sign don't exist in squares

44. Ishaan94

letme solve a lil

45. Ishaan94

:D

46. Akshay_Budhkar

i believe iit didnt want us to use the substitution method.... coz i have not found a way out till now after 3 months i gae the exam

47. Akshay_Budhkar

@face give it a try

48. Ishaan94

damn iit never let someone enter easily

49. Akshay_Budhkar

but the substitution method is alays there!!!!! lol!! the only weak point of iit jee.. lol!!!

50. Ishaan94

well what was the options

51. Akshay_Budhkar

*always

52. Ishaan94

lol

53. Akshay_Budhkar

its an integer type ques 0 to 9 all options

54. Ishaan94

aahhhh scary

55. Akshay_Budhkar

generally we mark 2 if we dun kno... 40 percent of ques hav ans 2 lol!!!

56. Ishaan94

Ok , Here I go

57. Akshay_Budhkar

ya go go bye. lol js kidding

58. Ishaan94

Okay Here we Have $x = a+b+c$ $y =a+bw+cw^2$ $z = a + bw^2+cw$ Now we have to find $\large{\frac{|x|^2 + |y|^2 + |z|^2 }{a^2 + b^2 +c^2}}$ $|z|^2 = z*(conj.z)$ $conj.y = a+bw^2 +cw$ $conj.z = a+bw+cw^2$ Okay So we have now $\frac{(a+b+c)^2 + 2(a + bw+cw^2)(a+bw^2+cw)}{a^2+b^2+c^2}$ $\frac{3(a^2+b^2+c^2) +2(ab+bc+ac) - 2(ab+bc+ac)}{a^2+b^2+c^2}$$3$ Beautiful Solution : )

59. Akshay_Budhkar

Beauty