Home
Class 12
MATHS
If p,q,r are positive integers and omega...

If p,q,r are positive integers and `omega` is the cube root of unity and `f(x)=x^(3p)+x^(31+1) x^(3r+2),` then what is `f(omega)` equal to ?

A

`omega`

B

`-omega^(2)`

C

`-omega`

D

0

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to evaluate the function \( f(x) = x^{3p} + x^{3q + 1} + x^{3r + 2} \) at \( x = \omega \), where \( \omega \) is a cube root of unity. ### Step-by-step Solution: 1. **Understanding Cube Roots of Unity**: The cube roots of unity are \( 1, \omega, \) and \( \omega^2 \), where: - \( \omega = e^{2\pi i / 3} \) - \( \omega^3 = 1 \) - \( 1 + \omega + \omega^2 = 0 \) 2. **Substituting \( \omega \) into the Function**: We need to evaluate \( f(\omega) \): \[ f(\omega) = \omega^{3p} + \omega^{3q + 1} + \omega^{3r + 2} \] 3. **Simplifying Each Term**: Since \( \omega^3 = 1 \), we can simplify the powers: - \( \omega^{3p} = (\omega^3)^p = 1^p = 1 \) - \( \omega^{3q + 1} = \omega^{3q} \cdot \omega = (\omega^3)^q \cdot \omega = 1^q \cdot \omega = \omega \) - \( \omega^{3r + 2} = \omega^{3r} \cdot \omega^2 = (\omega^3)^r \cdot \omega^2 = 1^r \cdot \omega^2 = \omega^2 \) 4. **Combining the Results**: Now substituting these simplified terms back into the function: \[ f(\omega) = 1 + \omega + \omega^2 \] 5. **Using the Property of Cube Roots of Unity**: From the property of cube roots of unity, we know: \[ 1 + \omega + \omega^2 = 0 \] 6. **Final Result**: Therefore, we conclude that: \[ f(\omega) = 0 \] ### Final Answer: \[ f(\omega) = 0 \]

To solve the problem, we need to evaluate the function \( f(x) = x^{3p} + x^{3q + 1} + x^{3r + 2} \) at \( x = \omega \), where \( \omega \) is a cube root of unity. ### Step-by-step Solution: 1. **Understanding Cube Roots of Unity**: The cube roots of unity are \( 1, \omega, \) and \( \omega^2 \), where: - \( \omega = e^{2\pi i / 3} \) - \( \omega^3 = 1 \) ...
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    NDA PREVIOUS YEARS|Exercise MCQs|40 Videos
  • CONICS - PARABOLA, ELLIPSE & HYPERBOLA

    NDA PREVIOUS YEARS|Exercise MATH|62 Videos

Similar Questions

Explore conceptually related problems

If omega!=1 is a cube root of unity,then roots of (x-2i)^(3)+i=0

If omega is complex cube root of unity ans x=omega^(2)-omega-2, then what is the vlaue of x^(2)+4x+7 ?

If 1,omega,omega^(2) denote the cube roots of unity,find the roots of (x+5)^(3)+27=0

If omega is a complex cube root of unity,then (x-y)(x omega-y)(x omega^(2)-y)=

If omega is complex cube root of unity then (3+5 omega+3 omega^(2))^(2)+(3+3 omega+5 omega^(2))^(2) is equal to

NDA PREVIOUS YEARS-COMPLEX NUMBERS-Multiple choice question
  1. The smallest positive integral value of n for which ((1-i)/(1+i))^(n) ...

    Text Solution

    |

  2. If alpha and beta are the complex cube roots of unity, then what is ...

    Text Solution

    |

  3. If p,q,r are positive integers and omega is the cube root of unity and...

    Text Solution

    |

  4. If z=(1+2i)/(2-i)-(2-i)/(1+2i), then what is the value of z^(2)+zbarz ...

    Text Solution

    |

  5. What is the argument of (1-sintheta)+icos theta ? (i=sqrt(-1))

    Text Solution

    |

  6. If A+iB =(4+2i)/(1-2i)where i=sqrt(-1) then what is the vlue of A ?

    Text Solution

    |

  7. If z=-barz, then which one of the following is correct?

    Text Solution

    |

  8. Consider the following statements : 1. (omega^(10)+1)^(7)+omega=0 ...

    Text Solution

    |

  9. The value of the sum sum(n=1)^(13) (i^(n)+i^(n+1))where i =sqrt(-1) is...

    Text Solution

    |

  10. What is the modulus of (sqrt2+i)/(sqrt2-i)where i =sqrt(-1)?

    Text Solution

    |

  11. What is sqrt(-10where I =sqrt(-1) equal to ?

    Text Solution

    |

  12. What is the argument of the complex number (-1-i) where i=sqrt(-1)?

    Text Solution

    |

  13. What is one of the square roots of 3+4i, where i=sqrt(-1) ?

    Text Solution

    |

  14. If P and Q are two compex numbers, then the modulus of the quotient o...

    Text Solution

    |

  15. Let z=x+iy where x,y are real variable i=sqrt(-1). If |2z-1|=|z-2|,the...

    Text Solution

    |

  16. If |z+barz|=|z-barz| then locus of z is

    Text Solution

    |

  17. What is the arugument of the complex number ((1+i) (2+i))/(3-i) whe...

    Text Solution

    |

  18. If z is a complex number such that |z|=4 and arg(z) =(5pi)/6 then z is...

    Text Solution

    |

  19. What is ((1+i)^(4n+5))/((1-i)^(4n+3)) equal to, where n is a natural n...

    Text Solution

    |

  20. If z=(-2(1+2i))/(3+i) where i=sqrt(-1) then argument theta(-pilt theta...

    Text Solution

    |