Home
Class 12
MATHS
For any integer k, let alpha h= cos. (k ...

For any integer k, let `alpha _h= cos. (k pi)/(7)+ I sin. (k pi)/(7) " where if "=sqrt(-1).` The value of the expression
`(Sigma_(k=1)^(12) |alpha_(k+1)-alpha_k|)/(Sigma_(k=1)^(3) |alpha_(4k+1)-alpha_(4k-2)|)`is

Text Solution

Verified by Experts

The correct Answer is:
4
Given `alpha_(k) = cos ((kpi)/(7)) + i sin ((k pi)/(7))`
`= cos ((2k pi)/(14)) + i sin ((2 kpi)/(14))`
`therefore alpha_(k)` are vertices of regular polygon having 14 sides .
Let the side length of regular polygon be `alpha`.
`therefore |alpha_(k + 1) - alpha_(k)| ` = length of a side of the regular polygon
`= alpha " " ... (i)`
and `|alpha_(4k-1) - alpha_(4k-2)|` = length of a side of the regular polygon
` = alpha " " .... (ii)`
`therefore ( sum_(h=1)^(12) |alpha_(k+1) - alpha_(k)|)/(sum_(h=1)^(3) |alpha_(4k-1) - alpha_(4k-2)|) = (12(a))/(3(a)) = 4`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COMPLEX NUMBERS

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 5 DE-MOIVRES THEOREM,CUBE ROOTS AND nth ROOTS OF UNITY ( OBJECTIVE QUESTIONS I)(ONLY ONE CORRECT OPTION)|12 Videos

  • COMPLEX NUMBERS

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 5 DE-MOIVRES THEOREM,CUBE ROOTS AND nth ROOTS OF UNITY ( MATCH THE COLUMNS )|1 Videos

  • COMPLEX NUMBERS

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 4 ROTATION OF A COMPLEX NUMBER (ANALYTICAL & DESCRIPTIVE QUESTIONS)|7 Videos

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 Integer Answer type Question|1 Videos

  • DEFINITE INTEGRATION

    IIT JEE PREVIOUS YEAR|Exercise LIMITS AS THE SUM|6 Videos

Similar Questions

Explore conceptually related problems

For any integer k, let alpha_(k)=cos((k pi)/(7))+i sin((k pi)/(7)), wherei =sqrt(-1) Value of the expression (sum_(k=1)^(12)| alpha_(k+1)-alpha_(k)|)/(sum_(k=1)^(3)| alpha_(4k-1)-alpha_(4k-2)|) is

The value of the expression Sigma_(k=0)^(27)k.^(27)C_(k)((1)/(3))^(k)((2)/(3))^(27-k) is equal to

Knowledge Check

  • The value of 5 * cot ( Sigma_(k =1)^(5) cot ^(-1) ( k^(2) + k + 1)) is equal to

    A
    `5/2`
    B
    7
    C
    `-7`
    D
    `7/2`

  • The value of sum_(k = 1)^(6) (sin "" ( 2 pi k)/( 7) - i cos "" ( 2 pi k)/( 7) ) is

    A
    i
    B
    `-1`
    C
    `-i`
    D
    0

  • The value of sum _(k = 1)^(6) ( sin "" ( 2 pi k)/( 7) - i cos "" ( 2 pi k)/( 7)) is

    A
    `-1`
    B
    0
    C
    `-i`
    D
    i

    • Similar Questions

    Explore conceptually related problems

    prod_ (k = 1) ^ (6) (sin ((2k pi) / (7)) - i cos ((2k pi) / (7))) =

    Given ( sin alpha)/( a) = ( cos alpha )/( b ) = ( tan alpha )/( c ) = k

    If an angle alpha is divided into two parts A&B such that A-B=x and tan A:tan B=k:1 Then the value of sin x is (A) (k+1)/(k-1)sin alpha (k)/(k+1)sin alpha (C) (k-1)/(k+1)sin alpha (D) tan alpha

    sum_ (k = 1) ^ (6) (sin, (2 pi k) / (7) -i cos, (2 pi k) / (7)) =?

    The value of sum_(k=1)^(10)((sin(2 pi k))/(11)-i(cos(2 pi k))/(11)) is

    Home
    Profile