Home
Class 12
MATHS
Let a,b and c be three real numbers sa...

Let a,b and c be three real numbers satisfying
[a b c ] `|(1,9,7),(8,2,7),(7,3,7)|`=[ 0 0 0 ] ….(i)
Let `omega` be a solution of `x^3-1=0` with `lim(omega) > 0` . If a=2 with b and c satisfying Eq.(i) then the value of `3/omega^4+ 1/omega^b + 1/omega^c` is :

A

`-2`

B

2

C

3

D

`-3`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`[abc][{:(1,9,7),(2,8,7),(7,3,7):}]=[0,0,0]`
`rArr [a+2b+7c, 9a+8b+3c, 7a+7b+7c]=[0 0 0]`
`rArr a+2b+7c=0, 9a+8b+3c=0`and `7a+7b+7c=0`
`rArr a=5,b=-6,c=1`
Clearly, `omega=-1/sqrt(2)+isqrt(3)/2` and it satisfies `omega^(3)=1, 1/omega=omega^(2)`
`therefore 3/omega^(a)+1/omega^(b)+3/omega^(c)=3/omega^(5)+1/omega^(-6)+3/omega^(1)=3omega+1+3omega^(2)`
`=3(omega^(2)+omega)+1=-3+1=-2`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|15 Videos

  • COMPLEX NUMBERS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|129 Videos

  • COMPLEX NUMBERS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|58 Videos

  • CIRCLES

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|53 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|87 Videos

Similar Questions

Explore conceptually related problems

Let a,b, and c be three real numbers satistying [a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0] Let omega be a solution of x^3-1=0 with Im(omega)gt0. I fa=2 with b nd c satisfying (E) then the vlaue of 3/omega^a+1/omega^b+3/omega^c is equa to (A) -2 (B) 2 (C) 3 (D) -3

Let omega be the solution of x^(3)-1=0 with "Im"(omega) gt 0 . If a=2 with b and c satisfying [abc][{:(1,9,7),(2,8,7),(7,3,7):}]=[0,0,0] , then the value of 3/omega^(a) + 1/omega^(b) + 1/omega^( c) is equal to

Let a,b, and c be three real numbers satistying [a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0] If the point P(a,b,c) with reference to (E), lies on the plane 2x+y+z=1 , the the value of 7a+b+c is (A) 0 (B) 12 (C) 7 (D) 6

Let a, b and c be three real numbers satisfying [a" "b" "c"][{:(1,9,7),(8,2,7),(7,9,7):}]=[0"" "0""" "0] and alpha and beta be the roots of the equation ax^(2) + bx + c = 0 , then sum_(n=0)^(oo) ((1)/(alpha)+(1)/(beta))^(n) , is

Let a,b, and c be three real numbers satistying [a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0] Let b=6, with a and c satisfying (E). If alpha and beta are the roots of the quadratic equation ax^2+bx+c=0 then sum_(n=0)^oo (1/alpha+1/beta)^n is (A) 6 (B) 7 (C) 6/7 (D) oo

Let p,q,r in R satisfies [p q r][(1,8,7),(9,2,3),(7,7,7)] = [0 0 0] .....(i) {:(,"List-I",,"List-I"),((P),"If the point M(p.q.r) with reference to (i) lies on the curve" 2x+y+z=1 then (7p+q+r) "is equal to",(1),-2),((Q),"Let" omega(ne1) "cube root of unity with" lm(omega) gt0.If p=2 "with q and r satisfying" (i) then (3/(omega^(p))+1/(omega^(q))+3/(omega^(r))) "is equal to ",(2),7),((R),"Let q=6 with p and r satisfying"(i). if alpha and beta "are roots of quadratic equation " px^(2)+qx+r=0 " then" Sigma_(n=0)^(oo) (1/(alpha)+1/(beta))^(n) " is equal to ",(3),6):}

Find the complex number satisfying the system of equations z^3+ omega^7=0a n dz^5omega^(11)=1.

If omega is an imaginary cube root of unity, then the value of |(a,b omega^(2),a omega),(b omega,c,b omega^(2)),(c omega^(2),a omega,c)| , is

If a, b, c are real numbers satisfying the condition a+b+c=0 then the rots of the quadratic equation 3ax^2+5bx+7c=0 are

If omega(!=1) is a cube root of unity, and (1+omega)^7= A + B omega . Then (A, B) equals

OBJECTIVE RD SHARMA ENGLISH-COMPLEX NUMBERS -Section I - Solved Mcqs
  1. The maximum value of |a r g(1/(1-z))|for|z|=1,z!=1 is given by.

    Text Solution

    |

  2. about to only mathematics

    Text Solution

    |

  3. Let a,b and c be three real numbers satisfying [a b c ] |(1,9,7),(...

    Text Solution

    |

  4. The set {R e((2i z)/(1-z^2)): zi sacom p l e xnu m b e r ,|z|=1,z=+-1}...

    Text Solution

    |

  5. about to only mathematics

    Text Solution

    |

  6. If |z(1)|=|z(2)| and arg (z(1))+"arg"(z(2))=0, then

    Text Solution

    |

  7. Let complex numbers alpha and 1/alpha lies on circle (x-x0)^2+(y-y0)^2...

    Text Solution

    |

  8. about to only mathematics

    Text Solution

    |

  9. Let S=S1 nn S2 nn S3, where s1={z in C :|z|<4}, S2={z in C :ln[(z-...

    Text Solution

    |

  10. Let S=S1 nn S2 nn S3, where s1={z in C :|z|<4}, S2={z in C :ln[(z-...

    Text Solution

    |

  11. Let z(k)=cos(2kpi)/10+isin(2kpi)/10,k=1,2,………..,9. Then, 1/10{|1-z(1)|...

    Text Solution

    |

  12. In Q. No. 121, 1-sum(k=1)^(9)cos(2kpi)/10 equals

    Text Solution

    |

  13. If z is a complex number such that |z|geq2 , then the minimum value...

    Text Solution

    |

  14. A complex number z is said to be unimodular if abs(z)=1. Suppose z(1) ...

    Text Solution

    |

  15. about to only mathematics

    Text Solution

    |

  16. f(n) = cot^2 (pi/n) + cot^2\ (2 pi)/n +...............+ cot^2\ ((n-1) ...

    Text Solution

    |

  17. If z(1) and z(2) are the complex roots of the equation (x-3)^(3) + 1=...

    Text Solution

    |

  18. If |z-1| =1 and arg(z)=theta, where z ne 0 and theta is acute, then (1...

    Text Solution

    |

  19. If z is a complex number lying in the first quadrant such that "Re"(z)...

    Text Solution

    |

  20. about to only mathematics

    Text Solution

    |