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Show that x^3-1=(x-1)(x-omega)(x-omega^2...

Show that `x^3-1=(x-1)(x-omega)(x-omega^2)`

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If omega is an imaginary cube root of unity, show that (x+omega+ omega^(2))(x-omega^(2)-omega^(4))(x+omega^(4)+omega^(8))(x-omega^(8)-omega^(16)).."to" 2n factors =(x^(2)-1)^(n)

If omega be an imaginary cube root of unity, show that, (1-omega)(1-omega^(2))(1-omega^(4))(1-omega^(5))=9

omega is an imagianry cube root of unity, show that, (1-omega^(2))(1-omega^(4))(1-omega^(8))(1-omega^(10))=9

The polynomial x^6+4x^5+3x^4+2x^3+x+1 is divisible by_______ where omega is one of the imaginary cube roots of unity. (a) x+omega (b) x+omega^2 (c) (x+omega)(x+omega^2) (d) (x-omega)(x-omega^2)

If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |[x/(1+omega), y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[(z)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

If omega be an imaginary cube root of unity, show that (1+omega-omega^(2))(1-omega+omega^(2))=4

show that tan^(-1)x+tan^(-1)((2x)/(1-x^2))=tan^(-1)((3x-x^3)/(1-3x^2)),|x|<1/sqrt3

If omega be an imaginary cube root or unity, prove that (x+y omega+ z omega^(2))^(4)+ (x omega+ y omega^(2)+z)^(4)+(x omega^(2)+y+ z omega)^(4)=0

Let omega=-1/2+i(sqrt(3))/2 . Then the value of the determinant |(1,1,1),(1,-1-omega^2,omega^2),(1,omega^2,omega^4)| is (A) 3omega (B) 3omega(omega-1) (C) 3omega^2 (D) 3omega(1-omega)

If D_1 =|{:(1,1,1),(1,omega,omega^2),(1,omega^2,omega):}| and D_2= |{:(1,1,omega),(1,1,omega^2),(omega^2,omega,1):}| show that , D_1=sqrt(3i)D_2 where omega=(-1+sqrt(3i))/2 and i=sqrt(-1) .

PATHFINDER-COMPLEX NUMBERS-QUESTION BANK
  1. Express 1/(1-i)^3 in the form A+iB

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  2. Find the square root of (i)

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  3. Show that x^3-1=(x-1)(x-omega)(x-omega^2)

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  4. Find 4th roots of 1

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  5. Express each of the following expressions as the sum of two squares: (...

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  6. Ifabs(z)gt1 then find the position of the complex no. of z in argand p...

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  7. If abs(z+2)+abs(z-2)le6 then find the maximum value of abs(z)

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  8. Ifz^2+abs(z)^2=0 then find the imaginary value of z

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  9. Show that in argand plane the complex number (1+4i),(2+7i)and (3+10i) ...

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  10. z1 and z2 are two different complex nos where z1ne0 z2ne0 prove that (...

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  11. If x=a+b ,y=aomega+bomega^2, z=aomega^2+bomega then show that ,x^3+y^3...

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  12. Find the modulus and amplitude of the complex nos (1+cosalpha+isinalph...

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  13. Express the complex nos (1+7i)/(2-i)^2 in polar form

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  14. Factorise a^3+b^3+c^3-3abc

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  15. Find the square roots : x-isqrt(x^(4)+x^(2)+1)

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  16. If omega is the imaginary cube root of 1 then prove that (a+bomega+com...

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  17. If y=sqrt(x^2+6x+8) then show that one of the value of sqrt(1+iy)+sqrt...

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  18. If x=cosalpha+isinalpha and 1+sqrt(1-y^2)=ny then show that y/2n(1+nx)...

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  19. If a=1/2(5-isqrt3) then find the value of a^3-6a^2+12a-8

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  20. Show that complex numbers (2+i3), (2-i3), (3-i2), (3+i2) are concyclic...

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