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If z1, z2 in C , then say which are tru...

If `z_1, z_2 in C ,` then say which are true and false `` `` `-` `` . `|z_1+z_2|^2=|z_1""|^2+|z_2|^2-2R e(z_1 z_2)` `|z_1-z_2|^2=|z_1""|^2-|z_2|^2-2R e(z_1 z_2)` `|z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2)` `|a z_1-b z_2|^2+|b z_1+a z_2|^2=(a^2+b^2)(|z_1|^2+|z_2|^2)` , where `a ,b in Rdot`

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RD SHARMA ENGLISH-COMPLEX NUMBERS-All Questions
  1. Evaluate 1+i^2+i^4+i^6+...+i^(2n)dot

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  2. Show that: {i^(19)+(1/i)^(25)}^2=-4 (ii) {i^(17)-(1/i)^(34)}^2=2i (ii...

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  3. If z1, z2 in C , then say which are true and false - ...

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  4. Prove that the following complex number is purely real: (i)((2+3i)/(3+...

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  5. Show that 1+i^(10)+i^(20)+i^(30) is a real number.

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  6. Find the values of following expression:1+i^2+i^4+i^6+i^8+doti^(20)

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  7. If (a+i b)(c+i d)(e+if)(g+i h)=A+i B prove that (a^2+b^2)(c^2+d^2)(e...

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  8. If z1, z2 are complex number such that (2z1)/(3z2) is purely imaginary...

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  9. If x=-5+2sqrt(-4) , find the value of x^4+9x^3+35 x^2-x+4.

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  10. If z is a complex number such that |z|=1, prove that (z-1)/(z+1) is pu...

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  11. If z=x+i y and w=(1-i z)/(z-i) , show that |w|=1 z is purely real.

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  12. If z=2-3i show that z^2-4z+13=0 and hence find the value of 4z^3-3z^...

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  13. If |z1|=|z2|=.......=|zn|=1, prove that |z1+z2+z3++zn|=1/(z1)+1/(z2)+...

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  14. Find the non-zero complex numbers z satisfying barz =i z^2dot

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  15. Show that if iz^3+z^2-z+i=0, then |z|=1

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  16. Solve the equation z^2 +|z|=0 , where z is a complex number.

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  17. If z1, z2a n dz3, z4 are two pairs of conjugate complex numbers, find ...

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  18. If zr=cos(pi/(3^r))+isin(pi/(3^r)),r=1,2,3, , prove that z1z2z3 zoo=i...

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  19. If xn=cospi/(2^n)+isinpi/(2^n) , prove that x1x2x3 xoo=-1.

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  20. Find the square root of the following: 5+12 i

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