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If z = x + iy lies in III quadrant, then...

If` z = x + iy` lies in III quadrant, then `bar z / z` also lies in III quadrant If:

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`z=x=iy` lies in third quadrant

`overline z/z=x-iy/x+iy=x-iy/x+iy xx x-iy/x-iy = (x-iy)^2/x^2+y^2=x^2-y^2-2ixy/x^2+y^2`

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RD SHARMA-COMPLEX NUMBERS-Solved Examples And Exercises
  1. The value of (1+i)^4+(1-i)^4 is

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  2. If the complex number z=x+i y satisfies the condition |z+1|=1, then ...

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  3. If z = x + iy lies in III quadrant, then bar z / z also lies in III qu...

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  4. If a=costheta+i sintheta, find the value of (1+a)/(1-a)dot

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  5. Let z1=r1(costheta1+isintheta1) and z2=r2(costheta2+isintheta2) be t...

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  6. For a positive integer n , find the value of (1-i)^n(1-1/i)^ndot

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  7. If (1+i)(1+2i)(1+3i)........(1+n i)=a+i b ,t h e n2xx5xx10 ....(1+n^2)...

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  8. Perform the indicated operation and find the result in the form a+i bd...

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  9. Let z1=2-i ,""""z2=-2+i . Find (i) Re ((z1z2)/( bar z1)) (ii) Im(1...

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  10. Which of the following is correct for any tow complex numbers z1a n dz...

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  11. If z1a n dz2 are two complex numbers such that |z1|=|z2|a n d arg(z1)...

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  12. Evaluate sum(n=1)^(13)(i^n+i^(n+1)), where n in Ndot

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  13. Find the value of 1+i^(2)+i^(4)+i^(6)+...+i^(2n),

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

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  15. Prove that the following complex numbers are purely real: (i)((2+3i)/...

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

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  17. Find the values of following expressions: i^(49)+i^(68)+i^(89)+i^(110)...

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

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

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