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Express the following complex numbers in...

Express the following complex numbers in the form `r(costheta+isintheta):` `1+itanalpha` (ii) `tanalpha-i` (iii) `1-sinalpha+icosalpha` (iv) `(1-i)/(cospi/3+is inpi/3)`

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The correct Answer is:
`sec alpha(cos alpha+isin alpha),0<`=`alpha<(pi)/(2)-sec alpha [cos(alpha-pi)+isin(alpha-pi],(pi)/(2),alpha<=pi`
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CBSE COMPLEMENTARY MATERIAL-COMPLEX NUMBERS AND QUADRATIC EQUATIONS -Short Answer Type Questions
  1. Prove that: x^4=4=(x+1+i)(x+1-i)(x-1+i(x-1-i)dot)

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  2. Show that |(z -2)/(z - 3)| = 2 represents a circle, find its centre ...

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  3. Find all non zero complex numbers z satisfying barz=iz^2

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  4. If i z^3+z^2-z+i=0 , where i=sqrt(-1) , then |z| is equal to 1 (b) 1/2...

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  5. If z1,z2 are complex numbers such that, (2z1)/(3z2) is purely imaginar...

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  6. If z1 and z2 are complex numbers such that, |1-bar z1z2|^(2)-|z1-z(2^(...

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  7. Find the number of solutions of z^2+|z|^2=0.

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  8. If z1,z2 are complex numbers such that, |(z1-3z2)/(3-z1.bar z2)|=1 and...

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  9. Evaluate x^(4) -4x^(3)+4x^(2)+8x+44, when x=3+2i

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  10. Complex number z1 and z2 satisfy z+barz=2|z-1| and arg (z1-z2) = pi/4 ...

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  11. Solve the following quadratic equation: \ 2x^2-(3+7i)x+(9i-3)=0

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  12. What is the locus of z , if amplitude of (z-2-3i) is pi/2?

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  13. If z=x+i ya n dw=(1-i z)/(z-i) , show that |w|=1 z is purely real.

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  14. Express the following complex numbers in the form r(costheta+isintheta...

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  15. Express the complex number in the form r(cos theta+isin theta) (ii)1-s...

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  16. If (1+i)/(1+2^2i)xx(1+3^2i)/(1+4^2i)xx.......xx(1+(2n-1)^2i)/(1+(2n)^2...

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  17. Find real values of xa n dy for which the complex numbers -3+i x^2ya n...

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  18. The complex numbers z1 z2 and z3 satisfying (z1-z3)/(z2-z3) =(1- i sqr...

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  19. If f(z)=(7-z)/(1-z^2) , where z=1+2i , then |f(z)| is (|z|)/2 (b) |z| ...

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  20. If z1,z2 and z3 are complex numbers such that |z1|=|z2|=|z3|= |1/z1+1/...

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