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If z=2-3i show that z^2-4z+13=0 and he...

If `z=2-3i` show that `z^2-4z+13=0` and hence find the value of `4z^3-3z^2+169.`

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To solve the problem, we will follow these steps: ### Step 1: Substitute the value of \( z \) Given \( z = 2 - 3i \), we need to substitute this into the expression \( z^2 - 4z + 13 \). ### Step 2: Calculate \( z^2 \) First, we calculate \( z^2 \): \[ ...
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RD SHARMA ENGLISH-COMPLEX NUMBERS-All Questions
  1. If z is a complex number such that |z|=1, prove that (z-1)/(z+1) is pu...

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

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

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

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

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

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

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

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

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

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

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  12. Find the square roots of -15-8idot

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

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  14. If ((a^2+1)/(2a-i))^2=x+i y , find the value of x^2+y^2dot

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  15. Let z1a n dz2 be two complex numbers such that ( z )1+i( z )2=0 and ...

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  16. If |z-5i|=|z+5i| , then the locus of zdot

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  17. If z1a n dz2 both satisfy z+barz r =2|z-1| and a r g(z1-z2)=pi/4 , t...

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  18. Show that the complex number z, satisfying the condition arg ((z -1)/(...

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  19. If z=(1+7i)/((2-i)^2) , then find |z|?

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  20. The value of (i^5+i^6+i^7+i^8+i^9)/(1+i) is

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