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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 need to show that \( z^2 - 4z + 13 = 0 \) for \( z = 2 - 3i \) and then find the value of \( 4z^3 - 3z^2 + 169 \). ### Step 1: Calculate \( z^2 \) Given \( z = 2 - 3i \), we first calculate \( z^2 \). \[ z^2 = (2 - 3i)^2 = 2^2 - 2 \cdot 2 \cdot 3i + (3i)^2 \] ...
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RD SHARMA-COMPLEX NUMBERS-Solved Examples And Exercises
  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, prove...

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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 roots of the following: 7-24 i (ii) 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=0a n d...

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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+\bar z =2|z-1| and a r g(z1-z2)=pi/4 , the...

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  18. Show that the complex number z , satisfying are (z-1)/(z+1)=pi/4 lies ...

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  19. If z=(1+7i)/((2-i)^2) , then (a)|z|=2 (b) |z|=1/2 c) a m p(z)=pi/4 (d)...

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  20. The value of (i^5+i^6+i^7+i^8+i^9)/(1+i) is (a) 1/2(1+i) (b) 1/2(1-...

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