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Show that 1+i^(10)+i^(20)+i^(30) is a re...

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

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To show that \( 1 + i^{10} + i^{20} + i^{30} \) is a real number, we will first evaluate the powers of \( i \). ### Step 1: Understand the powers of \( i \) The powers of \( i \) cycle every four terms: - \( i^0 = 1 \) - \( i^1 = i \) - \( i^2 = -1 \) ...
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RD SHARMA-COMPLEX NUMBERS-Solved Examples And Exercises
  1. Show that: {i^(19)+(1/i)^(25)}^2=-4 (ii) {i^(17)-(1/i)^(34)}^2=2i (ii...

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

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

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

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

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

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

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

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

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

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

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

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

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  15. If z1, z2a n dz3, z4 are two pairs of conjugate complex numbers, prove...

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

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

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  18. Find the square roots of the following: 7-24 i (ii) 5+12 i

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

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

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