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Prove that: (i) [((3+2i)/(2-5i))+((3-2...

Prove that:
(i) `[((3+2i)/(2-5i))+((3-2i)/(2+5i))]` is rational
(ii) `[((2-3i)/(3-4i))((2+3i)/(3+4i))]` is real.

Text Solution

Verified by Experts

The correct Answer is:
(i) `-(8)/(29)`.
(ii) `(13)/(25)`.
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MODERN PUBLICATION-COMPLEX NUMBERS-FREQUENTLY ASKED QUESTIONS
  1. Perform the indicated operation and give your anwer in the form x+iy, ...

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  2. Express each one of the following in the standard form a+i b :(1/(1-2i...

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  3. Prove that: (i) [((3+2i)/(2-5i))+((3-2i)/(2+5i))] is rational (ii)...

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

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  5. Show that a real value of x\ will satisfy the equation (1-i x)/(1+i x...

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  6. If y=sqrt(x^2+6x+8), show that one value of sqrt(1+i y)+sqrt(1-i y)[i=...

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  7. If a+i b=(c+i)/(c-i) , where c is real, prove that:a^2+b^2=1a n d b/a=...

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  8. If (a+ib)/(c+id)=x+iy, show that: x^(2)+y^(2)=(a^(2)+b^(2))/(c^(2)+d...

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  9. Find the conjugate and modulus of complex number 7-24i.

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  10. Find the conjugate of ((3-2i)(2+3i))/((1+2i)(2-i)).

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  11. Prove that for any complex number z, the product zoverline(z) is alway...

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  12. If z=((sqrt(5))/(2)+(i)/(2))^(5)+((sqrt(5))/(2)-(i)/(2))^(5), the prov...

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  13. Write the addtive inverse of the complex number -2+3i.

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  14. Find the additive inverse and reciprocal of complex number 3-4i.

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  15. Find the multiplicative inverse of the following: (i) 3+4i (ii) (5...

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  16. Inverse of (3+4i)/(4-5i)

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  17. If z=(1-i)^(6)+(1-i), then find the modulus of z (i.e., |z|) and multi...

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  18. Find the value of sqrt(3-4i).

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  19. Find the square roots of -48-14i.

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  20. Find the square roots of : ((2+3i)/(5-4i)+(2-3i)/(5+4i)).

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