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For any two complex numbers z1and z2, pr...

For any two complex numbers `z_1`and `z_2`, prove that `Re(z_1z_2)=Re(z_1) Re(z_2)-Im(z_1) Im(z_2)`.

Text Solution

Verified by Experts

Let `z1​=a+ib`
and `z2​=x+iy`
So,`z1​z2​=(a+ib)(x+iy)`
`=>z1​z2​=(ax−by+i(bx+ay))`
`=>Re(z1​z2​)=ax−by`
thus,`Re(z1​z2​)=Rez1​Rez2​−Imz1​Imz2​`
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NCERT-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-MISCELLANEOUS EXERCISE
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  2. Find the number of non-zero integral solutions of the equation |1-i|^...

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  3. If (a + i b) (c + i d) (e + if) (g + i h) = A + i B, then show that (a...

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  4. Let z1=2-i ,z2=-2+i. Find Re ((z1z2)/( bar z'1))

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  5. Find the modulus and argument of the complex number (1+2i)/(1-3i).

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  6. If z1=2-i ,z2=1+i ,find |(z1+z2+1)/(z1-z2+i)|

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  7. If a + i b =((x+i)^2)/(2x^2+1),prove that a^2+b^2=((x^2+1)^2)/((2x^2+...

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  8. If alphaand betaare different complex numbers with |beta|=1,then fin...

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  10. Find the modulus of (1+i)/(1-i)-(1-i)/(1+i).

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  11. If (x+i y)^3=u+i v ,then Find p if u/x+v/y=p(x^2-y^2).

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  12. Convert the following in the polar form : (i) (1+7i)/((2-i)^2) (ii) ...

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  13. Solve the equation :x^2-2x+3/2=0

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  14. Solve the equation : 3x^2-4x+(20)/3=0

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  15. Evaluate : [i^(18)+(1/i)^(25)]^3

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  16. Reduce (1/(1-4i)-2/(1+i))((3-4i)/(5+i))to the standard form.

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  17. For any two complex numbers z1and z2, prove that Re(z1z2)=Re(z1) Re(z2...

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  18. Solve the equation :21 x^2-28 x+10=0

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  19. Solve the equation :27 x^2-10 x+1=0

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  20. If ((1+i)/(1-i))^m=1,then find the least integral value of m.

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