If a1+ib1a2+ib2…an+ibn=A+iB then which of the following are (is) true

Let a1+ib1=r1eiθ1=r1cos⁡θ1+isin⁡θ1 where r12=u12+b12 and tan⁡θ1=b1a1 or θ1=tan−1⁡b1a1∴ r1eiθ1,r2eiθ2……=Reiθ or r1r2Γ…rneiθ1+θ2+…θn=Reiθ∴ r1r2…rn=R and θ1+θ2+…θn=θ.

If a1+ib1a2+ib2…an+ibn=A+iB then which of the following are (is) true

Let a1+ib1=r1eiθ1=r1cos⁡θ1+isin⁡θ1 where r12=u12+b12 and tan⁡θ1=b1a1 or θ1=tan−1⁡b1a1∴ r1eiθ1,r2eiθ2……=Reiθ or r1r2Γ…rneiθ1+θ2+…θn=Reiθ∴ r1r2…rn=R and θ1+θ2+…θn=θ.

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If $(a_{1} + i b_{1}) (a_{2} + i b_{2}) \dots (a_{n} + i b_{n}) = A + i B$ then which of the following are (is) true

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The amplitude of A+iB is sum of the amplitues of the individual complex numbers in left hand side

$\sqrt{A^{2} + B^{2} =} \sqrt{{a_{1}}^{2} + {a_{2}}^{2} + . . . + {a_{n}}^{2}} \times \sqrt{{b_{1}}^{2} + {b_{2}}^{2} + . . . + {b_{n}}^{2}}$

$(a_{1}^{2} + b_{1}^{2}) (a_{2}^{2} + b_{2}^{2}) \dots (a_{n}^{2} + b_{n}^{2}) = A^{2} + B^{2}$

$A = a_{1} \cdot a_{2} \cdot a_{3} \cdot \cdot \cdot \cdot a_{n}$

answer is B, D.

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Detailed Solution

Let $a_{1} + i b_{1} = r_{1} e^{i θ_{1}} = r_{1} (\cos θ_{1} + i \sin θ_{1})$ where $r_{1}^{2} = u_{1}^{2} + b_{1}^{2}$ and $\tan θ_{1} = \frac{b_{1}}{a_{1}}$ or $θ_{1} = \tan^{- 1} \frac{b_{1}}{a_{1}}$

$\begin{array}{l} ∴ r_{1} e^{i θ_{1}}, r_{2} e^{i θ_{2}} \dots \dots = {Re}^{i θ} \\ or r_{1} r_{2} Γ \dots r_{n} e^{i (θ_{1} + θ_{2} + \dots θ_{n})} = {Re}^{i θ} \\ ∴ r_{1} r_{2} \dots r_{n} = R and θ_{1} + θ_{2} + \dots θ_{n} = θ . \end{array}$