JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer The real values of  \[x\] and \[y\] for which the equation \[({{x}^{4}}+2xi)-(3{{x}^{2}}+yi)=\]\[(3-5i)+(1+2yi)\] is satisfied, are [Roorkee 1984]

    A) \[x=2,y=3\]

    B) \[x=-2,y=\frac{1}{3}\]

    C) Both (a) and (b)

    D)  None of these

    Correct Answer: C

    Solution :

    Given equation \[({{x}^{4}}+2xi)-(3{{x}^{2}}+yi)=(3-5i)+(1+2yi)\] \[\Rightarrow \,\,\,({{x}^{4}}-3{{x}^{2}})+i(2x-3y)=4-5i\] Equating real and imaginary parts, we get        \[{{x}^{4}}-3{{x}^{2}}=4\]   ......(i) and \[2x-3y=-5\] .....(ii) From (i) and (ii), we get \[x=\pm 2\]and \[y=3,\frac{1}{3}\] Trick: Put \[x=2,y=3\]and then \[x=-2,\]\[y=\frac{1}{3},\] we see that they both satisfy the given equation.

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