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COMPLEX NUMBERS AND QUADRATIC EQUATIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCOMPLEX NUMBERS AND QUADRATIC EQUATIONS1 Mark
MCQ
Choose the correct answer. $\sin\text{x}+\text{i}\cos2\text{x}$ and $\cos\text{x}-\text{i}\sin2\text{x}$ are conjugate to each other for:
  • A
    $\text{x}=\text{n}\pi$
  • B
    $\text{x}=\Big(\text{n}+\frac{1}{2}\Big)\frac{\pi}{2}$
  • C
    $\text{x}=0$
  • ✓
    no value of x
✓

Answer

Correct option: D.
no value of x
$\sin\text{x}+\text{i}\cos2\text{x}$ and $\cos\text{x}-\text{i}\sin2\text{x}$ are conjugate to each other.
$\Rightarrow\overline{\sin\text{x}+\text{i}\cos2\text{x}}=\cos\text{x}-\text{i}\sin2\text{x}$
$\Rightarrow{\sin\text{x}-\text{i}\cos2\text{x}}=\cos\text{x}-\text{i}\sin2\text{x}$
On comparing real and imaginary parts of both the sides, we get
$\sin\text{x}=\cos\text{x}$ and $\cos2\text{x}=\sin2\text{x}$
$\Rightarrow\tan\text{x}=1$ and $\tan2\text{x}=1$
Now, $\tan2\text{x}=1$
$\Rightarrow\frac{2\tan\text{x}}{1-\tan^{2}\text{x}}=1,$ which is not satisfied by $\tan\text{x}=1$
Hence, no value of x is possible.

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