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JEE Mains · Maths · STD 11 - 4.1 complex nubers

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

  1. A  both Statement \(I\) and Statement \(II\) are incorrect.
  2. B  Statement \(I\) is incorrect but Statement \(II\) is correct.
  3. C Statement \(I\) is correct but Statement \(II\) is incorrect.
  4. D  both Statement \(I\) and Statement \(II\) are correct.
Verified Solution

Answer & Solution

Correct Answer

(C) Statement \(I\) is correct but Statement \(II\) is incorrect.

Step-by-step Solution

Detailed explanation

Statement \(I\) : \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right|\) Since…
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