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Complex Numbers question

2024 · 5 Apr · Shift 1 · Q35
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  5. /2024 · 5 Apr · Shift 1 · Q35

Complex Numbers question

2024 · 5 Apr · Shift 1 · Q35

JEE MainMathematicsComplex NumbersMCQ+4 / −1
Consider the following two statements : Statement I: For any two non-zero complex numbers z1,z2,(∣z1∣+∣z2∣)∣z1∣z1∣+z2∣z2∣∣≤2(∣z1∣+∣z2∣), and z_1, z_2,(|z_1|+|z_2|)\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) \text {, and }z1​,z2​,(∣z1​∣+∣z2​∣)​∣z1​∣z1​​+∣z2​∣z2​​​≤2(∣z1​∣+∣z2​∣), and  Statement II : If x,y,zx, y, zx,y,z are three distinct complex numbers and a,b,c\mathrm{a}, \mathrm{b}, \mathrm{c}a,b,c are three positive real numbers such that a∣y−z∣=b∣z−x∣=c∣x−y∣\frac{\mathrm{a}}{|y-z|}=\frac{\mathrm{b}}{|z-x|}=\frac{\mathrm{c}}{|x-y|}∣y−z∣a​=∣z−x∣b​=∣x−y∣c​, then a2y−z+b2z−x+c2x−y=1\frac{\mathrm{a}^2}{y-z}+\frac{\mathrm{b}^2}{z-x}+\frac{\mathrm{c}^2}{x-y}=1y−za2​+z−xb2​+x−yc2​=1. Between the above two statements,
  1. A
    both Statement I and Statement II are incorrect.
  2. B
    Statement I is correct but Statement II is incorrect.
  3. C
    Statement I is incorrect but Statement II is correct.
  4. D
    both Statement I and Statement II are correct.
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