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Complex Numbers question

2018 · Shift 1 · Q19
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Complex Numbers question

2018 · Shift 1 · Q19

JEE AdvancedMathematicsComplex NumbersMultiple correct+4 / −1
For a non-zero complex number z, let arg(z) denote the principal argument with −π-\pi−π< arg(z) ≤π\le \pi≤π. Then, which of the following statement(s) is (are) FALSE?
  1. A
    arg(−-− 1 −-− i) = π4{\pi \over 4}4π​, where i = −1\sqrt { - 1}−1​
  2. B
    The function f : R →\to→ (−π-\pi−π, π\piπ), defined by f(t) = arg (−-− 1 + it) for all t ∈\in∈ R, is continuous at all points of R, where i =−1\sqrt { - 1}−1​.
  3. C
    For any two non-zero complex numbers z1 and z2, arg (z1z2)\left( {{{{z_1}} \over {{z_2}}}} \right)(z2​z1​​) −-− arg (z1) + arg(z2) is an integer multiple of 2 π\piπ.
  4. D
    For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition arg ((z−z1)(z2−z3)(z−z3)(z2−z1))=π\left( {{{(z - {z_1})({z_2} - {z_3})} \over {(z - {z_3})({z_2} - {z_1})}}} \right) = \pi((z−z3​)(z2​−z1​)(z−z1​)(z2​−z3​)​)=π, lies on a straight line.
View written solutionFree

Correct answer: A, B, D

The user wants me to identify the FALSE statements among the given options related to complex numbers.

Analysis of Option A

  1. Let the complex number be z=−1−iz = -1 - iz=−1−i. This corresponds to the point (−1,−1)(-1, -1)(−1,−1) in the Argand plane.
  2. The point (−1,−1)(-1, -1)(−1,−1) lies in the third quadrant.
  3. The principal argument θ=arg⁡(z)\theta = \arg(z)θ=arg(z) is given by tan⁡(θ)=yx=−1−1=1\tan(\theta) = \frac{y}{x} = \frac{-1}{-1} = 1tan(θ)=xy​=−1−1​=1. For a point in the third quadrant, the angle is θ=−π+π4=−3π4\theta = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}θ=−π+4π​=−43π​.
  4. The principal argument must be in the range (−π,π](-\pi, \pi](−π,π]. Our value θ=−3π4\theta = -\frac{3\pi}{4}θ=−43π​ lies in this range.
  5. The statement claims that arg⁡(−1−i)=π4\arg(-1 - i) = \frac{\pi}{4}arg(−1−i)=4π​. This is incorrect.
  6. Therefore, statement A is FALSE.

Analysis of Option B

  1. The function is given by f(t)=arg⁡(−1+it)f(t) = \arg(-1 + it)f(t)=arg(−1+it) for t∈Rt \in \mathbb{R}t∈R. The complex number z(t)=−1+itz(t) = -1 + itz(t)=−1+it lies on the vertical line with real part −1-1−1.
  2. To check for continuity, we can examine the behavior of the function at a potential point of discontinuity. The definition of the principal argument often has a jump discontinuity along the negative real axis. This corresponds to t=0t=0t=0 for z(t)=−1+itz(t) = -1 + itz(t)=−1+it.
  3. Let's evaluate the function at t=0t=0t=0: f(0)=arg⁡(−1+0i)=arg⁡(−1)=πf(0) = \arg(-1 + 0i) = \arg(-1) = \pif(0)=arg(−1+0i)=arg(−1)=π.
  4. Let's evaluate the right-hand limit as t→0+t \to 0^+t→0+: For t>0t>0t>0, the point −1+it-1+it−1+it is in the second quadrant. As t→0+t \to 0^+t→0+, the point approaches −1-1−1 from above the real axis. Thus, the argument approaches π\piπ. So, lim⁡t→0+f(t)=π\lim_{t \to 0^+} f(t) = \pilimt→0+​f(t)=π.
  5. Let's evaluate the left-hand limit as t→0−t \to 0^-t→0−: For t<0t<0t<0, the point −1+it-1+it−1+it is in the third quadrant. As t→0−t \to 0^-t→0−, the point approaches −1-1−1 from below the real axis. The principal argument is in (−π,π](-\pi, \pi](−π,π]. Thus, the argument approaches −π-\pi−π. So, lim⁡t→0−f(t)=−π\lim_{t \to 0^-} f(t) = -\pilimt→0−​f(t)=−π.
  6. Since the left-hand limit (−π)(-\pi)(−π) is not equal to the right-hand limit (π)(\pi)(π), the function f(t)f(t)f(t) is discontinuous at t=0t=0t=0.
  7. The statement claims the function is continuous at all points of R\mathbb{R}R, which is incorrect.
  8. Therefore, statement B is FALSE.

Analysis of Option C

  1. For any two non-zero complex numbers z1z_1z1​ and z2z_2z2​, the argument of their quotient is related to their individual arguments by the formula: arg⁡(z1z2)=arg⁡(z1)−arg⁡(z2)+2kπ\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) + 2k\piarg(z2​z1​​)=arg(z1​)−arg(z2​)+2kπ, for some integer k∈{−1,0,1}k \in \{-1, 0, 1\}k∈{−1,0,1}. The term 2kπ2k\pi2kπ is necessary to ensure the result is within the principal argument range (−π,π](-\pi, \pi](−π,π].
  2. The statement considers the expression arg⁡(z1z2)−arg⁡(z1)+arg⁡(z2)\arg\left(\frac{z_1}{z_2}\right) - \arg(z_1) + \arg(z_2)arg(z2​z1​​)−arg(z1​)+arg(z2​).
  3. Substituting the formula from step 1, we get: (arg⁡(z1)−arg⁡(z2)+2kπ)−arg⁡(z1)+arg⁡(z2)=2kπ(\arg(z_1) - \arg(z_2) + 2k\pi) - \arg(z_1) + \arg(z_2) = 2k\pi(arg(z1​)−arg(z2​)+2kπ)−arg(z1​)+arg(z2​)=2kπ.
  4. This shows that the expression is always an integer multiple of 2π2\pi2π.
  5. Therefore, statement C is TRUE.

Analysis of Option D

  1. The given condition is arg⁡((z−z1)(z2−z3)(z−z3)(z2−z1))=π\arg\left( \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} \right) = \piarg((z−z3​)(z2​−z1​)(z−z1​)(z2​−z3​)​)=π.
  2. The expression (z−z1)(z2−z3)(z−z3)(z2−z1)\frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}(z−z3​)(z2​−z1​)(z−z1​)(z2​−z3​)​ is the cross-ratio of the four complex numbers z,z2,z1,z3z, z_2, z_1, z_3z,z2​,z1​,z3​, often denoted as (z,z2;z1,z3)(z, z_2; z_1, z_3)(z,z2​;z1​,z3​).
  3. The condition that the argument of the cross-ratio is π\piπ implies that the cross-ratio is a negative real number.
  4. A fundamental theorem in complex analysis states that four distinct points z,z1,z2,z3z, z_1, z_2, z_3z,z1​,z2​,z3​ are concyclic (lie on a circle) or collinear (lie on a line) if and only if their cross-ratio is a real number.
  5. Since the cross-ratio is a negative real number, the point zzz must lie on the unique circle (or line) that passes through the three given distinct points z1,z2,z3z_1, z_2, z_3z1​,z2​,z3​.
  6. The statement claims that the locus of zzz lies on a straight line for any three distinct complex numbers z1,z2,z3z_1, z_2, z_3z1​,z2​,z3​. This is not generally true. If z1,z2,z3z_1, z_2, z_3z1​,z2​,z3​ are not collinear, they define a unique circle, and the locus of zzz is an arc of this circle.
  7. For example, let z1=1,z2=i,z3=−1z_1=1, z_2=i, z_3=-1z1​=1,z2​=i,z3​=−1. These points are not collinear and lie on the unit circle ∣z∣=1|z|=1∣z∣=1. The locus of zzz satisfying the condition is the lower semicircle of the unit circle, which does not lie on a straight line.
  8. The statement is only true in the special case where z1,z2,z3z_1, z_2, z_3z1​,z2​,z3​ are collinear. Since it does not hold for any three points, the statement is false.
  9. Therefore, statement D is FALSE.

Conclusion

The question asks for the FALSE statement(s). Based on the analysis, statements A, B, and D are false.

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