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JEE Advanced 2018 · Complex Numbers and Quadratic Equations

For a non-zero complex number zz, let arg⁡(z)\arg(z) denote the principal argument with −π<arg⁡(z)≤π-\pi<\arg(z)\le\pi. Then, which of the following statement(s) is (are) FALSE?

Answers: A. arg⁡(−1−i)=π4\arg(-1-i)=\frac{\pi}{4}, where i=−1i=\sqrt{-1} · B. The function f:R→(−π,π]f:\mathbb{R}\to(-\pi,\pi], defined by f(t)=arg⁡(−1+it)f(t)=\arg(-1+it) for all t∈Rt\in\mathbb{R}, is continuous at all points of R\mathbb{R}, where i=−1i=\sqrt{-1} · D. For any three given distinct complex numbers z1z_1, z2z_2 and z3z_3, the locus of the point zz satisfying the condition 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)=\pi, lies on a straight line

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Written solution

Given: z∈C, z≠0\text{Given: } z \in \mathbb{C},\ z \neq 0

arg⁡(z)∈(−π,π]  (principal argument)\arg(z) \in (-\pi, \pi]\ \text{ (principal argument)}

To find: which of the statements (1) to (4) are FALSE\text{To find: which of the statements }(1)\text{ to }(4)\text{ are FALSE}

(1) arg⁡(−1−i)=π4(1)\ \arg(-1-i) = \dfrac{\pi}{4}

(2) f(t)=arg⁡(−1+it) continuous on R(2)\ f(t) = \arg(-1+it) \text{ continuous on } \mathbb{R}

(3) arg⁡(z1z2)−arg⁡(z1)+arg⁡(z2)=2kπ, k∈Z(3)\ \arg\left(\dfrac{z_1}{z_2}\right) - \arg(z_1) + \arg(z_2) = 2k\pi,\ k \in \mathbb{Z}

Plan: test each of (1) to (4) in the Argand plane\text{Plan: test each of } (1)\text{ to }(4)\text{ in the Argand plane}

z=x+iy,arg⁡z=angle from the +x-axisz = x + iy,\quad \arg z = \text{angle from the } +x\text{-axis}

(1) Locate z=−1−i(1)\ \text{Locate } z = -1 - i

Re<0, Im<0⇒third quadrant\text{Re} < 0,\ \text{Im} < 0 \Rightarrow \text{third quadrant}

Third quadrant⇒arg⁡(−1−i)=−3π4\text{Third quadrant} \Rightarrow \arg(-1-i) = -\dfrac{3\pi}{4}

−3π4≠π4⇒(1) is FALSE-\dfrac{3\pi}{4} \neq \dfrac{\pi}{4} \Rightarrow (1)\ \text{is FALSE}

(2) z=−1+it: vertical line Re(z)=−1(2)\ z = -1 + it:\ \text{vertical line } \text{Re}(z) = -1

t<0→below the axis;t>0→abovet < 0 \to \text{below the axis};\quad t > 0 \to \text{above}

t→0+: arg⁡(−1+it)→πt \to 0^+:\ \arg(-1+it) \to \pi

t→0−: arg⁡(−1+it)→−πt \to 0^-:\ \arg(-1+it) \to -\pi

f(0)=arg⁡(−1)=πf(0) = \arg(-1) = \pi

lim⁡t→0−f(t)=−π≠π=f(0)\lim_{t \to 0^-} f(t) = -\pi \neq \pi = f(0)

f is discontinuous at t=0⇒(2) is FALSEf \text{ is discontinuous at } t = 0 \Rightarrow (2)\ \text{is FALSE}

(3) Recall: arg⁡ ⁣(z1z2)=arg⁡z1−arg⁡z2+2kπ(3)\ \text{Recall: } \arg\!\left(\dfrac{z_1}{z_2}\right) = \arg z_1 - \arg z_2 + 2k\pi

arg⁡ ⁣(z1z2)−arg⁡z1+arg⁡z2=2kπ\arg\!\left(\dfrac{z_1}{z_2}\right) - \arg z_1 + \arg z_2 = 2k\pi

2kπ is an integer multiple of 2π⇒(3) is TRUE2k\pi \text{ is an integer multiple of } 2\pi \Rightarrow (3)\ \text{is TRUE}

(4) Cross-ratio: (z−z1)(z2−z3)(z−z3)(z2−z1)(4)\ \text{Cross-ratio: } \dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}

Real  ⟺  z,z1,z2,z3 concyclic or collinear\text{Real} \iff z, z_1, z_2, z_3 \text{ concyclic or collinear}

Take z1=0, z2=1, z3=i\text{Take } z_1 = 0,\ z_2 = 1,\ z_3 = i

Not collinear⇒a unique circle passes through them\text{Not collinear} \Rightarrow \text{a unique circle passes through them}

arg⁡=π⇒one arc of this circle, not a line\arg = \pi \Rightarrow \text{one arc of this circle, not a line}

Locus is a circular arc, not a straight line\text{Locus is a circular arc, not a straight line}

⇒(4) is FALSE\Rightarrow (4)\ \text{is FALSE}

FALSE statements: (1), (2), (4)\text{FALSE statements: } (1),\ (2),\ (4)

(3) is TRUE(3)\ \text{is TRUE}

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