Exercise 3 -- Common to all candidates

In a vast plain, a network of sensors makes it possible to detect lightning and to produce an image of storm phenomena. The radar screen is divided into forty sectors denoted by a letter and a number between 1 and 5. The lightning sensor is represented by the center of the screen; five concentric circles corresponding to the respective radii $20, 40, 60, 80$ and 100 kilometres delimit five zones numbered from 1 to 5, and eight segments starting from the sensor delimit eight portions of the same angular opening, named in the trigonometric direction from A to H.
We assimilate the radar screen to a part of the complex plane by defining an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$:

• the origin O marks the position of the sensor;
• the abscissa axis is oriented from West to East;
• the ordinate axis is oriented from South to North;
• the chosen unit is the kilometre.

In the following, a point on the radar screen is associated with a point with affix $z$.

Part A

1. We denote $z_{P}$ the affix of the point P located in sector B3 on the graph. We call $r$ the modulus of $z_{P}$ and $\theta$ its argument in the interval $]-\pi ; \pi]$. Among the four following propositions, determine the only one that proposes a correct bound for $r$ and for $\theta$ (no justification is required):
   

   Proposition A	Proposition B	Proposition C	Proposition D
   \begin{tabular}{ c } $40 < r < 60$
   and
   $0 < \theta < \frac{\pi}{4}$

   &

   $20 < r < 40$

   and

   $\frac{\pi}{2} < \theta < \frac{3\pi}{4}$

   &

   $40 < r < 60$

   and

   $\frac{\pi}{4} < \theta < \frac{\pi}{2}$

   &

   $0 < r < 60$

   and

   $-\frac{\pi}{2} < \theta < -\frac{\pi}{4}$

   \hline \end{tabular}
   
2. A lightning impact is materialized on the screen at a point with affix $z$. In each of the two following cases, determine the sector to which this point belongs: a. $z = 70 \mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}$; b. $z = -45\sqrt{3} + 45\mathrm{i}$.

Part B

We assume in this part that the sensor displays an impact at point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. When the sensor displays the impact point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, the affix $z$ of the actual lightning impact point admits:

• a modulus that can be modeled by a random variable $M$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 5$;
• an argument that can be modeled by a random variable $T$ following a normal distribution with mean $\frac{\pi}{3}$ and standard deviation $\frac{\pi}{12}$.

We assume that the random variables $M$ and $T$ are independent. In the following, probabilities will be rounded to $10^{-3}$ near.

1. Calculate the probability $P(M < 0)$ and interpret the result obtained.
2. Calculate the probability $P(M \in ]40 ; 60[)$.
3. We admit that $P\left(T \in \left]\frac{\pi}{4} ; \frac{\pi}{2}\right[\right) = 0.819$. Deduce the probability that the lightning actually struck sector B3 according to this modeling.

Represent on the same figure $\tau _ { 0 } , \tau _ { 1 } , \tau$.

We recall that $\mathscr{C}$ was defined as the image of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
In this question, we seek a complex parametrization of $\mathscr{C}$, of the form $$z : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)}$$ where $\rho$ and $\theta$ are two continuous functions from $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ to $\mathbb{R}$, the function $\rho$ taking strictly positive values.
III.B.1) Calculate $\rho(t)$ for all $t \in \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. III.B.2) Represent on the calculator the parametrized arc $$\mathscr{G} : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}t}$$ and reproduce the curve roughly on the paper. What letter does this curve evoke? III.B.3) From the expression of $\gamma(t)$, calculate $\tan\theta(t)$. III.B.4) a) Represent the function $t \mapsto \arctan(2\tan t)$ on the part of the interval $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ on which this function is defined. b) Modify this function to determine the continuous function $\theta$ sought. The result will be verified by representing with the aid of the calculator the parametrized curve $z$. III.B.5) Indicate a sequence of Maple or Mathematica instructions allowing one to obtain this plot.

We define the applications: $$\alpha : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \frac{\pi}{4} + \frac{3\pi}{2n} \mathrm{E}\left(\frac{2n}{3\pi}\left(t - \frac{\pi}{4}\right)\right)$$ $$\omega : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \cos^2\left(\frac{2n}{3}\left(t - \frac{\pi}{4}\right)\right)$$ where $\mathrm{E}(x)$ denotes the integer part of the real number $x$.
III.C.1) Briefly study $\alpha$ and $\omega$, then represent on the same graph the two functions $t \mapsto \alpha(10, t)$ and $t \mapsto \omega(10, t)$. III.C.2) Represent the function $\psi : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, t \mapsto \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)$. III.C.3) We define the function: $$w : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, (n, t) \mapsto \rho(t)\left(1 + \psi(t)\omega(n, t)\right) \mathrm{e}^{\mathrm{i}\theta(\alpha(n, t))}$$ Identify which of the four graphics represents the function $t \mapsto w(40, t)$, and explain why. III.C.4) Write a sequence of Maple or Mathematica instructions allowing one to create the sequence of the first 100 curves (one may create an animation).

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Draw on a figure the circle $\mathcal{C}$ with center $O$ and radius $|z|$ and the points $M$ with affixe $z$ and $B$ with affixe $-|z|$. By considering well-chosen angles, show that $$\theta(z) = \operatorname{Arg}(z) = 2\operatorname{Arg}(z + |z|)$$ where $\operatorname{Arg}(z)$ denotes the principal determination of the argument of the complex number $z$.

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, $\arg(w)$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that $w = |w|(\cos\theta + i\sin\theta)$. Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations $\arg(z) > \arg(z+1)$ and $\arg(z) > \arg(z+i)$. Then $\cos(\arg(z))$ can take
(a) Any value in the set $(-1/2, 0) \cup (0, 1/2)$ but none from outside
(b) Any value in the interval $(-1, 0)$ but none from outside
(c) Any value in the interval $(0, 1)$ but none from outside
(d) Any value in the set $(-1, 0) \cup (0, 1)$ but none from outside.

Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]
(a) $\operatorname { Re } ( w ) > 0$
(b) $\operatorname { Im } ( w ) \geq 0$
(c) $| w | \leq 1$
(d) $| w | \geq 1$

Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.

The set of complex numbers $z$ satisfying the equation $$( 3 + 7i ) z + ( 10 - 2i ) \bar { z } + 100 = 0$$ represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point

The set of complex numbers $z$ satisfying the equation $$( 3 + 7i ) z + ( 10 - 2i ) \bar { z } + 100 = 0$$ represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point

If $z = x + i y$ is a complex number such that $\left| \frac { z - i } { z + i } \right| < 1$, then we must have
(A) $x > 0$
(B) $x < 0$
(C) $y > 0$
(D) $y < 0$.

Let $\Omega = \{ z = x + iy \in \mathbb{C} : |y| \leq 1 \}$. If $f(z) = z^{2} + 2$, then draw a sketch of $$f(\Omega) = \{ f(z) : z \in \Omega \}.$$ Justify your answer.

The locus of points $z$ in the complex plane satisfying $z ^ { 2 } + | z | ^ { 2 } = 0$ is
(A) a straight line
(B) a pair of straight lines
(C) a circle
(D) a parabola

Let $a, b, c$ be three complex numbers. The equation $$az + b\bar{z} + c = 0$$ represents a straight line on the complex plane if and only if
(A) $a = b$
(B) $\bar{a}c = b\bar{c}$
(C) $|a| = |b| \neq 0$
(D) $|a| = |b| \neq 0$ and $\bar{a}c = b\bar{c}$

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the Argand plane,
(a) a straight line. (B) a pair of intersecting straight lines. (C) a pair of distinct parallel straight lines. (D) a point.

Match the statements in Column-I with those in Column-II. [Note: Here $z$ takes values in the complex plane and $\operatorname { Im } z$ and $\operatorname { Re } z$ denote, respectively, the imaginary part and the real part of $z$.]
Column I
A) The set of points $z$ satisfying $| z - i | z \| = | z + i | z \mid$ is contained in or equal to
B) The set of points $z$ satisfying $| z + 4 | + | z - 4 | = 10$ is contained in or equal to
C) If $| w | = 2$, then the set of points $z = w - \frac { 1 } { w }$ is contained in or equal to
D) If $| w | = 1$, then the set of points $z = w + \frac { 1 } { w }$ is contained in or equal to
Column II p) an ellipse with eccentricity $\frac { 4 } { 5 }$ q) the set of points $z$ satisfying $\operatorname { Im } z = 0$ r) the set of points $z$ satisfying $| \operatorname { Im } z | \leq 1$ s) the set of points $z$ satisfying $| \operatorname { Re } z | \leq 2$ t) the set of points $z$ satisfying $| z | \leq 3$

If $z _ { 1 } , z _ { 2 }$ are complex numbers such that $\operatorname { Re } \left( z _ { 1 } \right) = \left| z _ { 1 } - 1 \right|$ and $\operatorname { Re } \left( z _ { 2 } \right) = \left| z _ { 2 } - 1 \right|$ and $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 6 }$ , then $\operatorname { Im } \left( z _ { 1 } + z _ { 2 } \right)$ is equal to :
(1) $2 \sqrt { 3 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { 2 } { \sqrt { 3 } }$

The region represented by $\{ z = x + i y \in C : | z | - \operatorname { Re } ( z ) \leq 1 \}$ is also given by the inequality
(1) $y ^ { 2 } \geq 2 ( x + 1 )$
(2) $y ^ { 2 } \leq 2 \left( x + \frac { 1 } { 2 } \right)$
(3) $y ^ { 2 } \leq \left( x + \frac { 1 } { 2 } \right)$
(4) $y ^ { 2 } \geq x + 1$

The equation $\arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { \pi } { 4 }$ represents a circle with:
(1) centre at $( 0,0 )$ and radius $\sqrt { 2 }$
(2) centre at $( 0,1 )$ and radius 2
(3) centre at $( 0 , - 1 )$ and radius $\sqrt { 2 }$
(4) centre at $( 0,1 )$ and radius $\sqrt { 2 }$

Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers such that $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 4 }$ and $z _ { 1 } , z _ { 2 }$ satisfy the equation $| z - 3 | = \operatorname { Re } ( z )$. Then the imaginary part $z _ { 1 } + z _ { 2 }$ is equal to

Let $z_1$ and $z_2$ be two complex numbers such that $\bar{z}_1 = i\bar{z}_2$ and $\arg\frac{z_1}{\bar{z}_2} = \pi$, then the argument of $z_1$ is
(1) $\arg z_2 = \frac{\pi}{4}$
(2) $\arg z_2 = -\frac{3\pi}{4}$
(3) $\arg z_1 = \frac{\pi}{4}$
(4) $\arg z_1 = -\frac{3\pi}{4}$

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z = 2 - i \left( 2 \tan \frac { 5 \pi } { 8 } \right)$, then $( r , \theta )$ is equal to
(1) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(2) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } \right)$
(3) $\left( 2 \sec \frac { 5 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(4) $\left( 2 \sec \frac { 11 \pi } { 8 } , \frac { 11 \pi } { 8 } \right)$

In a complex number plane, consider the complex numbers $z$ such that $z^3$ is a real number.
(1) Let $C$ be the figure formed by the set of complex numbers $z = x + iy$ satisfying the above condition. Since the arguments of the complex numbers $z$ satisfy $$\arg z = \frac{\pi}{\mathbf{M}}k \quad (k : \text{integer}),$$ figure $C$ consists of three straight lines represented in terms of $x$ and $y$ by the equations $$y = \mathbf{N}, \quad y = \sqrt{\mathbf{O}}\,x, \quad y = -\sqrt{\mathbf{O}}\,x.$$
(2) Suppose that on $C$ there exists only one complex number $z$ satisfying $|z - 1 - i| = r$. Then the value of $r$ is $$r = \frac{\sqrt{\mathbf{Q}} - \square\mathbf{R}}{\square}$$ and the value of $z$ is $$z = \frac{\mathbf{T} + \sqrt{\mathbf{U}}}{\square\mathbf{V}}(1 + \sqrt{\mathbf{W}}\,i).$$

Deduce the conditions for $z$ and, on the complex $z$ plane, draw the area of $z$ in which the imaginary part of the complex function $J(z) = e^{-i\alpha} z + e^{i\alpha} z^{-1}$ is positive. Here, $\alpha$ is a real number and $0 < \alpha < \pi/2$.