1. **Problem statement:** (a) Shade the region on the Argand diagram where complex numbers $z$ satisfy $$-\frac{\pi}{3} \leq \arg(z - 1 - 2i) \leq \frac{\pi}{3}$$ and $$\operatorname{Re}(z) \leq 3.$$ (b)(i) Shade the region where $$|z - 2 - 2i| \leq 1$$ and $$\arg(z - 4i) \geq -\frac{\pi}{4}.$$ 2. Shade the region where $$|z - 2i| \leq |z + 2 - i|$$ and $$0 \leq \arg(z + 1) \leq \frac{\pi}{4}.$$ 3. Given $u = -1 + \sqrt{7}i$ is a root of $$2x^3 + 3x^2 + 14x + k = 0,$$ find: (a) the value of $k$. (b) the other two roots. --- ### Step-by-step solutions: **1(a).** 1. The inequality $$-\frac{\pi}{3} \leq \arg(z - 1 - 2i) \leq \frac{\pi}{3}$$ means the argument of the vector from point $(1,2)$ to $z$ lies between $-60^\circ$ and $60^\circ$. 2. This defines a sector (wedge) with vertex at $(1,2)$ bounded by two rays at angles $-\frac{\pi}{3}$ and $\frac{\pi}{3}$ from the positive real axis shifted to $(1,2)$. 3. The condition $$\operatorname{Re}(z) \leq 3$$ restricts the region to the left of the vertical line $x=3$. 4. The shaded region is the intersection of this sector and the half-plane $x \leq 3$. **1(b)(i).** 1. The inequality $$|z - 2 - 2i| \leq 1$$ describes a circle centered at $(2,2)$ with radius $1$. 2. The inequality $$\arg(z - 4i) \geq -\frac{\pi}{4}$$ means the argument of the vector from $(0,4)$ to $z$ is at least $-45^\circ$. 3. This defines a half-plane bounded by the ray from $(0,4)$ at angle $-\frac{\pi}{4}$. 4. The shaded region is the intersection of the circle and this half-plane. **2.** 1. The inequality $$|z - 2i| \leq |z + 2 - i|$$ means points $z$ are closer or equidistant to $(0,2)$ than to $(-2,1)$. 2. The locus of points equidistant to these two points is the perpendicular bisector of the segment joining $(0,2)$ and $(-2,1)$. 3. The inequality selects the half-plane containing $(0,2)$. 4. The argument condition $$0 \leq \arg(z + 1) \leq \frac{\pi}{4}$$ means the vector from $(-1,0)$ to $z$ lies between $0$ and $45^\circ$. 5. The shaded region is the intersection of the half-plane closer to $(0,2)$ and the sector defined by the argument bounds. **3(a).** 1. Given $u = -1 + \sqrt{7}i$ is a root of $$2x^3 + 3x^2 + 14x + k = 0,$$ and coefficients are real, its conjugate $\overline{u} = -1 - \sqrt{7}i$ is also a root. 2. Substitute $u$ into the polynomial: $$2u^3 + 3u^2 + 14u + k = 0 \implies k = - (2u^3 + 3u^2 + 14u).$$ 3. Calculate powers: $$u^2 = (-1)^2 + 2 \cdot (-1) \cdot \sqrt{7}i + (\sqrt{7}i)^2 = 1 - 2\sqrt{7}i - 7 = -6 - 2\sqrt{7}i,$$ $$u^3 = u \cdot u^2 = (-1 + \sqrt{7}i)(-6 - 2\sqrt{7}i) = 6 + 2\sqrt{7}i - 6\sqrt{7}i - 2 \cdot 7 i^2 = 6 - 4\sqrt{7}i + 14 = 20 - 4\sqrt{7}i.$$ 4. Substitute: $$2u^3 = 40 - 8\sqrt{7}i,$$ $$3u^2 = -18 - 6\sqrt{7}i,$$ $$14u = -14 + 14\sqrt{7}i.$$ 5. Sum real parts: $40 - 18 - 14 = 8$. Sum imaginary parts: $-8\sqrt{7}i - 6\sqrt{7}i + 14\sqrt{7}i = 0$. 6. So, $$2u^3 + 3u^2 + 14u = 8,$$ hence $$k = -8.$$ **3(b).** 1. The polynomial is $$2x^3 + 3x^2 + 14x - 8 = 0.$$ 2. Since $u$ and $\overline{u}$ are roots, factor: $$2x^3 + 3x^2 + 14x - 8 = (x - u)(x - \overline{u})(2x + m)$$ for some $m$. 3. The quadratic factor from roots $u$ and $\overline{u}$ is: $$x^2 - 2\operatorname{Re}(u) x + |u|^2 = x^2 - 2(-1)x + ((-1)^2 + (\sqrt{7})^2) = x^2 + 2x + 8.$$ 4. Divide the cubic by this quadratic: $$\frac{2x^3 + 3x^2 + 14x - 8}{x^2 + 2x + 8} = 2x - 1.$$ 5. So the other root is from $$2x - 1 = 0 \implies x = \frac{1}{2}.$$ **Final answers:** - (3a) $$k = -8$$ - (3b) Other roots are $$-1 - \sqrt{7}i$$ and $$\frac{1}{2}.$$