Complex Numbers
Geometric Representation
Grade 11
Question:
<p>The set of points in an Argand diagram which satisfy both <span>|z| ≤ 4</span> and <span>0 ≤ \arg(z) ≤ \frac{π}{3}</span>, is</p>
<p>(a) a circle and a line</p>
<p>(b) a radius of a circle</p>
<p>(c) a sector of a circle</p>
<p>(d) an infinite part line</p>
Step-by-Step Solution
Key Concept: We need to interpret two conditions on complex numbers: |z| ≤ 4 describes all points within or on a circle of radius 4 centered at origin, while 0 ≤ arg(z) ≤ π/3 describes all points whose argument lies between 0 and π/3 (a wedge-shaped angular region). The intersection of these conditions forms a sector.
<p><strong>Step 1: Analyze the condition |z| ≤ 4</strong></p><p>This represents all complex numbers whose modulus (distance from origin) is at most 4. Geometrically, this is a closed disk (filled circle) with center at origin and radius 4, including the interior and boundary.</p><p><strong>Step 2: Analyze the condition 0 ≤ arg(z) ≤ π/3</strong></p><p>The argument of z is the angle that the line from origin to z makes with the positive real axis. This condition restricts z to the angular region between 0 radians (positive real axis) and π/3 radians (60°). This is a wedge-shaped region bounded by two rays from the origin.</p><p><strong>Step 3: Find the intersection of both conditions</strong></p><p>We need points that satisfy BOTH conditions simultaneously:</p><p>• Distance from origin ≤ 4 (within or on the circle)</p><p>• Angle with positive real axis between 0 and π/3 (within the angular wedge)</p><p><strong>Step 4: Identify the geometric shape</strong></p><p>The intersection of a disk with an angular wedge forms a <strong>sector of a circle</strong>. This is the region bounded by:</p><p>• Two radii: one along the positive real axis (arg = 0) and one at angle π/3</p><p>• The circular arc of radius 4 connecting these two radii</p><p>The sector includes all interior points and its boundary (since both inequalities are non-strict).</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C