This question tests your understanding of the correspondence between points on the complex plane and complex numbers. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The first coordinate is always the real part, the second is the imaginary part. For the point (-2, -7): (1) The x-coordinate -2 is the real part. (2) The y-coordinate -7 is the imaginary part. (3) The complex number is -2 + (-7)i = -2 - 7i. Remember that negative imaginary parts are written with subtraction: we write -2 - 7i, not -2 + (-7)i, for clarity! Choice C correctly identifies the complex number as -2 - 7i. Choice A incorrectly makes the imaginary part positive, giving -2 + 7i, which would correspond to point (-2, 7), not (-2, -7). Choice B swaps the real and imaginary parts, giving -7 - 2i for point (-7, -2). Choice D has the wrong sign on the real part, giving 2 - 7i for point (2, -7). Point to complex number recipe: (1) Point (a, b) corresponds to complex number a + bi. (2) First coordinate → real part, second coordinate → imaginary part. (3) If b is negative, write as a - |b|i for standard form. (4) Check: complex number a + bi plots back at point (a, b). The correspondence is one-to-one—every point has exactly one complex number!

This question tests your understanding of applying complex numbers to model physical quantities and calculating magnitude using the modulus. When a force F = 3 + 4i represents horizontal and vertical components, the magnitude |F| gives the total force strength using the Pythagorean theorem—just like finding the length of the hypotenuse! To find the magnitude of F = 3 + 4i: (1) Identify components: horizontal (real) = 3, vertical (imaginary) = 4. (2) Apply modulus formula: |F| = square root of (3 squared + 4 squared) = square root of (9 + 16) = square root of 25 = 5. (3) Physical interpretation: a force with 3 units horizontal and 4 units vertical has total magnitude 5 units. Choice A correctly calculates the magnitude as square root of (3 squared + 4 squared) = 5, recognizing this as a 3-4-5 right triangle—a classic Pythagorean triple! Choice C incorrectly adds 3 + 4 = 7, which would be like saying a 3-foot horizontal rope and 4-foot vertical rope stretched at right angles have total length 7 feet—but the diagonal distance is 5 feet, not 7! Complex numbers in physics: (1) Forces, velocities, and other vectors can be represented as complex numbers, (2) Real part = horizontal component, imaginary part = vertical component, (3) Modulus = magnitude (total strength/speed), (4) Argument = direction angle from positive horizontal. This representation makes vector addition simple (just add complex numbers) and connects algebra to physical applications!

This question tests your understanding of representing complex numbers geometrically on the complex plane and identifying the correct coordinates. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). For z = -3 + 4i, we identify the real part a = -3 and the imaginary part b = 4, so the complex number is plotted at the point (-3, 4): move 3 units left (negative real) and 4 units up (positive imaginary) from the origin. Choice A correctly identifies the point as (-3, 4), matching the real part -3 on the horizontal axis and imaginary part 4 on the vertical axis. Choice B incorrectly swaps the coordinates to (4, -3), putting the imaginary part on the horizontal axis and negating it—remember, real part goes horizontal, imaginary goes vertical! When plotting complex numbers, always remember: (1) Real part = horizontal coordinate (x-coordinate), (2) Imaginary part = vertical coordinate (y-coordinate), (3) The point (a, b) represents a + bi. This fundamental skill connects algebra to geometry, letting you visualize complex number operations as transformations on the plane!

This question tests your understanding of the geometric distance between two complex numbers on the plane, using $|z - w|$ as the modulus of their difference. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number $a + bi$ is plotted at point $(a, b)$, just like ordered pairs! For example, $3 + 2i$ goes at $(3, 2)$, and $-1 - 4i$ goes at $(-1, -4)$. The modulus (absolute value) of $a + bi$ equals square root of ($a$ squared + $b$ squared), which is the distance from the origin to point $(a, b)$ using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is the hypotenuse! For $z = 3 + 4i$ and $w = 1 + i$, $z - w = (3-1) + (4-1)i = 2 + 3i$, so $|z - w| = \sqrt{4 + 9} = \sqrt{13}$; geometrically, it's the distance between points $(3,4)$ and $(1,1)$. Choice A correctly subtracts components and uses the distance formula. Choice B adds instead of subtracting, but distance uses differences—think vector subtraction! To find distance between any two points $(a,b)$ and $(c,d)$, compute $\sqrt{(a-c)^2 + (b-d)^2}$; this is $|z - w|$—fantastic, you're connecting algebra and geometry beautifully!

This question tests your understanding of adding complex numbers geometrically as vector addition on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is the hypotenuse! For (2 + i) + (1 + 3i) = (2+1) + (1+3)i = 3 + 4i at (3,4); geometrically, add vectors tail-to-head from origin. Choice A correctly sums to 3 + 4i and plots at (3,4). Choice D has 3 + 4i but plots (4,3)—don't swap axes, real is horizontal! To add z1 = a + bi and z2 = c + di, sum reals a+c, imaginaries b+d, plot at (a+c, b+d); it's like vector addition—superb, you're adding with confidence!

This question tests your understanding of calculating the modulus of a complex number using its geometric representation on the complex plane. The modulus |a + bi| is $sqrt(a^2$ + $b^2$), representing the distance from the origin to the point (a, b); for example, for 6 - 8i at (6, -8), it's the hypotenuse of a right triangle with legs 6 and 8. Here, a = 6, b = -8 (note the sign doesn't affect squares), so |z| = sqrt(36 + 64) = sqrt(100) = 10; geometrically, it's 10 units from the origin in the fourth quadrant. Choice B correctly applies the formula with addition inside the square root and computes sqrt(100) = 10. A tempting distractor like Choice A subtracts instead of adding the squares, leading to a negative under the root, but remember, modulus uses $a^2$ + $b^2$ since both are squared positives—subtraction would be incorrect for distance! To find any modulus, identify a and b, square each $(a^2$ and $b^2$), add them, then take the square root—practice with positives and negatives, as signs vanish after squaring. Geometrically, visualize the point and draw the line to the origin; the length is the modulus—great job verifying this, keep building your skills!

This question tests your understanding of the geometric set where $|z| = 5$, the modulus as distance from the origin. $|z| = \sqrt{a^2 + b^2} = 5$ means all points (a, b) exactly 5 units from (0, 0), forming a circle of radius 5 centered at the origin. Geometrically, it's every complex number whose distance to zero is 5, like points on the circumference. Choice A correctly describes this as a circle of radius 5 at the origin. A tempting distractor like Choice B shifts the center to (5, 0), but $|z| = 5$ is centered at origin, not offset—$|z - 5|$ would be centered at 5 + 0i. To describe sets like $|z| = r$, recognize it's a circle radius r at origin; for $|z - c| = r$, it's centered at c. This is key for geometric interpretations—keep practicing, you've got a strong foundation!

This question tests your understanding of calculating the modulus for a complex number in the second quadrant on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is the hypotenuse! For z = -2 + 3i at (-2,3), modulus = sqrt(4 + 9) = sqrt(13), the distance from (0,0) to (-2,3). Choice A correctly squares both and adds. Choice B adds without squaring or rooting, but modulus requires squares for Pythagorean—don't forget! Identify quadrant by signs (negative real, positive imag = second), but modulus ignores direction, just distance—terrific, you're navigating the plane with ease!

This question tests your understanding of representing complex numbers geometrically on the complex plane and finding their modulus (absolute value) as the distance from the origin. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is hypotenuse! To find the distance from origin to z = -5 - 12i: (1) Identify real part a = -5 and imaginary part b = -12. (2) Apply distance formula: |z| = √((-5)² + (-12)²) = √(25 + 144) = √169 = 13. (3) The point (-5, -12) is 13 units from the origin—this forms a 5-12-13 Pythagorean triple! Choice A correctly gives 13 as the distance. Choice B gives √119, which would come from an arithmetic error, choice C gives 17 which might come from incorrectly adding 5 + 12, and choice D shows √(25 - 144) which incorrectly subtracts instead of adds the squared terms—remember, the formula uses a² + b², not a² - b²! When finding modulus, always add the squares of both parts: (1) Square the real part, (2) Square the imaginary part, (3) Add these squares together, (4) Take the square root of the sum. The signs of a and b don't matter after squaring since (-5)² = 25 and (-12)² = 144 are both positive. This distance formula works because complex numbers form a right triangle with the axes, making the Pythagorean theorem our perfect tool!

This question tests your understanding of complex number addition and its geometric interpretation as vector addition on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! When adding complex numbers, we add their real parts and imaginary parts separately, which corresponds to vector addition on the plane. For z₁ = 1 + 2i and z₂ = 4 - 3i: (1) Add real parts: 1 + 4 = 5. (2) Add imaginary parts: 2i + (-3i) = -1i. (3) Therefore, z₁ + z₂ = 5 - 1i, which plots at point (5, -1). Geometrically, this is like placing the tail of vector z₂ at the head of vector z₁—the sum is the vector from origin to the final point! Choice A correctly gives (5, -1) as the coordinates of the sum. Choice B gives (-3, 6), which might come from subtracting instead of adding. Choice C gives (-5, 1), which has both coordinates negated. Choice D gives (1, 5), which incorrectly swaps the real and imaginary parts. Complex addition follows the parallelogram rule: if you draw z₁ and z₂ as vectors from the origin, then complete the parallelogram, the diagonal from the origin represents z₁ + z₂. This visual approach makes complex arithmetic intuitive—addition is just "tip-to-tail" vector addition, making navigation on the complex plane as easy as following directions on a map!