Algebra 2 Quiz: Geometric Representations Of Complex Numbers
20 questions · exam conditions
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Geometric Representations Of Complex NumbersQuestion 1 of 20

Let z=4−3iz=4-3i. The complex conjugate z‾\overline{z} is obtained by reflecting the point for zz across the real axis. Which statement correctly describes z‾\overline{z} and its point on the complex plane?​

z‾=4+3i\overline{z}=4+3i, plotted at (4,3)(4,3)
z‾=−4−3i\overline{z}=-4-3i, plotted at (−4,−3)(-4,-3)
z‾=−4+3i\overline{z}=-4+3i, plotted at (−4,3)(-4,3)
z‾=3−4i\overline{z}=3-4i, plotted at (3,−4)(3,-4)
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Algebra 2 Quiz

Algebra 2 Quiz: Geometric Representations Of Complex Numbers

Practice Geometric Representations Of Complex Numbers in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Geometric Representations Of Complex Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let z=4−3iz=4-3i. The complex conjugate z‾\overline{z} is obtained by reflecting the point for zz across the real axis. Which statement correctly describes z‾\overline{z} and its point on the complex plane?​

  1. z‾=4+3i\overline{z}=4+3i, plotted at (4,3)(4,3) (correct answer)
  2. z‾=−4−3i\overline{z}=-4-3i, plotted at (−4,−3)(-4,-3)
  3. z‾=−4+3i\overline{z}=-4+3i, plotted at (−4,3)(-4,3)
  4. z‾=3−4i\overline{z}=3-4i, plotted at (3,−4)(3,-4)
Explanation: This question tests your understanding of complex conjugates and their geometric reflection on the complex plane. The conjugate of a + bi is a - bi, which reflects the point (a, b) over the real axis to (a, -b); for z = 4 - 3i at (4, -3), the conjugate is 4 + 3i at (4, 3). Geometrically, this flips the sign of the imaginary part while keeping the real part, mirroring across the horizontal axis. Choice A correctly gives \bar{z} = 4 + 3i at (4, 3). A tempting distractor like Choice B negates both parts to -4 - 3i at (-4, -3), but conjugation only changes the imaginary sign, not the real—it's not a full inversion! To find the conjugate, change the sign of the i term only; plot by keeping x-coordinate and flipping y-coordinate over the x-axis. Both z and \bar{z} have the same modulus since reflection preserves distance—wonderful, you're grasping this reflection idea!

Question 2

A force in a plane is represented by the complex number F=3+4iF=3+4i (real part is the horizontal component, imaginary part is the vertical component). What is the magnitude of the force, interpreted as the modulus ∣F∣=32+42|F|=\sqrt{3^2+4^2}?

  1. ∣F∣=7|F|=7
  2. ∣F∣=1|F|=1
  3. ∣F∣=5|F|=5 (correct answer)
  4. ∣F∣=32+42=25=25|F|=\sqrt{3^2+4^2}=\sqrt{25}=25
Explanation: This question tests your understanding of interpreting the modulus of a complex number as the magnitude of a force vector on the 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 F = 3 + 4i, components 3 horizontal and 4 vertical, magnitude = sqrt(9 + 16) = sqrt(25) = 5, like vector length. Choice C correctly gives 5. Choice D computes sqrt(25) but says =25— that's forgetting sqrt(25)=5, not 25! Treat forces as vectors: magnitude is modulus, direction by angle—outstanding, you're applying this to physics brilliantly!

Question 3

Plotting complex numbers uses the point (a,b)(a,b) to represent a+bia+bi on the complex plane. If z=2+iz=2+i, what complex number corresponds to the point obtained by multiplying by ii (i.e., iziz), and what geometric transformation does this represent?

  1. iz=2−iiz=2-i; reflection across the real axis
  2. iz=−1+2iiz=-1+2i; rotation 90∘90^\circ counterclockwise about the origin (correct answer)
  3. iz=1−2iiz=1-2i; rotation 90∘90^\circ clockwise about the origin
  4. iz=−2−iiz=-2-i; rotation 180∘180^\circ about the origin
Explanation: This question tests your understanding of multiplying by i and its geometric effect as a rotation 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 + i at (2,1), iz = i(2 + i) = 2i + i² = 2i - 1 = -1 + 2i at (-1,2); this is a 90° counterclockwise rotation, as multiplying by i swaps and changes signs: real becomes -imag, imag becomes real. Choice B correctly computes -1 + 2i and identifies the rotation. Choice C has 1 - 2i, which would be clockwise—multiplying by -i does that, not i! Multiplying by i rotates 90° CCW: new point (-b, a) from (a,b); modulus stays the same—amazing, you're grasping transformations like a pro!

Question 4

On the complex plane (real axis horizontal, imaginary axis vertical), the complex number z=−3+2iz = -3 + 2i is represented by the point (a,b)(a,b) where aa is the real part and bb is the imaginary part. Which point should be plotted for zz?

  1. (−3,2)(-3, 2) (correct answer)
  2. (2,−3)(2, -3)
  3. (3,2)(3, 2)
  4. (−3,−2)(-3, -2)
Explanation: This question tests your understanding of representing complex numbers geometrically on the complex plane by plotting their real and imaginary parts as 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 + 2i, identify the real part a = -3 and the imaginary part b = 2, so plot at (-3, 2): start at the origin, move 3 units left (negative real), then 2 units up (positive imaginary), and mark the point. Choice A correctly identifies the point as (-3, 2) by assigning the real part to the horizontal axis and the imaginary part to the vertical axis. Choice B might tempt you if you switch the axes, plotting (2, -3) by mistakenly putting the imaginary part on the horizontal axis, but remember, real is horizontal (like x), imaginary is vertical (like y)—the order is crucial! To plot any complex number a + bi: (1) Draw axes with horizontal labeled 'Real' and vertical 'Imaginary,' (2) Move a units along the real axis (right for positive, left for negative), then b units along the imaginary axis (up for positive, down for negative), and plot the point. Keep practicing this, and you'll get great at visualizing complex numbers geometrically—you've got this!

Question 5

Consider the set of all complex numbers z=a+biz=a+bi such that ∣z∣=5|z|=5, where ∣z∣=a2+b2|z|=\sqrt{a^2+b^2}. What does this set represent geometrically on the complex plane?

  1. A line segment from (0,0)(0,0) to (5,0)(5,0)
  2. A circle of radius 5 centered at the origin (correct answer)
  3. A circle of radius 5 centered at (5,0)(5,0)
  4. The vertical line a=5a=5
Explanation: This question tests your understanding of the geometric set of complex numbers with a fixed modulus, like |z| = 5. 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! The set |z| = 5 means all points (a,b) where sqrt(a² + b²) = 5, which is a circle of radius 5 centered at the origin (0,0), including points like 5 + 0i, 0 + 5i, etc. Choice B correctly describes this circle. Choice A is just the segment to (5,0), but the set includes all directions at distance 5—not a line! For |z| = r, it's always a circle radius r at origin; vary a and b with a² + b² = r²—excellent, you're visualizing loci perfectly!

Question 6

Let z=2−5iz=2-5i. Which statement correctly describes the geometric relationship between zz and its conjugate z‾\overline{z} on the complex plane?​

  1. z‾\overline{z} is a reflection of zz across the imaginary axis.
  2. z‾\overline{z} is a 90° counterclockwise rotation of zz about the origin.
  3. z‾\overline{z} is a reflection of zz across the real axis. (correct answer)
  4. z‾\overline{z} is the point obtained by swapping the coordinates of zz.
Explanation: This question tests your understanding of complex conjugates and their geometric representation on the complex plane. The conjugate of a complex number z = a + bi is defined as z̄ = a - bi, which changes the sign of the imaginary part while keeping the real part the same. For z = 2 - 5i, the conjugate is z̄ = 2 + 5i: the point (2, -5) becomes (2, 5), which is a reflection across the real axis (horizontal axis)—imagine folding the complex plane along the real axis, and z lands on z̄! Choice B correctly identifies that z̄ is a reflection of z across the real axis, as the real parts match but imaginary parts have opposite signs. Choice A incorrectly states reflection across the imaginary axis, which would change the sign of the real part (giving -2 - 5i), not the imaginary part—that's a different transformation! The conjugate operation has beautiful geometric properties: (1) It reflects points across the real axis (horizontal axis), (2) Points on the real axis are their own conjugates (since b = 0), (3) The modulus stays the same: |z| = |z̄| because distance from origin is preserved by reflection. Understanding conjugates geometrically helps visualize why z × z̄ always gives a real number (it's |z| squared) and why conjugates appear in pairs as roots of polynomials with real coefficients!

Question 7

A complex number z=1+3iz=1+3i is represented by the point (1,3)(1,3) on the complex plane. If you add w=2−iw=2-i, the sum z+wz+w corresponds to adding the vectors. What point represents z+wz+w?​

  1. (3, 2)(3,\,2) (correct answer)
  2. (−1, 4)(-1,\,4)
  3. (2, 4)(2,\,4)
  4. (3, 4)(3,\,4)
Explanation: This question tests your understanding of complex number addition as vector addition on the complex plane. When adding complex numbers z = a + bi and w = c + di, we get (a + c) + (b + d)i, which corresponds to adding the vectors from origin to each point—just add corresponding coordinates! To find z + w where z = 1 + 3i and w = 2 - i: (1) Add real parts: 1 + 2 = 3. (2) Add imaginary parts: 3 + (-1) = 2. (3) Result: z + w = 3 + 2i, represented by point (3, 2). Geometrically, this is vector addition: from origin to (1, 3), then add displacement vector (2, -1) to reach (3, 2). Choice A correctly identifies the sum as the point (3, 2), properly adding both real and imaginary components. Choice B gives (-1, 4), which would be from subtracting real parts but adding imaginary parts—remember to add both parts consistently! The vector interpretation of complex addition: (1) Each complex number is a vector from origin to its point, (2) Adding complex numbers = adding vectors tip-to-tail, (3) The sum's coordinates are the sums of the corresponding coordinates. This geometric view explains why complex addition is commutative (z + w = w + z) and helps visualize translations: adding a fixed complex number w to any z translates z by the vector w!

Question 8

A complex number z=a+biz=a+bi is plotted as the point (a,b)(a,b) on the complex plane. What is the modulus ∣z∣|z| of z=6−8iz=6-8i? (Recall ∣a+bi∣=a2+b2|a+bi|=\sqrt{a^2+b^2}.)

  1. 1414
  2. 1010 (correct answer)
  3. 100\sqrt{100}
  4. 28\sqrt{28}
Explanation: 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 modulus of 6 - 8i: (1) Identify real part a = 6 and imaginary part b = -8 (the coefficient of i). (2) Apply modulus formula: |6 - 8i| = √(6² + (-8)²) = √(36 + 64) = √100 = 10. (3) Geometric interpretation: the point (6, -8) is 10 units from origin. To plot this number: start at origin, move 6 units right (positive real), then 8 units down (negative imaginary), mark the point. The modulus 10 is the straight-line distance from origin to this point. Choice C correctly calculates the modulus as 10, which equals √100 shown in choice D—both are correct representations of the same value! Choice A gives √28, which would be the modulus of a different complex number like 2 + 2√6i, while choice B gives 14, which might come from incorrectly adding 6 + 8 instead of using the Pythagorean theorem. Modulus (distance from origin) recipe: (1) From a + bi, identify a and b (watch signs!). (2) Square both: a² and b² (both positive after squaring). (3) Add: a² + b². (4) Take square root: √(a² + b²). That's the modulus! The modulus is always non-negative (it's a distance), and it tells us how far the complex number is from zero on the complex plane!

Question 9

The complex number z=3−4iz = 3 - 4i is represented as a point on the complex plane. When zz is multiplied by ii, the resulting point is obtained by which geometric transformation of the original point?

  1. A rotation of 90°90° counterclockwise about the origin (correct answer)
  2. A reflection across the real axis followed by a scaling by factor ii
  3. A translation by 44 units left and 33 units up from the origin
  4. A rotation of 90°90° clockwise about the origin with magnitude preserved
Explanation: Multiplying a complex number by ii corresponds to a 90°90° counterclockwise rotation about the origin. We can verify: z⋅i=(3−4i)⋅i=3i−4i2=3i+4=4+3iz \cdot i = (3-4i) \cdot i = 3i - 4i^2 = 3i + 4 = 4 + 3i. The original point (3,−4)(3, -4) becomes (4,3)(4, 3), which is indeed a 90°90° counterclockwise rotation. Choice B incorrectly describes reflection and scaling. Choice C describes translation, which is incorrect. Choice D has the wrong direction of rotation.

Question 10

Two complex numbers u=1+iu = 1 + i and v=3−iv = \sqrt{3} - i are multiplied together. If this multiplication is interpreted geometrically as a composition of scaling and rotation, what is the total angle of rotation applied to uu?

  1. 15°15° obtained by adding the individual arguments of both complex numbers
  2. −15°-15° obtained by subtracting the argument of vv from the argument of uu
  3. 75°75° obtained by adding 45°45° and 30°30° from the respective arguments
  4. −30°-30° obtained by finding the argument of vv and applying it as rotation (correct answer)
Explanation: In complex multiplication, the geometric effect is scaling by the modulus and rotating by the argument of the multiplier. Here uu is being multiplied by vv, so we need the argument of v=3−iv = \sqrt{3} - i. Since vv is in the fourth quadrant with tan⁡−1(−13)=−30°\tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) = -30°, the rotation angle is −30°-30°. Choice A incorrectly adds both arguments. Choice B subtracts in wrong order. Choice C incorrectly uses 30°30° instead of −30°-30° for vv's argument.

Question 11

Two complex numbers z1=2+3iz_1 = 2 + 3i and z2=1−2iz_2 = 1 - 2i are added geometrically on the complex plane. If the parallelogram method is used to represent this addition, what are the coordinates of the vertex opposite to the origin?

  1. (3,1)(3, 1) representing the sum of the two complex numbers (correct answer)
  2. (1,5)(1, 5) representing twice the first complex number minus the second
  3. (4,2)(4, 2) representing the sum of both complex numbers doubled
  4. (2,4)(2, 4) representing the midpoint of the diagonal connecting the endpoints
Explanation: In the parallelogram method for complex addition, we place vectors representing z1z_1 and z2z_2 tail-to-tail at the origin. The vertex opposite the origin is located at the sum z1+z2=(2+3i)+(1−2i)=3+iz_1 + z_2 = (2+3i) + (1-2i) = 3+i, which corresponds to point (3,1)(3,1). Choice B gives an incorrect calculation. Choice C incorrectly doubles the sum. Choice D confuses the opposite vertex with the midpoint of a diagonal.

Question 12

A complex number w=−1−4iw=-1-4i is plotted on the complex plane with coordinates (a,b)(a,b) corresponding to a+bia+bi. What are the coordinates of the point for ww, and what is the modulus ∣w∣=a2+b2|w|=\sqrt{a^2+b^2}?

  1. Point (−1,−4)(-1,-4) and ∣w∣=17|w|=\sqrt{17} (correct answer)
  2. Point (−4,−1)(-4,-1) and ∣w∣=17|w|=\sqrt{17}
  3. Point (1,4)(1,4) and ∣w∣=15|w|=\sqrt{15}
  4. Point (−1,4)(-1,4) and ∣w∣=5|w|=\sqrt{5}
Explanation: This question tests your understanding of plotting negative complex numbers on the complex plane and calculating their modulus 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 the hypotenuse! For w = -1 - 4i, real part a = -1 (left on horizontal axis) and imaginary part b = -4 (down on vertical axis), so plot at (-1, -4); modulus is sqrt(1 + 16) = sqrt(17), the distance to origin. Choice A correctly gives point (-1,-4) and modulus sqrt(17), aligning with the signs and formula. Choice B swaps to (-4,-1), but real is horizontal and imaginary vertical—order matters, don't reverse them! When handling negatives, remember: negative real means left of origin, negative imaginary means below; for modulus, squaring makes them positive, so it's always sqrt of sum of squares—great job verifying signs, you're building strong skills!

Question 13

On the complex plane (real axis horizontal, imaginary axis vertical), the complex number z=−3+4iz = -3 + 4i is represented by the point (a,b)(a,b) where z=a+biz=a+bi. What is the modulus ∣z∣|z| (the distance from the origin), and which point represents zz?​

  1. Point (−3,4)(-3,4) and ∣z∣=5|z|=5 (correct answer)
  2. Point (4,−3)(4,-3) and ∣z∣=5|z|=5
  3. Point (−3,4)(-3,4) and ∣z∣=7|z|=7
  4. Point (3,−4)(3,-4) and ∣z∣=5|z|=5
Explanation: This question tests your understanding of representing complex numbers geometrically on the complex plane and finding their modulus 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 + 4i goes at (-3, 4), and the modulus |z| equals sqrt(a2a^2 + b2b^2), which is the distance from the origin using the Pythagorean theorem. For z = -3 + 4i, identify a = -3 and b = 4, so plot at (-3, 4) and |z| = sqrt(9 + 16) = sqrt(25) = 5; geometrically, it's 5 units from the origin in the second quadrant. Choice A correctly identifies the point (-3, 4) and modulus 5. A tempting distractor like Choice B swaps the coordinates to (4, -3), but remember, real part is horizontal (x-like) and imaginary is vertical (y-like)—don't reverse them! To plot any complex number a + bi, draw axes with real horizontal and imaginary vertical, move a units horizontally (positive right, negative left) and b units vertically (positive up, negative down), then mark the point; for modulus, always use sqrt(a2a^2 + b2b^2), which is always non-negative as it's a distance—keep practicing, you've got this!

Question 14

The complex plane below shows the real axis (horizontal) and imaginary axis (vertical). Points PP and QQ are plotted at P(3,2)P(3,2) and Q(−1,−4)Q(-1,-4). Which pair of complex numbers matches points PP and QQ?

(Recall: the point (a,b)(a,b) represents a+bia+bi.)​

  1. P=2+3iP=2+3i and Q=−1−4iQ=-1-4i
  2. P=3−2iP=3-2i and Q=−1+4iQ=-1+4i
  3. P=2+3iP=2+3i and Q=−4−iQ=-4-i
  4. P=3+2iP=3+2i and Q=−1−4iQ=-1-4i (correct answer)
Explanation: This question tests your understanding of matching complex numbers to their points on the complex plane. The point (a, b) represents a + b i, with a real (horizontal) and b imaginary (vertical); so P at (3, 2) is 3 + 2i, Q at (-1, -4) is -1 - 4i. Geometrically, (3, 2) is right 3, up 2; (-1, -4) is left 1, down 4. Choice B correctly pairs P = 3 + 2i and Q = -1 - 4i. A tempting distractor like Choice A swaps reals and imaginaries to 2 + 3i and -4 - i, but real is x-coordinate, imaginary is y—don't switch! To match, read x as real part, y as imaginary part, including signs; practice plotting back and forth. This skill is foundational for complex geometry—keep practicing, you're on the right track!

Question 15

Let z=4−3iz=4-3i. The complex conjugate z‾\overline{z} is obtained by reflecting the point for zz across the real axis. Which statement correctly describes z‾\overline{z} and its point on the complex plane?

  1. z‾=4+3i\overline{z}=4+3i, plotted at (4,3)(4,3) (correct answer)
  2. z‾=−4−3i\overline{z}=-4-3i, plotted at (−4,−3)(-4,-3)
  3. z‾=−4+3i\overline{z}=-4+3i, plotted at (−4,3)(-4,3)
  4. z‾=3−4i\overline{z}=3-4i, plotted at (3,−4)(3,-4)
Explanation: This question tests your understanding of complex conjugates and their geometric reflection on the complex plane. The conjugate of a + bi is a - bi, which reflects the point (a, b) over the real axis to (a, -b); for z = 4 - 3i at (4, -3), the conjugate is 4 + 3i at (4, 3). Geometrically, this flips the sign of the imaginary part while keeping the real part, mirroring across the horizontal axis. Choice A correctly gives \bar{z} = 4 + 3i at (4, 3). A tempting distractor like Choice B negates both parts to -4 - 3i at (-4, -3), but conjugation only changes the imaginary sign, not the real—it's not a full inversion! To find the conjugate, change the sign of the i term only; plot by keeping x-coordinate and flipping y-coordinate over the x-axis. Both z and \bar{z} have the same modulus since reflection preserves distance—wonderful, you're grasping this reflection idea!

Question 16

The complex plane below shows the real axis (horizontal) and imaginary axis (vertical). Points PP and QQ are plotted at P(3,2)P(3,2) and Q(−1,−4)Q(-1,-4). Which pair of complex numbers matches points PP and QQ?

(Recall: the point (a,b)(a,b) represents a+bia+bi.)

  1. P=2+3iP=2+3i and Q=−4−iQ=-4-i
  2. P=3−2iP=3-2i and Q=−1+4iQ=-1+4i
  3. P=2+3iP=2+3i and Q=−1−4iQ=-1-4i
  4. P=3+2iP=3+2i and Q=−1−4iQ=-1-4i (correct answer)
Explanation: This question tests your understanding of matching complex numbers to their points on the complex plane. The point (a, b) represents a + b i, with a real (horizontal) and b imaginary (vertical); so P at (3, 2) is 3 + 2i, Q at (-1, -4) is -1 - 4i. Geometrically, (3, 2) is right 3, up 2; (-1, -4) is left 1, down 4. Choice B correctly pairs P = 3 + 2i and Q = -1 - 4i. A tempting distractor like Choice A swaps reals and imaginaries to 2 + 3i and -4 - i, but real is x-coordinate, imaginary is y—don't switch! To match, read x as real part, y as imaginary part, including signs; practice plotting back and forth. This skill is foundational for complex geometry—keep practicing, you're on the right track!

Question 17

On the complex plane (real axis horizontal, imaginary axis vertical), the complex number z=−3+4iz = -3 + 4i is represented by the point (a,b)(a,b) where z=a+biz=a+bi. What is the modulus ∣z∣|z| (the distance from the origin), and which point represents zz?

  1. Point (−3,4)(-3,4) and ∣z∣=5|z|=5 (correct answer)
  2. Point (4,−3)(4,-3) and ∣z∣=5|z|=5
  3. Point (−3,4)(-3,4) and ∣z∣=7|z|=7
  4. Point (3,−4)(3,-4) and ∣z∣=5|z|=5
Explanation: This question tests your understanding of representing complex numbers geometrically on the complex plane and finding their modulus 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 + 4i goes at (-3, 4), and the modulus |z| equals sqrt(a2a^2 + b2b^2), which is the distance from the origin using the Pythagorean theorem. For z = -3 + 4i, identify a = -3 and b = 4, so plot at (-3, 4) and |z| = sqrt(9 + 16) = sqrt(25) = 5; geometrically, it's 5 units from the origin in the second quadrant. Choice A correctly identifies the point (-3, 4) and modulus 5. A tempting distractor like Choice B swaps the coordinates to (4, -3), but remember, real part is horizontal (x-like) and imaginary is vertical (y-like)—don't reverse them! To plot any complex number a + bi, draw axes with real horizontal and imaginary vertical, move a units horizontally (positive right, negative left) and b units vertically (positive up, negative down), then mark the point; for modulus, always use sqrt(a2a^2 + b2b^2), which is always non-negative as it's a distance—keep practicing, you've got this!

Question 18

Let z=−2+7iz=-2+7i and w=1−1iw=1-1i. What is the distance between the points representing zz and ww on the complex plane?

  1. 32+62=45\sqrt{3^2+6^2}=\sqrt{45}
  2. 58\sqrt{58}
  3. 73\sqrt{73} (correct answer)
  4. 10\sqrt{10}
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using 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! To find the distance between two complex numbers z and w, we calculate |z - w|, which gives the straight-line distance between their corresponding points. For z = -2 + 7i at point (-2, 7) and w = 1 - 1i at point (1, -1): (1) First find z - w = (-2 + 7i) - (1 - 1i) = -2 + 7i - 1 + 1i = -3 + 8i. (2) Then calculate |z - w| = |-3 + 8i| = √((-3)² + 8²) = √(9 + 64) = √73. (3) This represents the distance between points (-2, 7) and (1, -1) on the complex plane. Choice A correctly gives √73 as the distance. Choice B gives √58, which might come from a calculation error. Choice C gives √10, which is too small for points this far apart. Choice D shows the calculation √(3² + 6²) = √45, but this uses incorrect values—the difference is -3 + 8i, not 3 + 6i! The distance formula for complex numbers mirrors the familiar distance formula from coordinate geometry: d = √((x₂-x₁)² + (y₂-y₁)²). This connection shows how complex numbers unify algebra and geometry, letting us solve geometric problems using algebraic tools and visualize algebraic concepts geometrically!

Question 19

On the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number z=3+2iz=3+2i is represented by the point (a,b)(a,b) where aa is the real part and bb is the imaginary part. Which point represents zz, and what is its modulus ∣z∣=a2+b2|z|=\sqrt{a^2+b^2}?

  1. Point (−3,2)(-3,2) and ∣z∣=5|z|=5
  2. Point (3,2)(3,2) and ∣z∣=13|z|=\sqrt{13} (correct answer)
  3. Point (2,3)(2,3) and ∣z∣=13|z|=\sqrt{13}
  4. Point (3,−2)(3,-2) and ∣z∣=5|z|=\sqrt{5}
Explanation: 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 the hypotenuse! For z = 3 + 2i, identify real part a = 3 and imaginary part b = 2, so plot at (3, 2); the modulus is sqrt(9 + 4) = sqrt(13), and geometrically, that's the straight-line distance from (0,0) to (3,2). Choice B correctly identifies the point (3,2) and modulus sqrt(13), matching the standard plotting and formula. Choice A swaps the coordinates to (2,3), but remember, real part is horizontal (x-like), imaginary is vertical (y-like)—don't mix them up! To plot any complex number a + bi: draw axes with 'Real' horizontal and 'Imaginary' vertical, move a units horizontally (right for positive, left for negative) and b units vertically (up for positive, down for negative), then mark the point; for modulus, always compute sqrt(a² + b²), which is always non-negative like any distance—keep practicing, you've got this!

Question 20

On the complex plane (real axis horizontal, imaginary axis vertical), the complex number z=−3+4iz = -3 + 4i is represented by the point (a,b)(a,b).

Which point corresponds to zz, and what is ∣z∣|z| (the distance from the origin), where ∣a+bi∣=a2+b2|a+bi|=\sqrt{a^2+b^2}?

  1. (4,−3)(4,-3) and ∣z∣=5|z|=5
  2. (−3,4)(-3,4) and ∣z∣=7|z|=7
  3. (−3,4)(-3,4) and ∣z∣=5|z|=5 (correct answer)
  4. (3,4)(3,4) and ∣z∣=5|z|=5
Explanation: 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 right triangle, modulus is hypotenuse! To plot z = -3 + 4i: (1) Identify real part a = -3 and imaginary part b = 4 (the coefficient of i). (2) Plot at point (-3, 4): start at origin, move 3 units left (negative real), then 4 units up (positive imaginary). (3) Calculate modulus: |z| = square root of ((-3) squared + 4 squared) = square root of (9 + 16) = square root of 25 = 5. The point is 5 units from the origin! Choice C correctly identifies the point as (-3, 4) and calculates |z| = 5 using the distance formula. Choice B incorrectly swaps the coordinates to (4, -3) instead of (-3, 4)—remember, real part goes horizontal, imaginary vertical! Choice B also miscalculates |z| = 7 by adding |-3| + |4| instead of using the Pythagorean theorem. Choice D has the wrong sign on the real part. Plotting complex numbers recipe: (1) From a + bi, a is horizontal coordinate, b is vertical. (2) Plot at (a, b). (3) Modulus = square root of (a squared + b squared). Always check your signs carefully—negative real means left, negative imaginary means down!