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Score band 700-800 · Additional Topics in Math

Complex Numbers drills for 700-800

400500600700800

Targeted practice for students currently scoring in the 700-800 range, drilling exclusively on complex numbers.

What a 700-800 scorer needs from this topic

The 700 to 800 band is the elite tier of SAT Math. Reaching it requires zero careless errors and confident solves on the hardest one or two questions in each module. Mastery-band drills focus on the hardest released questions: multi-concept problems that combine two or three skills in one stem, abstract algebraic manipulation with parameters instead of numbers, geometry questions that require a clever construction, and data-analysis questions that test conceptual understanding rather than computation. At this level, speed matters as much as accuracy, because the only way to leave time for the hardest questions is to dispatch the easy and medium ones in well under a minute each.

For Complex Numbers specifically, students in the 700-800 band need to focus on the question patterns the College Board uses at this difficulty level. Add, multiply, and simplify complex number expressions. The questions below are pulled from the ScoreReady question bank and filtered to the 700-800 band based on difficulty calibration that matches publicly released College Board practice materials.

Drill these untimed first. Once you can produce a clean worked solution on paper for every question without notes, switch to timed mode and aim for under 75 seconds per question. That pace is roughly the average time per question on the actual SAT Math section, and it leaves time for the harder questions you will see at the end of each module.

Practice set

  1. 700-800 hard Complex Conjugates and Division

    What is the product (4 - 1i)(4 + 5i) expressed in the form a + bi?

    1. A 21 + 16i
    2. B 16 + -5i
    3. C 21 + 17i
    4. D 22 + 16i
    Worked solution

    Answer: A — 21 + 16i

    Use FOIL and replace i^2 with -1. (4 + -1i)(4 + 5i) = 16 + 20i + -4i + -5i^2 = 16 + 16i + -5(-1) = 21 + 16i.

  2. 700-800 hard Powers of i

    What is the product (-1 - 1i)(-3 - 1i) expressed in the form a + bi?

    1. A 2 + 5i
    2. B 3 + 4i
    3. C 2 + 4i
    4. D 3 + 1i
    Worked solution

    Answer: C — 2 + 4i

    Use FOIL and replace i^2 with -1. (-1 + -1i)(-3 + -1i) = 3 + 1i + 3i + 1i^2 = 3 + 4i + 1(-1) = 2 + 4i.

  3. 700-800 hard Adding and Subtracting Complex Numbers

    What is the product (1 - 5i)(5 - 4i) expressed in the form a + bi?

    1. A 5 + 20i
    2. B -15 - 29i
    3. C -15 + -28i
    4. D -14 - 29i
    Worked solution

    Answer: B — -15 - 29i

    Use FOIL and replace i^2 with -1. (1 + -5i)(5 + -4i) = 5 + -4i + -25i + 20i^2 = 5 + -29i + 20(-1) = -15 + -29i.

  4. 700-800 hard Multiplying Complex Numbers

    What is the product (-2 - 6i)(-6 + 1i) expressed in the form a + bi?

    1. A 19 + 34i
    2. B 18 + 34i
    3. C 18 + 35i
    4. D 12 + -6i
    Worked solution

    Answer: B — 18 + 34i

    Use FOIL and replace i^2 with -1. (-2 + -6i)(-6 + 1i) = 12 + -2i + 36i + -6i^2 = 12 + 34i + -6(-1) = 18 + 34i.

  5. 700-800 hard Complex Conjugates and Division

    What is the product (5 - 2i)(-3 + 4i) expressed in the form a + bi?

    1. A -6 + 26i
    2. B -7 + 26i
    3. C -7 + 27i
    4. D -15 + -8i
    Worked solution

    Answer: B — -7 + 26i

    Use FOIL and replace i^2 with -1. (5 + -2i)(-3 + 4i) = -15 + 20i + 6i + -8i^2 = -15 + 26i + -8(-1) = -7 + 26i.

  6. 700-800 hard Powers of i

    What is the product (1 - 5i)(1 - 1i) expressed in the form a + bi?

    1. A -3 - 6i
    2. B -4 - 6i
    3. C 1 + 5i
    4. D -4 + -5i
    Worked solution

    Answer: B — -4 - 6i

    Use FOIL and replace i^2 with -1. (1 + -5i)(1 + -1i) = 1 + -1i + -5i + 5i^2 = 1 + -6i + 5(-1) = -4 + -6i.

How to use these drills to climb a band

Climbing from one score band to the next requires a different study mix than climbing within a band. Within a band, you are mostly fixing careless errors and pattern-recognizing the question types you already understand. Climbing to the next band means adding new question types to your toolbox — patterns you currently recognize but cannot solve fluently. The 700–800 set in this drill is exactly that toolbox for students currently in the 600–700 range.

The single most reliable indicator that you are ready to move up a band is being able to explain a worked solution to someone else, in your own words, without referring to notes. Practice this with one classmate or one parent per week. The act of teaching exposes the gaps your timed solves did not.

Other 700-800 drills