Complex numbers: polar form and De Moivre Test

{"questions":[{"content":"Enter the modulus of $-8+15i$. [[input-complex-numbers-polar-form-and-de-moivre-test-1]]","widgets":{"input-complex-numbers-polar-form-and-de-moivre-test-1":{"type":"input","answer":"17"}},"hints":[],"explanation":"The squared modulus is 64 + 225 = 289.","id":"complex-numbers-polar-form-and-de-moivre-test-1","evidence_kind":"objective-concept-test","correct_feedback":"The squared modulus is 64 + 225 = 289.","incorrect_feedback":"Square both components before adding: a negative real part still contributes a positive square. The modulus is the positive square root of the sum."},{"content":"Find the principal argument of $-\\sqrt{3}-i$.\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-2]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-2":{"type":"choice","options":["$-5π/6$","$π/6$","$-π/6$"],"answer":[0],"choice_feedback":["The point is in quadrant III; the principal argument is −5π/6.","An argument of π/6 has both components positive. Here both components are negative, so the point is in quadrant III.","An argument of −π/6 has positive real part. Keep the reference angle π/6 but choose the principal angle in quadrant III."]}},"hints":[],"explanation":"The point is in quadrant III; the principal argument is −5π/6.","id":"complex-numbers-polar-form-and-de-moivre-test-2","evidence_kind":"objective-concept-test","correct_feedback":"The point is in quadrant III; the principal argument is −5π/6.","feedback_visual":{"type":"argand","points":[[-1.7320508075688772,-1,"z = −√3 − i"]],"caption":"Both components are negative. The principal argument is measured clockwise through 5π/6, giving −5π/6."}},{"content":"For $z=3(\\cos(π/6)+i\\sin(π/6))$, find $z^3$.\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-3]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-3":{"type":"choice","options":["$27i$","$9i$","$-27$"],"answer":[0],"choice_feedback":["Cubing gives modulus 27 and argument π/2.","Cubing a modulus of 3 gives 27, not 9. The angle π/2 is correct, so the answer should be positive imaginary with that larger modulus.","The argument is multiplied by 3, giving π/2 rather than π. This points along the positive imaginary axis."]}},"hints":[],"explanation":"Cubing gives modulus 27 and argument π/2.","id":"complex-numbers-polar-form-and-de-moivre-test-3","evidence_kind":"objective-concept-test","correct_feedback":"Cubing gives modulus 27 and argument π/2."},{"content":"Find $(1+i)^6$.\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-4]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-4":{"type":"choice","options":["$-8i$","$8i$","$-8$"],"answer":[0],"choice_feedback":["The modulus becomes (√2)^6 = 8 and the argument becomes 3π/2.","Six times the argument π/4 is 3π/2, not π/2. The result points down the imaginary axis.","The modulus is 8, but the argument is 3π/2. A negative real result would require argument π."]}},"hints":[],"explanation":"The modulus becomes (√2)^6 = 8 and the argument becomes 3π/2.","id":"complex-numbers-polar-form-and-de-moivre-test-4","evidence_kind":"objective-concept-test","correct_feedback":"The modulus becomes (√2)^6 = 8 and the argument becomes 3π/2."},{"content":"If $z/w=2(\\cos(π/3)+i\\sin(π/3))$ and $w=3i$, find $z$.\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-5]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-5":{"type":"choice","options":["$-3\\sqrt{3}+3i$","$3+3\\sqrt{3}i$","$3\\sqrt{3}-3i$"],"answer":[0],"choice_feedback":["The product has modulus 6 and argument 5π/6.","Multiplication by 3i rotates by π/2 as well as scaling by 3. Add this rotation to π/3 before finding components.","The product has argument 5π/6, in quadrant II. Your signs put it in quadrant IV instead."]}},"hints":[],"explanation":"The product has modulus 6 and argument 5π/6.","id":"complex-numbers-polar-form-and-de-moivre-test-5","evidence_kind":"objective-concept-test","correct_feedback":"The product has modulus 6 and argument 5π/6."},{"content":"A student finds only one cube root of $8i$ by dividing π/2 by 3. Which additional arguments are needed in [0,2π)?\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-6]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-6":{"type":"choice","options":["$5π/6$ and $3π/2$","$π/3$ and $π/2$","$7π/6$ and $11π/6$"],"answer":[0],"choice_feedback":["Start from π/6 and add 2π/3 twice to obtain all three roots.","Successive cube roots differ by 2π/3, not π/6. Start with π/6 and add a third of a full turn.","Those angles do not differ from π/6 by multiples of 2π/3. Cubing each candidate argument must give π/2 modulo 2π."]}},"hints":[],"explanation":"Start from π/6 and add 2π/3 twice to obtain all three roots.","id":"complex-numbers-polar-form-and-de-moivre-test-6","evidence_kind":"objective-concept-test","correct_feedback":"Start from π/6 and add 2π/3 twice to obtain all three roots.","deeper_explanation":"Use all arguments of $8i$: $π/2+2kπ$. Dividing by 3 for k = 0, 1, 2 gives $π/6$, $5π/6$ and $3π/2$.","feedback_visual":{"type":"argand","points":[],"caption":"The three cube roots of 8i have modulus 2. The numbered points show all three directions, each separated by 2π/3.","roots":{"count":3,"radius":2,"phase":0.5235987755982988}}},{"content":"A nonzero number $z$ has principal argument π/8. What is the least positive integer n for which $z^n$ is negative real? [[input-complex-numbers-polar-form-and-de-moivre-test-7]]","widgets":{"input-complex-numbers-polar-form-and-de-moivre-test-7":{"type":"input","answer":"8"}},"hints":[],"explanation":"The first negative real direction occurs when nπ/8 = π.","id":"complex-numbers-polar-form-and-de-moivre-test-7","evidence_kind":"objective-concept-test","correct_feedback":"The first negative real direction occurs when nπ/8 = π.","incorrect_feedback":"A negative real result first occurs when the argument reaches an odd multiple of π. Set nπ/8 = π and check that no smaller positive integer works."},{"content":"The roots of $z^3=27(\\cos(π/3)+i\\sin(π/3))$ form a triangle on an Argand diagram. Find its side length.\u003Cbr\u003E[[choice-complex-numbers-polar-form-and-de-moivre-test-8]]","widgets":{"choice-complex-numbers-polar-form-and-de-moivre-test-8":{"type":"choice","options":["$3\\sqrt{3}$","$3$","$9$"],"answer":[0],"choice_feedback":["The roots lie on a circle of radius 3, separated by 120°. Each chord has length 2(3)sin(60°).","The roots have radius 3, but the side of the triangle is a chord between different roots, not a radius.","A side length of 9 exceeds the diameter 6, so it cannot join two points on this circle. Use the 120° angle between adjacent roots."]}},"hints":[],"explanation":"The roots lie on a circle of radius 3, separated by 120°. Each chord has length 2(3)sin(60°).","id":"complex-numbers-polar-form-and-de-moivre-test-8","evidence_kind":"objective-concept-test","correct_feedback":"The roots lie on a circle of radius 3, separated by 120°. Each chord has length 2(3)sin(60°).","deeper_explanation":"Join two neighboring roots to the origin to form an isosceles triangle with sides 3 and included angle 120°. Bisecting it gives half the chord as $3\\sin60°$, so the whole chord is $3\\sqrt3$.","feedback_visual":{"type":"argand","points":[],"caption":"The dashed triangle joins adjacent roots. Each root has radius 3, while each triangle side is the longer chord 3√3.","roots":{"count":3,"radius":3,"phase":0.3490658503988659,"join":true}}}],"mix":0,"tinkumo_module_mode":"test"}