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Numbers Ultimate Exam
Question 1. What is the value of (i^{7})? A) (-i) B) (i) C) (-1) D) (1) Answer: A Explanation: Powers of (i) repeat every four: (i^{1}=i,;i^{2}=-1,;i^{3}=- i,;i^{4}=1). Thus (i^{7}=i^{4}i^{3}=1\cdot(-i)=-i). Question 2. If (z=3-4i), what is (\overline{z})? A) (3+4i) B) (-3+4i) C) (-3-4i) D) (3-4i) Answer: A Explanation: The complex conjugate changes the sign of the imaginary part: ( overline{3-4i}=3+4i). Question 3. Which of the following numbers is purely imaginary? A) (5) B) (-2i) C) (3+0i) D) (0) Answer: B Explanation: A purely imaginary number has zero real part; (-2i) satisfies this. Question 4. Find the modulus of (z=6+8i). A) (10) B) (14) C) (2) D) (\sqrt{100}) Answer: A Explanation: (|z|=\sqrt{6^{2}+8^{2}}=\sqrt{36+64}=10). Question 5. The principal argument of (z=-1-i) (in radians) is: A) (-\pi/4) B) (-3\pi/4) C) (3\pi/4) D) (\pi/4) Answer: B Explanation: The point lies in quadrant III; (\tan\theta =1) so (\theta = -3\pi/4) (principal value in ((-π,π])). Question 6. Which identity holds for any complex numbers (z_1, z_2)?
Numbers Ultimate Exam
A) (|z_1+z_2|=|z_1|+|z_2|) B) (|z_1z_2|=|z_1||z_2|) C) (\arg(z_1z_2)=\arg z_1- arg z_2) D) (\overline{z_1+z_2}= \overline{z_1}\overline{z_2}) Answer: B Explanation: Modulus multiplicative property: (|z_1z_2|=|z_1||z_2|). The other statements are false in general. Question 7. Express (5-12i) in polar form (r(\cos\theta+i\sin\theta)). A) (13(\cos -\tan^{-1}\tfrac{12}{5}+i\sin -\tan^{-1}\tfrac{12}{5})) B) (13(\cos \tan^{-1}\tfrac{12}{5}+i\sin \tan^{-1}\tfrac{12}{5})) C) (13(\cos \pi-\tan^{- 1}\tfrac{12}{5}+i\sin \pi-\tan^{-1}\tfrac{12}{5})) D) (13(\cos \pi+\tan^{-1} tfrac{12}{5}+i\sin \pi+\tan^{-1}\tfrac{12}{5})) Answer: C Explanation: Modulus (r=\sqrt{5^{2}+12^{2}}=13). The point is in quadrant IV, so (\theta=-\tan^{-1}(12/5)) or equivalently (\pi-\tan^{-1}(12/5)). Option C gives the correct representation. Question 8. Compute (\frac{1+i}{1-i}) and write the result in standard form. A) (0) B) (i) C) (-i) D) (1) Answer: B Explanation: Multiply numerator and denominator by the conjugate of denominator: (\frac{1+i}{1-i}\cdot\frac{1+i}{1+i}=\frac{(1+i)^{2}}{1+1}= frac{1+2i-1}{2}=i). Question 9. Which complex number satisfies (|z-2i|=3)? A) (2+3i) B) (-2+3i) C) (2-3i) D) (5i) Answer: A Explanation: Distance from (z) to (0+2i) must be 3. For (z=2+3i), (|(2+3i)-2i|=| 2+i|=\sqrt{4+1}= \sqrt{5}\neq3). Actually none of the options are correct; we need a correct one. The correct answer is (2+5i) because (|(2+5i)-2i|=|2+3i|= sqrt{13}\neq3). Realizing the mistake, we replace the options. Corrected: A) (2+3i) B) (2+i) C) (2+2i) D) (2+4i) Answer: B
Numbers Ultimate Exam
A) All three roots are real. B) One root is (1) and the other two are complex conjugates. C) None of the roots lie on the unit circle. D) The roots are (1,, -1,, i). Answer: B Explanation: Solving (z^{3}=1) gives (z=1,, e^{2\pi i/3},, e^{4\pi i/3}); the non-real roots are complex conjugates and all lie on the unit circle. Question 14. Using De Moivre’s theorem, compute ((\cos\frac{\pi}{5}+i\sin frac{\pi}{5})^{4}). A) (\cos\frac{4\pi}{5}+i\sin\frac{4\pi}{5}) B) (\cos\frac{2\pi}{5}+i\sin\frac{2 pi}{5}) C) (\cos\frac{8\pi}{5}+i\sin\frac{8\pi}{5}) D) (\cos\frac{3\pi}{5}+i sin\frac{3\pi}{5}) Answer: A Explanation: De Moivre: ((\cos\theta+i\sin\theta)^{n}= \cos n\theta + i\sin n theta). Here (n=4), (\theta=\pi/5) → (4\theta=4\pi/5). Question 15. If (z=2e^{i\pi/6}), what is (z^{3}) in rectangular form? A) (8\left(\frac{\sqrt{3}}{2}+ \frac{i}{2}\right)) B) (8\left(\frac{1}{2}+ \frac{i sqrt{3}}{2}\right)) C) (8\left(\frac{\sqrt{3}}{2}- \frac{i}{2}\right)) D) (8 left(-\frac{1}{2}+ \frac{i\sqrt{3}}{2}\right)) Answer: B Explanation: (z^{3}=2^{3}e^{i\pi/2}=8(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}) = 8(0+i\cdot1)=8i). Wait none match. Let's recompute: (\pi/6 *3 = \pi/2). So (z^{3}=8e^{i\pi/2}=8i). In rectangular form that's (0+8i). None of the options give that; we replace options: A) (8i) B) (-8i) C) (8) D) (-8) Answer: A Explanation: As shown, (z^{3}=8i). Question 16. Which of the following is the correct formula for dividing (z_1) by (z_2) using conjugates? A) (\displaystyle\frac{z_1}{z_2}= \frac{z_1\overline{z_2}}{|z_2|^{2}}) B) ( displaystyle\frac{z_1}{z_2}= \frac{\overline{z_1}z_2}{|z_1|^{2}}) C) (\
Numbers Ultimate Exam
displaystyle\frac{z_1}{z_2}= \frac{z_1z_2}{|z_2|^{2}}) D) (\displaystyle frac{z_1}{z_2}= \frac{\overline{z_1}}{\overline{z_2}}) Answer: A Explanation: Multiply numerator and denominator by (\overline{z_2}) to rationalize: (\frac{z_1}{z_2}=\frac{z_1\overline{z_2}}{z_2\overline{z_2}}= frac{z_1\overline{z_2}}{|z_2|^{2}}). Question 17. Determine the argument of the complex number (-\sqrt{3}+i). A) (-\pi/6) B) (\pi/6) C) (5\pi/6) D) (-5\pi/6) Answer: C Explanation: The point lies in quadrant II. (\tan\theta = \frac{1}{-\sqrt{3}} = - frac{1}{\sqrt{3}}); reference angle is (\pi/6); thus (\theta = \pi - \pi/6 = 5\pi/6). Question 18. Which of the following expressions equals (\Re(z)) for any complex (z=a+bi)? A) (\frac{z+\overline{z}}{2}) B) (\frac{z-\overline{z}}{2i}) C) (|z|) D) ( arg(z)) Answer: A Explanation: (z+\overline{z}= (a+bi)+(a-bi)=2a); dividing by 2 yields (a=\Re(z)). Question 19. Find all solutions of (z^{2}= -4). A) (2i,;-2i) B) (2,;-2) C) (4i,;-4i) D) (1\pm i) Answer: A Explanation: (z^{2}=-4) ⇒ (z=\pm\sqrt{-4}= \pm 2i). Question 20. If (z=1+i), compute (|z|^{2}). A) (2) B) (1) C) (\sqrt{2}) D) (0) Answer: A Explanation: (|z|^{2}= (1)^{2}+(1)^{2}=2). Question 21. Which point lies on the circle (|z-3i|=5)?
Numbers Ultimate Exam
Explanation: Cross-multiply: (z-2 = i(z+2) \Rightarrow z-2 = iz +2i \Rightarrow z-iz = 2+2i \Rightarrow z(1-i)=2(1+i)). Multiply numerator and denominator by (1+i): (z = \frac{2(1+i)}{1-i}=2\frac{(1+i)(1+i)}{(1-i)(1+i)} =2\frac{(1+2i-1)}{2}=2i). Question 25. The set ({z\in\mathbb{C}: \arg(z)=\frac{\pi}{2}}) corresponds to: A) Positive real axis B) Positive imaginary axis C) Negative real axis D) Negative imaginary axis Answer: B Explanation: Argument (\pi/2) points straight upward on the imaginary axis (positive imaginary values). Question 26. If (z_1=4e^{i\pi/3}) and (z_2=2e^{i\pi/6}), find (\frac{z_1} {z_2}) in exponential form. A) (2e^{i\pi/2}) B) (2e^{i\pi/6}) C) (8e^{i\pi/2}) D) (2e^{i\pi/12}) Answer: A Explanation: Divide moduli: (4/2=2). Subtract arguments: (\pi/3-\pi/6=\pi/6). Wait that gives (2e^{i\pi/6}). Actually compute: (\pi/3 - \pi/6 = \pi/6). So answer should be (2e^{i\pi/6}). Option B matches. Answer: B Explanation: As above. Question 27. Which of the following is a root of the polynomial (z^{4}+4=0)? A) (1+i) B) (\sqrt{2},e^{i\pi/4}) C) ( \sqrt{2},e^{i3\pi/4}) D) (2i) Answer: C Explanation: (z^{4} = -4 = 4e^{i\pi}). Fourth roots: ( \sqrt{2}e^{i(\pi+2k\pi)/4}= \sqrt{2}e^{i(\pi/4 + k\pi/2)}) for (k=0,1,2,3). For (k=1) we get (\sqrt{2}e^{i3\pi/4}). Question 28. The sum of the complex numbers (z_1=2-3i) and (z_2=-5+4i) is: A) (-3+i) B) (-3-i) C) (7-7i) D) (-7+7i) Answer: A
Numbers Ultimate Exam
Explanation: Add real parts: (2+(-5)=-3). Add imaginary parts: (-3i+4i=i). So (- 3+i). Question 29. Which of the following is equivalent to (\cos\theta + i\sin\theta)? A) (e^{-\theta}) B) (e^{i\theta}) C) (\ln(\theta)) D) (\theta^{i}) Answer: B Explanation: Euler’s identity: (e^{i\theta}= \cos\theta + i\sin\theta). Question 30. Find the principal value of (\log ( -i )) where the complex logarithm is defined as (\log z = \ln|z| + i\Arg(z)). A) (0 - i\frac{\pi}{2}) B) (0 + i\frac{\pi}{2}) C) (\ln 1 + i\frac{3\pi}{2}) D) (\ln 1 - i\frac{3\pi}{2}) Answer: B Explanation: (|-i|=1) so (\ln|z|=0). Principal argument of (-i) is (-\pi/2) but the principal range ((-π,π]) gives (-\pi/2). However many textbooks choose (-\pi/2); the answer B uses (+\pi/2) which is incorrect. Correct answer should be (0 - i frac{\pi}{2}). So we adjust: Answer: A Explanation: As shown. Question 31. Which of the following complex numbers lies on the line ( Im(z)=2\Re(z))? A) (1+2i) B) (2+4i) C) (-3-6i) D) All of the above Answer: D Explanation: For each, imaginary part equals twice the real part: (2=2\cdot1), (4=2\cdot2), (-6=2\cdot(-3)). Hence all satisfy the line equation. Question 32. If (z) satisfies (|z|=2) and (\arg(z)=\frac{5\pi}{6}), what is (z) in rectangular form? A) (-\sqrt{3}+i) B) (-\sqrt{3}+i) multiplied by 2? C) (-\sqrt{3}+i) times 2? Actually compute: (z=2(\cos5π/6 + i\sin5π/6)=2(-\sqrt3/2 + i\frac12)= -\sqrt3 + i). Thus answer: (-\sqrt{3}+i).
Numbers Ultimate Exam
Explanation: (32 = 32e^{i0}). Fifth roots: (32^{1/5}=2) and arguments ((0+2k pi)/5). Hence (2e^{2k\pi i/5}) for (k=0,\dots,4). Question 37. Which complex number is the image of (z=1+i) after a rotation of (90^{\circ}) counter-clockwise about the origin? A) (-1+i) B) (-1-i) C) (i-1) D) (-i+1) Answer: B Explanation: Rotation by (90^{\circ}) multiplies by (e^{i\pi/2}=i). (i(1+i)=i -1 = -1 + i). Wait compute: (i(1+i)=i + i^{2}=i-1 = -1 + i). That's option A. So answer A. Answer: A Explanation: Multiplying by (i) yields (-1+i). Question 38. If (z) satisfies (z^{2}=6-8i), what is (|z|)? A) (\sqrt{10}) B) (\sqrt{5}) C) (2) D) (\sqrt{13}) Answer: A Explanation: Modulus of (z^{2}) equals (|z|^{2}). (|6-8i| = sqrt{6^{2}+8^{2}} = 10). Hence (|z|^{2}=10) ⇒ (|z|=\sqrt{10}). Question 39. Which of the following is the correct expression for (\sin(3\theta)) using De Moivre’s theorem? A) (3\sin\theta - 4\sin^{3}\theta) B) (4\sin^{3}\theta - 3\sin\theta) C) (\sin theta\cos^{2}\theta) D) None of the above. Answer: A Explanation: De Moivre yields (\sin 3\theta = 3\sin\theta -4\sin^{3}\theta). Question 40. Evaluate (\displaystyle\frac{(2+i)(3-2i)}{5}) and write in standard form. A) (\frac{1}{5}+ \frac{8}{5}i) B) (\frac{8}{5}+ \frac{1}{5}i) C) (\frac{4} {5}+ \frac{7}{5}i) D) (\frac{7}{5}+ \frac{4}{5}i) Answer: B
Numbers Ultimate Exam
Explanation: Numerator: ((2+i)(3-2i)=6-4i+3i-2i^{2}=6 -i +2 =8 - i). Divide by 5 → ( \frac{8}{5} - \frac{1}{5}i). Wait sign: we got (8-i). So standard form is ( frac{8}{5} - \frac{1}{5}i). None of the options match; adjust: A) (\frac{8}{5} - \frac{1}{5}i) B) (\frac{8}{5}+ \frac{1}{5}i) C) (\frac{1} {5}+ \frac{8}{5}i) D) (\frac{1}{5}- \frac{8}{5}i) Answer: A Explanation: As computed. Question 41. The complex number (z) satisfies (|z-2i|=|z+2i|). Which of the following must be true? A) (\Re(z)=0) B) (\Im(z)=0) C) (|z|=2) D) None of the above. Answer: A Explanation: Points equidistant from (2i) and (-2i) lie on the real axis (( Re(z)=0)). Actually the perpendicular bisector of the segment joining (2i) and (- 2i) is the real axis, so (\Im(z)=0). Wait the segment is vertical; its midpoint is 0, and the bisector is the horizontal line (real axis) meaning (\Im(z)=0). So answer B. Answer: B Explanation: Set equality leads to ((x)^2+(y-2)^2 = x^{2}+(y+2)^{2}) ⇒ ((y- 2)^{2}=(y+2)^{2}) ⇒ (y=0). Question 42. Which of the following is the correct polar representation of the complex number (-2-2i)? A) (2\sqrt{2},e^{i3\pi/4}) B) (2\sqrt{2},e^{i5\pi/4}) C) (2\sqrt{2},e^{i pi/4}) D) (2\sqrt{2},e^{i7\pi/4}) Answer: B Explanation: Modulus (r=2\sqrt{2}). The point is in quadrant III, angle (\theta = pi + \pi/4 = 5\pi/4). Question 43. Find the sum of the three cube roots of unity. A) (0) B) (1) C) (-1) D) (3) Answer: A
Numbers Ultimate Exam
A) (4) B) (-4) C) ( \pm4) D) (0) Answer: C Explanation: (a=3). Then (b^{2}=|z|^{2}-a^{2}=25-9=16) ⇒ (b= \pm4). Question 49. The transformation (w = (z-1)/(z+1)) maps the unit circle (|z|=1) onto which geometric set? A) Imaginary axis B) Real axis C) Unit circle again D) A line through the origin with slope 1 Answer: A Explanation: The Möbius transformation sends the unit circle to the imaginary axis because for (|z|=1), (\overline{z}=1/z) leading to (\overline{w} = -w), i.e., purely imaginary. Question 50. Which of the following is the principal value of (\sqrt{-9})? A) (3i) B) (-3i) C) (3) D) (-3) Answer: A Explanation: Principal square root takes argument half of (\pi): (\sqrt{-9}= sqrt{9}e^{i\pi/2}=3i). Question 51. Evaluate ((\cos\theta + i\sin\theta)^{2}) using De Moivre’s theorem. A) (\cos 2\theta + i\sin 2\theta) B) (\cos^{2}\theta - \sin^{2}\theta + i2\sin theta\cos\theta) C) Both A and B D) None of the above Answer: C Explanation: De Moivre gives (\cos2\theta + i\sin2\theta). Expanding using double-angle identities yields the expression in B; thus both are correct. Question 52. Find all solutions of (z^{3}+8=0). A) (-2,; 1\pm i\sqrt{3}) B) (-2,; 2e^{i\pi/3},; 2e^{i5\pi/3}) C) (-2,; 2e^{i pi/3},; 2e^{i\pi}) D) (-2,; 2e^{i\pi/3},; 2e^{i5\pi/3}) Answer: D
Numbers Ultimate Exam
Explanation: Rewrite as (z^{3} = -8 = 8e^{i\pi}). Cube roots: (2e^{i(\pi+2k\pi)/3}) for (k=0,1,2). Gives (-2) (k=1), (2e^{i\pi/3}) (k=0), and (2e^{i5\pi/3}) (k=2). Question 53. Which of the following points is the image of (z=3+4i) after a dilation by factor 2 about the origin? A) (6+8i) B) (1.5+2i) C) (-6-8i) D) (3+4i) Answer: A Explanation: Dilation multiplies the complex number by the real factor 2. Question 54. If (z) satisfies (\arg(z)= -\frac{3\pi}{4}) and (|z|= \sqrt{2}), then (z) equals: A) (-1 - i) B) (1 + i) C) (-1 + i) D) (1 - i) Answer: A Explanation: (\sqrt{2}(\cos(-3\pi/4)+i\sin(-3\pi/4)) = \sqrt{2}\left(-\frac{\sqrt{2}} {2} - i\frac{\sqrt{2}}{2}\right)= -1 - i). Question 55. Which expression gives the distance between two complex numbers (z_1) and (z_2) in the Argand plane? A) (|z_1 - z_2|) B) (|z_1| - |z_2|) C) (\arg(z_1) - \arg(z_2)) D) (|\Re(z_1)- Re(z_2)|) Answer: A Explanation: Distance is the modulus of the difference. Question 56. The set ({z: |z| = \Re(z)}) represents which geometric figure? A) Circle centered at ((\frac12,0)) with radius (\frac12) B) Line (x=0) C) Ray making (45^{\circ}) with real axis D) No points except the origin Answer: A Explanation: Let (z = x+iy). Condition (\sqrt{x^{2}+y^{2}} = x). Square both sides: (x^{2}+y^{2}=x^{2}) ⇒ (y^{2}=0) and (x\ge0). Actually this gives (y=0) and (x\ge0). But also the equation yields a circle? Let's re-evaluate: (|z|=x) with (x\ge0). Squaring: (x^{2}+y^{2}=x^{2}) ⇒ (y^{2}=0) ⇒ (y=0). So the set is the non-negative real axis, not a circle. So answer is C? Not correct. Let's
Numbers Ultimate Exam
A) ((z+2i)(z-2i)) B) ((z+2)(z-2)) C) ((z+i\sqrt{2})(z-i\sqrt{2})) D) Irreducible over (\mathbb{C}) Answer: A Explanation: Roots are (z=\pm 2i); thus factorization ((z-2i)(z+2i)). Question 61. Compute ((\cos\theta + i\sin\theta)^{0}). A) (0) B) (1) C) (\cos 0 + i\sin 0) D) Both B and C Answer: D Explanation: Any non-zero number to the zero power is 1; Euler form also gives ( cos0+i\sin0=1). Question 62. Which of the following is the correct expression for the nth roots of a non-zero complex number (w = re^{i\phi})? A) (w^{1/n}= r^{1/n}e^{i\phi/n}) B) (w^{1/n}= r^{1/n}e^{i(\phi+2k\pi)/n},;k=0,\dots,n-1) C) (w^{1/n}= r^{n}e^{i n\phi}) D) None of the above. Answer: B Explanation: The nth roots are given by (r^{1/n}e^{i(\phi+2k\pi)/n}) for each integer (k). Question 63. If (z=3e^{i\pi/2}), what is its rectangular form? A) (0+3i) B) (-3) C) (3) D) (0-3i) Answer: A Explanation: (e^{i\pi/2}=i); thus (z=3i). Question 64. Which of the following complex numbers lies on the line making an angle of (30^{\circ}) with the positive real axis and at a distance of 4 from the origin? A) (4\cos30^{\circ}+i4\sin30^{\circ}) B) (4\cos30^{\circ}-i4\sin30^{\circ}) C) (-4\cos30^{\circ}+i4\sin30^{\circ}) D) (4\sin30^{\circ}+i4\cos30^{\circ}) Answer: A Explanation: Polar to rectangular conversion with radius 4 and angle (π/6).
Numbers Ultimate Exam
Question 65. Find the value of (\displaystyle\frac{(1+i)^{4}}{(1-i)^{4}}). A) (1) B) (-1) C) (i) D) (-i) Answer: B Explanation: ((1+i)^{2}=2i) ⇒ ((1+i)^{4}= (2i)^{2}= -4). Similarly ((1- i)^{2}= -2i) ⇒ ((1-i)^{4}= (-2i)^{2}= -4). Ratio ((-4)/(-4)=1). Wait compute again: Actually ((1-i)^{2}= (1)^2 -2i + i^{2}=1-2i-1 = -2i). Square again: ((- 2i)^{2}=4i^{2}= -4). So both numerator and denominator are (-4) → ratio =1. So answer A. Answer: A Explanation: As shown. Question 66. The set of points satisfying (\Im(z^{2})=0) is: A) The real axis B) The imaginary axis C) The union of lines (y = x) and (y = - x) D) A circle centered at origin Answer: C Explanation: For (z=x+iy), (z^{2}=x^{2}-y^{2}+2ixy). Imaginary part zero ⇒ (xy=0). Actually (\Im(z^{2}) = 2xy). Setting to zero gives (x=0) or (y=0) (axes). Wait that is axes not lines y=±x. So correct answer is A or B? Let's compute: If (2xy=0) then either (x=0) (imaginary axis) or (y=0) (real axis). So union of axes. Not among options. Replace options: A) Real axis B) Imaginary axis C) Union of real and imaginary axes D) None of the above. Answer: C Explanation: As derived. Question 67. Which of the following is the correct conversion of (z= - sqrt{3}+i) to polar form? A) (2e^{i5\pi/6}) B) (2e^{i\pi/6}) C) (2e^{i -\pi/6}) D) (2e^{i -5\pi/6}) Answer: A Explanation: Modulus (= \sqrt{(\sqrt{3})^{2}+1^{2}}=2). Argument is in quadrant II: (\pi - \tan^{-1}(1/\sqrt{3}) = \pi - \pi/6 = 5\pi/6).
Numbers Ultimate Exam
Answer: A Explanation: Differentiate: (dz/d\theta = 2(-\sin\theta + i\cos\theta)=2i(\cos\theta + i\sin\theta)= i z). Question 73. Which of the following is the correct expression for the distance between the points representing (z_{1}=1+2i) and (z_{2}=4-3i)? A) (\sqrt{34}) B) (\sqrt{13}) C) (\sqrt{25}) D) (\sqrt{20}) Answer: A Explanation: Difference (z_{1}-z_{2}= (1-4)+(2+3)i = -3+5i). Modulus (\sqrt{(- 3)^{2}+5^{2}}=\sqrt{9+25}= \sqrt{34}). Question 74. The set ({z: \Re(z^{2})=1}) corresponds to which curve? A) Hyperbola B) Circle C) Pair of straight lines D) Ellipse Answer: C Explanation: Let (z=x+iy). Then (z^{2}=x^{2}-y^{2}+2ixy). Real part equals 1 ⇒ (x^{2}-y^{2}=1), which is a hyperbola, not a pair of lines. Actually that's a hyperbola. So answer A. Answer: A Explanation: Equation (x^{2}-y^{2}=1) defines a rectangular hyperbola. Question 75. Which of the following complex numbers is a solution to (z^{4}=16)? A) (2) B) (-2) C) (2i) D) All of the above (with appropriate arguments) Answer: D Explanation: Fourth roots of (16=16e^{i0}) are (2e^{i\pi k/2}) for (k=0,1,2,3): (2, 2i, -2, -2i). Hence all listed. Question 76. If (z = \frac{3+4i}{5}), what is (\arg(z)) (principal value)? A) (\tan^{-1}\frac{4}{3}) B) (\tan^{-1}\frac{3}{4}) C) (-\tan^{-1}\frac{4} {3}) D) (0) Answer: A
Numbers Ultimate Exam
Explanation: Argument depends only on ratio of imaginary to real parts; scaling by 5 does not change it. Question 77. Which of the following is the correct expression for the product of two complex numbers in polar form (z_{1}=r_{1}e^{i\theta_{1}}) and (z_{2}=r_{2}e^{i\theta_{2}})? A) ((r_{1}+r_{2})e^{i(\theta_{1}+\theta_{2})}) B) ((r_{1}r_{2})e^{i( theta_{1}+\theta_{2})}) C) ((r_{1}r_{2})e^{i(\theta_{1}-\theta_{2})}) D) ((r_{1}+r_{2})e^{i(\theta_{1}-\theta_{2})}) Answer: B Explanation: Polar multiplication multiplies moduli and adds arguments. Question 78. Find the principal argument of (z = -\sqrt{2} - i\sqrt{2}). A) (-\frac{3\pi}{4}) B) (\frac{5\pi}{4}) C) (\frac{3\pi}{4}) D) (-\frac{\pi} {4}) Answer: A Explanation: The point is in quadrant III; reference angle (\pi/4). Principal argument is (-\pi + \pi/4 = -3\pi/4). Question 79. Which of the following is true for any complex number (z\neq0)? A) (\displaystyle\frac{z}{|z|}=e^{i\arg(z)}) B) (\displaystyle\frac{z}{|z|}= cos(\arg z)+i\sin(\arg z)) C) Both A and B D) Neither A nor B Answer: C Explanation: Dividing by its modulus yields a unit-modulus number with argument (\arg(z)), which equals (e^{i\arg(z)} = \cos\arg(z)+i\sin\arg(z)). Question 80. If (z=4e^{i\pi/3}) and (w=2e^{i\pi/6}), compute ( displaystyle\frac{z}{w}) in polar form. A) (2e^{i\pi/2}) B) (2e^{i\pi/12}) C) (2e^{i\pi/6}) D) (2e^{i\pi/3}) Answer: A Explanation: Moduli divide: (4/2=2). Arguments subtract: (\pi/3 - \pi/6 = \pi/6). Wait that's (\pi/6) not (\pi/2). So correct answer should be (2e^{i\pi/6}) (option C).
