Math 1316 Exam 3. if u = 4, c. ÄuÄ = isin π Ë 5 34, , 5 34, 3
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1 Math 36 Exam 3 Multiple Choice Identify the choice that best completes the statement or answers the question.. Find the component form of v if ÄÄ= v 0 and the angle it makes with the x-axis is 50. 0,0 3 5, , 0 d. 5 3,5 e. 5,5. final Given that Force = 50 pounds and Force = 90 pounds, find the angle between the forces if the magnitude of the resultant force is 90 pounds. Round answer to the nearest degree d. 66 e Given u = 3i 4j and v = i + j, find u v. 6 d. 4 e Use the dot product to find the magnitude of u if u = 4,. ÄuÄ = 9 ÄuÄ = d. ÄuÄ = 3 e. ÄuÄ = 7 7 ÄuÄ = 4 5. Find the angle between the vectors u and v if u = i 3j and v = 4i 3j. Round answer to two decimal places d e Find the angle between the vectors u and v. d. e. 7. Determine whether u are v and orthogonal, parallel, or neither. u = 7, 3, v = 5, parallel neither orthogonal 8. Find the trigonometric form of i. 3 cos π 3 + isin π 6 cos π isin π 3 e. 3 cos π 6 + isin π 6 3 cos π 3 + isin π 3 d. 6 cos π 6 + isin π 6 9. Find the projection of u onto v if u =,, v = 5, 3. 34, , , 5 34 d. 5 34, 34 e. 34, 34
2 0. A 700-pound trailer is sitting on an exit ramp inclined at 44 on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round answer to two decimal places pounds pounds pounds d pounds e pounds. A force of 70 pounds is exerted along a rope attached to a crate at an angle of 30 above the horizontal. The crate is moved 3 feet. How much work has been accomplished? Round answer to one decimal place.,40.0 foot-pounds,338. foot-pounds,939.9 foot-pounds d.,586.5 foot-pounds e.,089.9 foot-pounds. Find the trigonometric form of the complex number shown below. 5i 5 cos π + isin π 5 cos 3π + isin 3π d. 5 cos π 4 + isin π 4 5( cosπ + isinπ ) e. 5( cos0+ isin0) d. 3. Plot the following complex number. 4. Find the standard form of the complex number shown below. Round answers to two decimal places. 4( cos40 +isin40 ) i i i d i e i 5. Find the absolute value of the complex number + i d. 5 e. 3
3 6. Write the complex number shown below in trigonometric form. d. e. 7. Use DeMoivre's Theorem to find the indicated power of the following complex number i ,683,87 d. 46,656 e. 6,56 8. Use DeMoivre's Theorem to find the indicated power of the following complex number. ( 3 + 3i) ,96 d. 34 e. 34 d. 9. Represent the complex number below graphically. 3
4 0. Find the standard form of the complex number shown below. 9 cos 5π 6 + isin 5π i i i d i e i. Perform the operation shown below and leave the result in trigonometric form. È 3 cos π 3 + isin π È 3 cos5π 7 + isin 5π 7 È 6 cos π 3 + isin π È 6 cos 0π 0π È + isin 5 cos 9π 9π + isin 3 È d. 6 cos 9π 9π È + isin e. 5 cos 0π 0π + isin. Find the cube roots of the following complex number. Write each of the roots in standard form. 79 9, e. 9, i, i, i i 9, 3. Find all solutions to the following equation i, i 9 d. 9, 9 x 3 8 = 0 x =, x =, + 3 i, 3 i x =, 3 i, + 3 i d. x = e. x =, + 3 i, 3 i 4. Find all solutions to the following equation. x i = x = 9i, x = 9i, i, i, i x = 9i, i d. x = 9i e. x = 9 9 i, i 5. Find the trigonometric form of 9 + 9i. 9 cos 3π + isin 3π 9 cos 3π + isin 3π 9 cos 3π 4 + isin 3π 4 d. 9 cos 3π 4 + isin 3π e. 36 cos 3π isin 3π 8 4
5 6. Use DeMoivre's Theorem to find the indicated power of the following complex number. È cos π 3 + isin π d. 4 e Represent the following complex number graphically. 4 cos( 35 )+ isin( 35 ) 7. Find the fourth roots of + 3 roots in trigonometric form. w = cos( 5 )+ isin( 5 ) i. Write the w = cos( 05 )+ isin( 05 ) = cos( 95 )+ isin( 95 ) = cos( 85 )+ isin( 85 ) w = cos( 0 )+ isin( 0 ) w = cos( 00 )+ isin( 00 ) = cos( 90 )+ isin( 90 ) = cos( 80 )+ isin( 80 ) w = cos( 5 )+ isin( 5 ) w = cos( 5 )+ isin( 5 ) = cos( 05 )+ isin( 05 ) = cos( 95 )+ isin( 95 ) d. w = cos( 5 )+ isin( 5 ) w = cos( 95 )+ isin( 95 ) = cos( 85 )+ isin( 85 ) = cos( 75 )+ isin( 75 ) e. w = cos( 0 )+ isin( 0 ) w = cos( 0 )+ isin( 0 ) = cos( 00 )+ isin( 00 ) = cos( 90 )+ isin( 90 ) 5
6 d. 9. Find the standard form of the complex number 9 cos π π + isin. Round values to the nearest hundredth i i i d i e i 30. Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form i i i 5 3 5i d i e i 3. Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. È 5 cos 9π + isin 9π i 5i 9535 d. 9535i e Short Answer 3. Given u = 5i + j and v = 4i + j, determine u 6v. 33. Find a unit vector in the direction of w = 4, Let w be a vector with initial point ( 8, 0) and terminal point ( 9,8). Write w as a linear combination of the standard unit vectors i and j. 35. Find the magnitude and direction angle of v = cos0 i + sin0 j. 36. Simplify and write the following complex number in standard form. ( 6 4i) ( 3 + 8i) + 5i 37. Simplify and write the answer in 8i standard form. 6
7 38. Evaluate the power of i. 46. Find the component form of vector v. i Find the indicated power. Write the answer in standard form. È ( cos 40 +isin 40 ) Plot the point on a polar coordinate system. ( 3, 90 ) 4. Transform the given coordinates to the indicated ordered pair. 5, 5 3 to ( r, θ ) 4. Find a polar form of the equation x y =. 43. Find a rectangular form of the equation r = 6cosθ. 47. Find the magnitude of vector v with initial point ( 7, ) and terminal point ( 4,4). 48. Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.] u + v 44. Find a rectangular form of the equation θ = 5π Write the following expression as a complex number in standard form. 4 7i 6i 49. Find a rectangular form of the equation r 8cosθ + sinθ ( ) = 5. 7
8 ID: A Math 36 Exam 3 Answer Section MULTIPLE CHOICE. D. A 3. E 4. B 5. A 6. A 7. B 8. B 9. B 0. A. C. C 3. A 4. E 5. C 6. B 7. D 8. E 9. B 0. E. D. A 3. B 4. C 5. D 6. C 7. A 8. A 9. E 30. B 3. D SHORT ANSWER 3. u 6v = 34i 4j 33. 3, w = i + 8j 35. ÄÄ= v ; θ = 0
9 ID: A i i i i ( 0, 60 ) 4. r ( cosθ ) = 43. x + y 6x = y = x i 46. v = 4, ÄÄ= v y = 8x + 5
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