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TRIGONOMETRY a course in year 3, Exams of Trigonometry

University of GhanaTrigonometry
Prof. John Sam

Describe the content and the topics covered. Between 30 and 1000 characters

Typology: Exams

2023/2024

Uploaded on 04/13/2024

mark-aggrey
mark-aggrey ๐Ÿ‡ฌ๐Ÿ‡ญ

1 document

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Download TRIGONOMETRY a course in year 3 and more Exams Trigonometry in PDF only on Docsity! Page 1 of 7 INSTRUCTION: ANSWER ALL QUESTIONS BY CIRCLING THE RIGHT ANSWER ON THE QUESTION PAPER.EACH QUESTION ANSWERED CORRECTLY ATTRACTS ONE MARK.ROUGH WORK SHOULD BE DONE ON THE SHEET PROVIDED. PE,ES,MN,MR,GL,GM PLEASE IF YOU FEEL YOUR ANSWER IS NOT AMONG THE MULTIPLE CHOICE A- D.WRITE YOUR OWN ANSWER 1. Write โˆš9 in simplest form in terms of real numbers and ํ‘– a. -3 b. 3 c. 3i d - 3i 2. Solve the equation ํ‘ง + z โˆ’ 2= 0, writing the answer in z=a + bi a. 1,โˆ’ + โˆš ํ‘–,โˆ’ โˆ’ โˆš b. 1,โˆ’ + โˆš ํ‘–, โˆ’ โˆš c. 1,โˆ’ โˆ’ โˆš ํ‘–,โˆ’ โˆ’ โˆš d. 1,โˆ’ + โˆš ํ‘–,โˆ’ + โˆš 3. Express 2i(iโˆ’1) + ํ‘– (2+i) in the form a+bi a. 1+4i b. 1โˆ’4ํ‘– c. โˆ’1 โˆ’ 4ํ‘– d. โˆ’1 + 4ํ‘– 4. Find the real and imaginary parts of (4 + 3ํ‘–) a. 44,117 b. 44,โˆ’117 c. โˆ’44,117 d. โˆ’44,โˆ’117 5. Write down the complex conjugate of ( ) a. ( ) b. ( ) c. ( ) d. ( ) 6. Put this expression ( )( ) into a+bi form. a. + ํ‘– b โˆ’ + c. โˆ’ โˆ’ ํ‘– d. โˆ’ ํ‘– + 7. Evaluate โˆ’ and write whether is purely imaginary or purely real. a. , imaginary, difference of complex conjugate b. , imaginary, sum of complex conjugate d. , imaginary, product of complex conjugate c. , imaginary, addition of complex conjugate 8. Simplify the complex number + . Find the modulus and argument of the result. UNIVERSITY OF MINES AND TECHNOLOGY, TARKWA FIRST SEMESTER EXAMINATIONS, DEC 2014 COURSE NO: MATH 153 COURSE NAME: LINEAR AND TRIGNOMENTARY CLASS: TIME: 3 HOURS Name: __________________________________________ Index Number: _______________ Print to PDF without this message by purchasing novaPDF (http://www.novapdf.com/) Page 2 of 7 a. + ํ‘–,modulus โˆš , argument= b. โˆ’ ํ‘–, modulus โˆš , argument= c. + ํ‘–, modulus โˆš , argument= d. โˆ’ + ํ‘–, modulus โˆš , argument= 9. Convert 3โˆ  โˆ’ into Cartesian form a. โˆ’ โˆš + ํ‘– b. โˆš + ํ‘– c. โˆ’ โˆš โˆ’ ํ‘– d. โˆ’ โˆš + ํ‘– 10. If |ํ‘ง | =5, Argํ‘ง = , |ํ‘ง | =3, Argํ‘ง = . Find the Cartesian form of ํ‘ง and ํ‘ง a. ( โˆš ), ( ) โˆš b. ( โˆš ), ( ) โˆš c. ( โˆš ), ( ) โˆš d. โˆ’ ( โˆš ), ( ) โˆš 11. Using the values in question 10 find the value of |ํ‘ง ํ‘ง | a. 18 b. 16 c. 17 d. 15 12. If ํ‘ง = 3โˆ  , ํ‘ง = 2โˆ ( ), ํ‘ง = ( ) find the polar form of a. โˆ’ โˆ ( ) ํ‘. โˆ ( ) c. โˆ ( ) d. โˆ ( ) 13. Express this โ„ in the form a+bi a. โˆš + โˆš ํ‘– b. โˆš โˆ’ โˆš c. โˆš โˆ’ โˆš d. โˆš โˆ’ โˆš 14. What is the value of ( ) ( ) ( ) ( ) a. cos(38ํœƒ)โˆ’ ํ‘– sin(38ํœƒ) b. cos(38ํœƒ) + ํ‘– sin(38ํœƒ) c. cos(โˆ’38ํœƒ) โˆ’ ํ‘– sin(โˆ’38ํœƒ) d. cos(38ํœƒ) โˆ’ ํ‘– sin(โˆ’38ํœƒ) 15. Determine the fourth root of โˆš3 + ํ‘– a. 2 โˆ ( ) b. 2 โˆ (โˆ’ ) c. โˆ’2 โˆ ( ) d. 2 โˆ (โˆ’ ) 16. Express 3ํ‘ฅ โˆ’ ํ‘ฆ + 2ํ‘ง = 1 in matrix form 2ํ‘ฅ + 2ํ‘ฆ + 3ํ‘ง = 2 โˆ’3ํ‘ฅ โˆ’ ํ‘ง = โˆ’3 a. 3 โˆ’1 2 2 2 3 โˆ’3 0 โˆ’1 ํ‘ฅ ํ‘ฆ ํ‘ง = 1 โˆ’2 โˆ’3 b. 3 โˆ’1 2 2 2 โˆ’3 โˆ’3 0 โˆ’1 ํ‘ฅ ํ‘ฆ ํ‘ง = 1 2 โˆ’3 c . 3 โˆ’1 2 2 2 3 โˆ’3 0 โˆ’1 ํ‘ฅ ํ‘ฆ ํ‘ง = 1 2 โˆ’3 d. 3 โˆ’1 2 2 โˆ’2 3 โˆ’3 0 โˆ’1 ํ‘ฅ ํ‘ฆ ํ‘ง = 1 2 โˆ’3 Use A = โˆ’1 2 3 4 0 2 โˆ’1 1 2 and B = 0 1 3 2 โˆ’1 4 3 โˆ’1 2 to answer 17,18,19 and 20 17. Evaluate โˆ‘ ํ‘Ž ํ‘ ,โˆ‘ ํ‘Ž ํ‘ Print to PDF without this message by purchasing novaPDF (http://www.novapdf.com/) Page 5 of 7 a. 7ํ‘ฅ โˆ’ 5ํ‘ฆ + 18 b โˆ’7ํ‘ฅ โˆ’ 5ํ‘ฆ โˆ’ 18 c โˆ’7ํ‘ฅ โˆ’ 5ํ‘ฆ + 18 d. โˆ’7ํ‘ฅ + 5ํ‘ฆ + 18 34. Evaluate ํ‘ฅ ํ‘ฆ 1 4 โˆ’2 1 โˆ’1 5 1 a. โˆ’12ํ‘Ž โˆ’ 23ํ‘ โˆ’ 3ํ‘ b. โˆ’12ํ‘Ž โˆ’ 23ํ‘ + 3ํ‘ c. โˆ’12ํ‘Ž + 23ํ‘ โˆ’ 3ํ‘ d. 12ํ‘Ž โˆ’ 23ํ‘ โˆ’ 3ํ‘ 35. What are the zeros of ํ‘ฅ โˆ’ 3 a. (ํ‘ฅ โˆ’ 2) ํ‘ฅ + 1 โˆ’ โˆš3ํ‘– ํ‘ฅ โˆ’ 1 + โˆš3ํ‘– b. (ํ‘ฅ โˆ’ 2) ํ‘ฅ + 1 โˆ’ โˆš3ํ‘– (ํ‘ฅ + 1 + โˆš3ํ‘–) c. (ํ‘ฅ โˆ’ 2) ํ‘ฅ + 1 โˆ’ โˆš3ํ‘– (ํ‘ฅ โˆ’ 1 โˆ’ โˆš3ํ‘–) d (ํ‘ฅ โˆ’ 2) ํ‘ฅ โˆ’ 1 โˆ’ โˆš3ํ‘– (ํ‘ฅ โˆ’ 1 + โˆš3ํ‘–) 36. What are the factors of ํ‘ฅ โˆ’ 2ํ‘ฅ + 9ํ‘ฅ โˆ’ 18 a. (ํ‘ฅ โˆ’ 2)(ํ‘ฅ โˆ’ 3ํ‘–)(ํ‘ฅ โˆ’ 3ํ‘–) b. (ํ‘ฅ + 2)(ํ‘ฅ โˆ’ 3ํ‘–)(ํ‘ฅ โˆ’ 3ํ‘–) c. (ํ‘ฅ โˆ’ 2ํ‘–)(ํ‘ฅ โˆ’ 3ํ‘–)(ํ‘ฅ โˆ’ 3) d. (ํ‘ฅ โˆ’ 2)(ํ‘ฅ โˆ’ 3ํ‘–)(ํ‘ฅ + 3ํ‘–) 37. What are the factors of ํ‘ฅ โˆ’ 1 a. (ํ‘ฅ + 1) ํ‘ฅ + โˆ’ โˆš ํ‘– (ํ‘ฅ + + โˆš ํ‘–)(ํ‘ฅ โˆ’ 1) ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘– (ํ‘ฅ โˆ’ + โˆš ํ‘–) b. (ํ‘ฅ + 1) ํ‘ฅ + โˆ’ โˆš ํ‘– (ํ‘ฅ + + โˆš ํ‘–)(ํ‘ฅ โˆ’ 1) ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘– (ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘–) c. (ํ‘ฅ โˆ’ 1) ํ‘ฅ + โˆ’ โˆš ํ‘– (ํ‘ฅ + + โˆš ํ‘–)(ํ‘ฅ โˆ’ 1) ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘– (ํ‘ฅ โˆ’ + โˆš ํ‘–) d. (ํ‘ฅ + 1) ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘– (ํ‘ฅ + + โˆš ํ‘–)(ํ‘ฅ โˆ’ 1) ํ‘ฅ โˆ’ โˆ’ โˆš ํ‘– (ํ‘ฅ โˆ’ + โˆš ํ‘–) 38. Find vector a joining points P and Q where point P has co-ordinates (4,-1,3) and point Q has co-ordinates (2,5,0).Also find |ํ‘Ž|,the magnitude or the norm of a a. โˆ’2ํ‘– + 6ํ‘— โˆ’ 3ํ‘˜, 7 b. โˆ’2ํ‘– โˆ’ 6ํ‘— โˆ’ 3ํ‘˜, 7 c. โˆ’2ํ‘– + 6ํ‘— + 3ํ‘˜, 7 d. 2ํ‘– + 6ํ‘— โˆ’ 3ํ‘˜, 7 39. If ํ‘ = 2ํ‘– + ํ‘— โˆ’ ํ‘˜ and ํ‘ž = ํ‘– โˆ’ 3ํ‘— + 2ํ‘˜ determine ํ‘.ํ‘ž a. 6 b. 7 c. 5 d. 4 40. What is the value of |ํ‘ + ํ‘ž| in question 39 a. โˆš6 b. โˆš7 c. โˆš14 d. โˆš13 41. Determine the angle between vectors ํ’ํ’‚ and ํ’ํ’ƒ when ํ‘œํ‘Ž = ํ‘– + 2ํ‘– โˆ’ 3ํ‘˜ and ํ‘œํ‘ = 2ํ‘– โˆ’ ํ‘— + 4ํ‘˜ a. 134.4 or 225.6 b. 124.4 or 225.6 c. 134.4 or 235.6 d. 134.4 or 125.6 42. Find the direction cosines of 3ํ‘– + 2ํ‘— + ํ‘˜ a. โˆš , โˆš , โˆš b. โˆš , โˆš , โˆš c. โˆš , โˆš , โˆš d. โˆš , โˆš , โˆš 43. If the vectors ํ‘Ž = ํ‘– + 4ํ‘— โˆ’ 2ํ‘˜ and ํ‘ = 2ํ‘– โˆ’ ํ‘— + 3ํ‘˜, what is the value of ํ‘Ž ร— ํ‘ a. 10ํ‘– โˆ’ 7ํ‘— โˆ’ 9ํ‘˜ b. 10ํ‘– โˆ’ 7ํ‘— + 9ํ‘˜ c. 10ํ‘– + 7ํ‘— + 9ํ‘˜ d. โˆ’10ํ‘– โˆ’ 7ํ‘— + 9ํ‘˜ 44. What is the value of |ํ‘Ž ร— ํ‘| using question 43. a. โˆš233 b. โˆš230 c. โˆš232 d. โˆš231 Print to PDF without this message by purchasing novaPDF (http://www.novapdf.com/) Page 6 of 7 45. If ํ‘ = 4ํ‘– + ํ‘— โˆ’ 2ํ‘˜,ํ‘ž = 3ํ‘– โˆ’ 2ํ‘— + ํ‘˜ and ํ‘Ÿ = ํ‘– โˆ’ 2ํ‘˜ find (ํ‘ โˆ’ 2ํ‘ž) ร— ํ‘Ÿ a. 10ํ‘– โˆ’ 8ํ‘— โˆ’ 5ํ‘˜ b. 10ํ‘– + 8ํ‘— โˆ’ 5ํ‘˜ c. 10ํ‘– โˆ’ 8ํ‘— + 5ํ‘˜ d โˆ’10ํ‘– โˆ’ 8ํ‘— โˆ’ 5ํ‘˜ 46. What is the value of ํ‘ ร— (2ํ‘Ÿ ร— 3ํ‘ž) using question 45 a. 48(2ํ‘– โˆ’ 2ํ‘— + 3ํ‘˜) b. โˆ’48(2ํ‘– + 2ํ‘— + 3ํ‘˜) c. โˆ’48(2ํ‘– โˆ’ 2ํ‘— โˆ’ 3ํ‘˜) d. โˆ’48(2ํ‘– โˆ’ 2ํ‘— + 3ํ‘˜) 47. The equation = = represents a straight line, express this equation in vector form. a. ํ‘Ÿ = (1 + 4ํœ†)ํ‘– + (2ํœ† โˆ’ 1)ํ‘— + (4 + 3ํœ†)ํ‘˜ b. ํ‘Ÿ = (1 + 4ํœ†)ํ‘– + (2ํœ† + 1)ํ‘— + (4 โˆ’ 3ํœ†)ํ‘˜ c. ํ‘Ÿ = (1 + 4ํœ†)ํ‘– + (2ํœ† โˆ’ 1)ํ‘— + (4 โˆ’ 3ํœ†)ํ‘˜ d. ํ‘Ÿ = (1โˆ’ 4ํœ†)ํ‘– + (2ํœ† โˆ’ 1)ํ‘— + (4 โˆ’ 3ํœ†)ํ‘˜ 48. Determine the vector equation of the line through the point with position vector 2ํ‘– + 3ํ‘— โˆ’ ํ‘˜ which is parallel to the vector ํ‘– โˆ’ 2ํ‘— + 3ํ‘˜ a. ํ‘Ÿ = (2 + ํœ†)ํ‘– + (3โˆ’ 2ํœ†)ํ‘— + (3ํœ† โˆ’ 1)ํ‘˜ b. ํ‘Ÿ = (2 + ํœ†)ํ‘– + (3 + 2ํœ†)ํ‘— + (3ํœ† โˆ’ 1)ํ‘˜ c. ํ‘Ÿ = (2 + ํœ†)ํ‘– + (3 โˆ’ 2ํœ†)ํ‘— + (3ํœ† + 1)ํ‘˜ d. ํ‘Ÿ = (2 + ํœ†)ํ‘– โˆ’ (3โˆ’ 2ํœ†)ํ‘— + (3ํœ† โˆ’ 1)ํ‘˜ 49. Find the point on the line corresponding to ํœ† = 3 in the resulting equation of part in question 48 a. ํ‘Ÿ = 5ํ‘– + 3ํ‘— + 8ํ‘˜ b. ํ‘Ÿ = 5ํ‘– โˆ’ 3ํ‘— + 8ํ‘˜ c. ํ‘Ÿ = 5ํ‘– โˆ’ 3ํ‘— โˆ’ 8ํ‘˜ d. ํ‘Ÿ = โˆ’5ํ‘– โˆ’ 3ํ‘— + 8ํ‘˜ 50. Using question 48 express the vector equation of the line in standard Cartesian form. a. ํ‘ฅ โˆ’ 2 = = = ํœ† b. ํ‘ฅ โˆ’ 2 = = = ํœ† c. ํ‘ฅ + 2 = = = ํœ† d. ํ‘ฅ โˆ’ 2 = = = ํœ† 51. If cos ํ‘ฅ = determine the value of sin ํ‘ฅ a. b. c. d. 52. If sin ํœƒ = 0.625 and cosํœƒ = 0.500 determine the value of ํ‘ํ‘œํ‘ ํ‘’ํ‘ํœƒ a. 2.00 b. 1.600 c. 1.25 d. 0.80 53. Solve triangle XYZ given โˆ ํ‘‹ = 90 ,โˆ ํ‘Œ = 23.17 and ํ‘Œํ‘ = 20.0ํ‘šํ‘š.Determine also its area in two decimal places a. 72.61ํ‘šํ‘š b. 72.60ํ‘šํ‘š c. 71.61ํ‘šํ‘š d. 72.62ํ‘šํ‘š 54. A surveyoy measures the angle of elavation of the a perpendicular buildingas 19 .He moves 120m nearer the building and finds the angle of elevation is now 47 .Determine the height of the building in two decimal places. a. 60.75m b. 60.73m c. 60.85m d. 60.83m 55. Evaluate . . . . leave your answer in four significant figures. a. 1.710 b. โˆ’1.710 c. 1.720 d. 1.720 56. Evaluate cos 75 leave your in surd form Print to PDF without this message by purchasing novaPDF (http://www.novapdf.com/) Page 7 of 7 a. (โˆš6โˆ’ โˆš2) b. (โˆš6 + โˆš2) c. (โˆ’โˆš6 โˆ’ โˆš2) d. โˆ’ (โˆš6โˆ’ โˆš2) 57. If ํ‘ = cosํœƒ What is the value a. ํ‘กํ‘Žํ‘› ํœƒ b. ํ‘ํ‘œํ‘  ํœƒ c. ํ‘ ํ‘–ํ‘› ํœƒ d. ํ‘ํ‘œํ‘ก ํœƒ 58. What is the value of ํœƒ in the equations ํ‘ฅ = ํ‘Ž cosํœƒ, ํ‘ฆ = ํ‘ sin ํœƒ a. โˆ’ + = 1 b. + = 1 c. โˆ’ = 1 d. + = โˆ’1 59. If sin ํœƒ = what is the of sin ํœƒ and tanํœƒ giving your answer in surd form and that ํœƒ is acute. a. โˆ’ โˆš , โˆš b. โˆš ,โˆ’ โˆš c. โˆš , โˆš d. โˆš , โˆš 60. Given that cos 45 = โˆš ,evaluate sin 22.5 a. ( โˆš ) b. ( โˆš ) c. ( โˆš ) d. โˆ’ ( โˆš ) EXAMINERS: I. B. NABUBIE AND MRS C. CHRISTIANA. NYARKO Print to PDF without this message by purchasing novaPDF (http://www.novapdf.com/)

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