MATH1061 Practice Exam, University of Sydney PDF

Chem1911 Sample Quiz 1 (i) 210 1. What is the decay product resulting from the emission of an alpha particle from 85 At ? 207...

Chem1911 Sample Quiz 1 (i) 210 1. What is the decay product resulting from the emission of an alpha particle from 85 At ? 207 210 206 206 206 a) 82 Pb b) 86 Rn c) 83 Bi d) 81 Tl e) 85 At 2. Which nuclide is needed to balance the following nuclear reaction? 235 92 U + 1 0 n ® ? + 96 39 Y + 3 01 n 139 138 137 136 135 a) 53 I b) 53 I c) 53 I d) 53 I e) 53 I 3. Only one of the following isotopes of strontium undergoes radioactive decay by b– emission? Which one is it? 83 86 87 88 90 a) 38 Sr b) 38 Sr c) 38 Sr d) 38 Sr e) 38 Sr 4. For which one of the following atoms or ions would the 2s and 2p orbitals have the same energy? a) O2– b) H c) He d) Li+ e) F6+ 5. Which of the following electron excitations of the hydrogen atom requires light of the shortest wavelength? a) n = 2 to n = 3 b) n = 3 to n = 4 c) n = 4 to n = 20 d) n = 5 to n = 100 e) n = 4 to n = 1000 6. How many nodes does a 5s atomic orbital have? a) 0 planar nodes and 0 spherical nodes b) 3 planar nodes and 2 spherical nodes c) 1 planar node and 1 spherical node d) 0 planar nodes and 4 spherical nodes e) 2 planar nodes and 3 spherical nodes 7. The 1s1 3p1 ® 1s2 transition of He is at 54 nm. Which of the following statements is correct? a) The 1s1 2p1 ® 1s2 transition of He is at a longer wavelength than 54 nm. b) The 1s1 2p1 ® 1s2 transition of He is at a shorter wavelength than 54 nm. c) The 1s1 2p1 ® 1s2 transition of He is also at 54 nm. d) No deduction about the 1s1 2p1 ® 1s2 transition of He can be made. 8. The half-life of 14C is 5730 years. Which of the following can be usefully dated using 14C dating methods? a) dinosaur bones (70 million years old) b) 15th century paintings c) rocks that are 2 billion years old d) early human ancestor remains (approximately 2 million years old) e) a corpse in a murder investigation (less than 2 years old) 9. Which one of the following sets of quantum numbers is valid? n l ml ms a) 3 1 0 0 b) 1 1 0 –½ c) 3 3 –2 +½ d) 1 1 1 0 e) 5 4 3 +½ 10. Reference: http://firstyear.chem.usyd.edu.au/LabManual/W5.pdf When computed on a calculator, the algebraic expression 0.350 kg ´ 141 J has a (0.921 m + 68 m) value of 0.716037202. Expressed to the appropriate number of significant figures, this is: a) 0.7 kg J m–1 b) 0.71 kg J m–1 c) 0.72 kg J m–1 d) 0.716 kg J m–1 e) 0.71604 kg J m–1 11. Which of the following statements about molecular orbitals for homonuclear diatomic molecules is incorrect? a) Sigma orbitals are symmetric to rotation about the internuclear axis. b) Sigma orbitals have a planar node normal to the internuclear axis between the atomic nuclei. c) Sigma orbitals have a symmetrical distribution of electron density around the two nuclei. d) Pi orbitals have a planar node along the internuclear axis. e) Sigma* orbitals are higher in energy than their corresponding sigma orbitals. 12. What is the hybridisation of the designated atoms in the following compound? Q a) P = sp2, Q = sp2, R = sp2 O b) P = sp2, Q = sp2, R = sp3 P NH N c) P = sp2, Q = sp3, R = sp2 d) P = sp3, Q = sp3, R = sp2 NH NH2 N e) P = sp3, Q = sp2, R = sp R 13. Identify the correct Lewis structure of H2O2. Ensure that all charges and lone pairs are shown. H O O H H O O H H O O H H O O H a) b) c) d) 14. How many non-bonding electron pairs (lone pairs) are around the Cl atom in HCl? a) 0 b) 1 c) 2 d) 3 e) 4 15. Identify the functional group labelled A in the molecule below? a) Alcohol b) Carboxylic acid c) Amine d) Ester e) Ether Correct answers: 1C, 2C, 3E, 4B, 5A, 6D, 7A, 8B, 9E, 10C, 11B, 12B, 13B, 14D, 15C  Chem1911 Sample Quiz 1 (ii) 144 1. What is the decay product resulting from electron capture by the 61 Pm nuclide? 144 144 145 144 144 a) 60 Pm b) 62 Pm c) 60 Nd d) 60 Nd e) 62 Sm 2. Which nuclide is needed to balance the following nuclear reaction? 233 92 U + 1 0 n ® ? + 101 42 Mo + 3 01 n 132 131 130 129 128 a) 50 Sn b) 50 Sn c) 50 Sn d) 50 Sn e) 50 Sn 3. Only one of the following isotopes of gallium does not undergo radioactive decay via electron capture. Which one is it? 69 68 67 66 65 a) 31 Ga b) 31 Ga c) 31 Ga d) 31 Ga e) 31 Ga 4. For which one of the following atoms or ions would the 2s and 2p orbitals have the same energy? a) O2– b) H– c) He d) Be2+ e) N6+ 5. Which of the following electron excitations of the hydrogen atom requires light of the longest wavelength? a) n = 2 to n = 3 b) n = 3 to n = 4 c) n = 4 to n = 20 d) n = 5 to n = 100 e) n = 4 to n = 1000 6. How many nodes does a 2p atomic orbital have? a) 0 planar nodes and 0 spherical nodes b) 0 planar nodes and 1 spherical nodes c) 1 planar nodes and 0 spherical nodes d) 1 planar node and 1 spherical node e) 2 planar nodes and 2 spherical nodes 7. The 1s1 3p1 ® 1s2 transition of He is at 54 nm. Which of the following statements is correct? a) The 1s1 2p1 ® 1s2 transition of He is at a longer wavelength than 54 nm. b) The 1s1 2p1 ® 1s2 transition of He is at a shorter wavelength than 54 nm. c) The 1s1 2p1 ® 1s2 transition of He is also at 54 nm. d) No deduction about the 1s1 2p1 ® 1s2 transition of He can be made. 8. The half-life of 14C is 5730 years. Which of the following can be usefully dated using 14C dating methods? a) dinosaur bones (70 million years old) b) 15th century paintings c) rocks that are 2 billion years old d) early human ancestor remains (approximately 2 million years old) e) a corpse in a murder investigation (less than 2 years old) 9. Which one of the following sets of quantum numbers is valid? n l ml ms a) 4 4 3 +½ b) 2 1 0 –½ c) 3 2 –2 +1 d) 1 1 1 0 e) 3 1 0 0 10. Reference: http://firstyear.chem.usyd.edu.au/LabManual/W5.pdf When computed on a calculator, the algebraic expression has a value of 0.596443966. Expressed to the appropriate number of significant figures, this is: a) 0.5 kg J m–1 b) 0.6 kg J m–1 c) 0.59 kg J m–1 d) 0.60 kg J m–1 e) 0.596 kg J m–1 11. Which one of the following statements about molecular orbitals for homonuclear diatomic molecules is incorrect? a) Sigma orbitals are symmetric to rotation about the internuclear axis. b) Pi orbitals have a planar node normal to the internuclear axis between the atomic nuclei. c) Sigma orbitals have a symmetrical distribution of electron density around the two nuclei. d) Pi orbitals have a planar node along the internuclear axis. e) Pi* orbitals are higher in energy than their corresponding pi orbitals. 12. What is the hybridisation of the designated atoms in the following compound? Q a) P = sp2, Q = sp2, R = sp2 P b) P = sp2, Q = sp2, R = sp3 c) P = sp, Q = sp2, R = sp2 N C O CH2 CH3 d) P = sp, Q = sp3, R = sp2 e) P = sp, Q = sp2, R = sp3 R 13. Identify the correct Lewis structure of CN-. Ensure that all charges and lone pairs are shown. C N C N C N C N a) b) c) d) 14. How many non-bonding electron pairs (lone pairs) are around the B atom in BF3? a) 0 b) 1 c) 2 d) 3 e) 4 15. Consider the molecule below. What is the name of the functional group labelled B? a) Amine b) Alcohol c) Amide d) Carboxylic acid e) Ether Correct answers: 1D, 2C, 3A, 4E, 5D, 6C, 7A, 8B, 9B, 10D, 11B, 12E, 13A, 14A, 15B Quiz S Semester 1, 2025 The University of Sydney Faculties of Arts, Economics, Education, Engineering and Science MATH1061: Mathematics 1A Time allowed: 40 Minutes This booklet contains 7 pages. Name:................................... SID:................................... Signature:................................. Day:................................. Time:................................... Room:................................... ˆ No calculators ˆ No reference material or additional paper ˆ Every question has exactly one correct answer ˆ Record your answers on the multiple choice answer sheet ˆ Record your quiz version on the multiple choice answer sheet Quiz Version: S Quiz S Semester 1, 2025 page 2 of 7 Questions Question 1 Points: 1 Which of the following statements is not true? (A) Z ⊂ R. (B) N ⊂ Z. (C) Z ⊂ Q. (D) R ⊂ Q. Question 2 Points: 1 Let f and g be f : R → R, f (x) = 1 − x2 , g : N → R, g(x) = ln(x), where N is the natural domain of the function g. What are the domain and range of g ◦ f ? (A) Domain: [−1, 1], Range: (−∞, 0]. (B) Domain: (−1, 1), Range: (−∞, 0]. (C) Domain: [−1, 1], Range: [0, ∞). (D) Domain: (−1, 1), Range: [0, ∞). turn to page 3 Quiz S Semester 1, 2025 page 3 of 7 Question 3 Points: 1 The function tanh(x) is given by ex − e−x tanh(x) =. ex + e−x Calculate lim tanh(x). x→−∞ (A) 0 (B) 1 (C) −1 (D) The limit does not exist. Question 4 Points: 1 Let f (x) = x6 sin(x) + cosh(6x). What is f ′ (x)? (A) x6 cos(x) + 6x5 sin(x) + 6 sinh(6x) (B) x6 cos(x) + 6x5 sin(x) − 6 sinh(6x) (C) x6 sin(x) + 6x5 cos(x) + 6 sinh(6x) (D) x6 cos(x) + 6x5 sin(x) − 6 sinh(6x) turn to page 4 Quiz S Semester 1, 2025 page 4 of 7 Question 5 Points: 1 Let f (x) = (2x)x. Calculate f ′ (x). (A) x(2x)x−1 (B) x(2x)x−1 ln(2x) (C) (2x)x ln(2x) (D) (2x)x (1 + ln(2x)) Question 6 Points: 1 Calculate the three-term Taylor polynomial about x = 0 for f (x), where 1 f (x) =. 1 − 2x (A) −1 − 2x − 8x2 (B) −1 + 2x − 8x2 (C) −1 + 2x − 4x2 (D) −1 − 2x − 4x2 turn to page 5 Quiz S Semester 1, 2025 page 5 of 7 Question 7 Points: 1 Let z = 1 − 3i , w = 2 + i. Calculate wz. (A) 5 + 5i (B) 5 − 5i (C) −1 + 5i (D) −1 − 5i Question 8 Points: 1 Let a, b ∈ R and let u = [1, 2, 1], v = [b, a, 2] and w = [a, −b, −4]. For what values of a and b do we have 6u − v + w = 0? (A) a = 1, b = 3 (B) a = −9, b = 3 (C) a = 3, b = 9 (D) There are no values of a and b which satisfy this. turn to page 6 Quiz S Semester 1, 2025 page 6 of 7 Question 9 Points: 1 Let ℓ represent the line in R2 that passes through the point (1, 1) and is perpendicular to the vector [3, 2]. Which of the following is a parametric equation for ℓ? (A) ℓ : [3, 2] + t[1, 1], t ∈ R. (B) ℓ : [3, 2] + t[−1, 1], t ∈ R. (C) ℓ : [1, 1] + t[3, 2], t ∈ R. (D) ℓ : [1, 1] + t[−2, 3], t ∈ R. Question 10 Points: 1 Which of the following parametric equations represents the line ℓ in R3 that passes through the point (1, 2, 5) and is perpendicular to both of the vectors u = [1, 1, 1] and v = [2, 0, 0]? (A) ℓ : [1, 2, 5] + t[0, 2, −2], t ∈ R. (B) ℓ : [1, 2, 5] + t[0, 1, 1], t ∈ R. (C) ℓ : [1, 2, 5] + t[2, 0, 0], t ∈ R. (D) ℓ : [1, 2, 5] + t[3, 1, 1], t ∈ R. turn to page 7 Quiz S Semester 1, 2025 page 7 of 7 Question 11 Points: 1 Determine whether the line [x, y, z] = [1, 2, −2] + t[2, 1, 2], t ∈ R intersects the plane x + 2y + 3z = 19, and if so, find the intersection. (A) The line does not intersect the plane. (B) The line intersects the plane at (6, 2, 3). (C) The line intersects the plane at (5, 4, 2). (D) The line intersects the plane at (5, 1, 4). Question 12 Points: 1 Which of the following is equal to sin(4θ)? (A) 4 cos3 (θ) sin(θ) − 4 cos(θ) sin2 (θ) (B) 4 sin3 (θ) cos(θ) − 4 sin(θ) cos3 (θ) (C) cos4 (θ) − 6 cos2 (θ) sin2 (θ) + sin4 (θ) (D) cos4 (θ) + 6 cos2 (θ) sin2 (θ) + sin4 (θ) This is the end of the quiz paper  lOMoARcPSD|44658817 MATH1061-Practice-2024 Mathematics 1A (University of Sydney) Scan to open on Studocu Studocu is not sponsored or endorsed by any college or university Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS MATH1061 MATHEMATICS lA June 2024 LECTURERS: Nathan Brownlowe, Christopher Lustri, Brad Roberts, Haotian Wu TIME ALLOWED: Reading time - 10 minutes; Writing time - 2 hours EXAM CONDITIONS: This is a closed-book examination - no material permitted. Writing is not permitted at all during reading time. Family Name: SID:........................... Other Names: Seat Number:................. Please check that your examination paper is complete (34 pages) and indicate by signing below. I have checked the examination paper and affirm it is complete. Signature:.................................................. Date: MARKER'S USE ONLY This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination. There are 20 questions. The questions are of equal value. All questions may be attempted. Answers to the Multiple Choice questions must be entered on the Multiple Choice Answer Sheet before the end of the examination. The Extended Answer Section is worth 50% of the total examination. There are 4 questions. The questions are of equal value. All questions may be attempted. Working must be shown. Non-programmable calculators may be used, as long as they have a University of Sydney approval sticker on them. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. PAGE 1 OF 34 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 2 OF 34 Multiple Choice Section In each question, choose at most one option. Your answers must be entered on the Multiple Choice Answer Sheet. 1. Let f: ffi.----+ [1, oo), where f(x) = cosh(x). Which statement is true? (a) f is injective but not surjective. (b) f is surjective but not injective. (c) f is bijective. (d) f is neither surjective nor injective. 2. Evaluate the following limit: 2x - 2ex-l lim----- x---+1 x 2 - 2x + 1 (a) 0 (b) 1 (c) -1 (d) The limit does not exist 3. If f : ffi. ----+ ffi. such that f(x) = 2x 3 - 3x 2 - 12x + 4, what is the minimum value for J(x)? (a) 11 (b) -11 (c) -16 (d) The function has no minimum value Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 3 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 4 OF 34 4. Calculate 1 x(sin(x)) 2 (a ) --+--- cos(x) (cos(x)) 2 (b) cos\x) - sin~x) 1 xtan(x) (c) cos(x) + cos(x) 1 x sin(x) (d) cos(x) cos(x) 5. A function is given by ke-x X :s; 7f g (X ) ={ sin(2x) x > 1r where k is a constant. What value of k makes this a continuous function? (a) There is no value of k for which this is continuous (b) 1 (c) -1 (d) 0 6. State the first three nonzero terms of the Taylor series for 1x ( e- 82 ) ds about x = 0. x3 x5 (a) X + 3 + lQ x3 x5 (b) X - 3 + 10 x4 x6 (c) x2 + - +-6 2 x4 x6 (d) x2 - 2 +6 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 5 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 6 OF 34 7. The following three plots contain a function f (x), and its first and second derivative f'(x) and f"(x). Which plot corresponds to f(x), f'(x), and f"(x)? - 10 _, 10 (a) f(x): Plot 1, f'(x): Plot 2, f"(x): Plot 3 (b) f(x): Plot 1, f'(x): Plot 3, f"(x): Plot 2 (c) f(x): Plot 3, f'(x): Plot 1, f"(x): Plot 2 (d) f(x): Plot 3, f'(x): Plot 2, f"(x): Plot 1 8. Calculate JJl cos(x) + 2sin(x) dx (a) Jl + 2sin(x) + C (b) Jl + ~sin(x) + C 1 (C) -----==== +C J1+2sin(x) (d) 1 +C 2Jl + 2 sin(x) 9. What is the length of the curve defined by y = cosh(x) from x = 0 to x = ln(2)? 3 (a) 4 (b) 2 (c) 4 (d) 4 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 7 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 8 OF 34 10. The volume generated by rotating the curve vx around the x-axis from Oto 4 is (a) 21r (b) 41r (c) 81r (d) l61r 11. Let S be the set S := {z E C : Re z 2: 0 and Im z ::; 0}. Which quadrant of the complex plane can be represented by the set {iz: z ES}? (a) 1st (b) 2nd (c) 3rd (d) 4th 12. Let v = [2, 6, -1] and w = [-3, a,~]. Find the value of a so that v and w are parallel. (a) 9 (b) -9 2 (c) 3 (d) -~ 3 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 9 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 10 OF 34 13. Let u, v, w E IR2 be nonzero vectors with u v = 0. Which one of the following is not necessarily true? (a) proju(w) · projv(w) = 0. (b) w = proju(w) + projv(w). (c) Either proju(w) = w or projv(w) = w. (d) If proju(w) = 0, then projv(w) = w. 14. Consider the lines with parametric equations X = 3+t X = -3- 2s y = -5 + 2t t E IR and y = -2 - 4s s E IR z=5-t z = l + 2s Which one of the following is true? (a) The lines intersect at the point (7, 3, -9) (b) The lines do not intersect (c) The lines intersect at the point (-2, -15, 0) (d) The lines are the same line 15. Which one of the following vectors is parallel to the plane 2x - y + 5z = 3? (a) [1, -1, 0] (b) [-2, 1, 1] (c) [2, -1, 5] (d) [1, 1, 1] Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 11 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 12 OF 34 16. Suppose A is a 3 x 3 matrix, B is a 3 x 2 matrix, and C is a 2 x 3 matrix. Which one of the following is true? (a) ACE is defined (b) (BC) 2 is a 2 x 2 matrix (c) A 2 + BC is a 3 x 3 matrix (d) B(A - BC) is defined 1 1 ol 1 7. If A = [ l 1 1 , which one of the following is true? 0 1 1 (a) A is not invertible (b) A- 1 = [~ 1 1 1 ~] -1 (c) A- 1 = [ ~1 1 -1 ~1] 1 (d) A- 1 = [11 Tl -1 1 18. Suppose B is a square matrix with det(B) = 2 and det (- ~B) = ~- Then B is what size? (a) 2 X 2 (b) 4 x 4 (c) 8 X 8 (d) There is not enough information to determine Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 13 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 14 OF 34 19. Let A 1 = P D 1 P- 1 and A2 = P D 2 P- 1 , where D1 = [t ~] , D2 = [ ~] , and P = [ t] · Then (A 1 A2 ) 5 is which of the following? [25 35 - 25] (a) 0 35 [-25 25-35] (b) 0 -35 (c) [205 31]5 (d) l [305 25o 20. Suppose A is a 3 x 3 matrix which has three distinct eigenvalues. Which one of the following is not necessarily true? (a) The characteristic polynomial of A has three distinct roots (b) A is invertible (c) E>.(A) involves exactly one parameter for every eigenvalue,\ (d) There is a 3 x 3 invertible matrix P such that p-l AP is diagonal Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 15 OF 34 This page is left blank for your working. Working for the Multiple Choice section will not be marked. End of Multiple Choice Section Make sure that your answers are entered on the Multiple Choice Answer Sheet THE EXTENDED ANSWER SECTION BEGINS ON THE NEXT PAGE Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 16 OF 34 Extended Answer Section There are four questions in this section, each with a number of parts. Write your answers in the space provided below each part. There is extra space at the end of the paper. 1. (a) Consider the function f(x) = e-x - e-2x (i) State the natural domain of f and find any vertical or horizontal asymptotes. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 17 OF 34 ( ii) Find any critical points and identify the intervals on which f is increasing or decreasing. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 18 OF 34 ( iii) Find any points of inflection of f and identify the intervals of positive and negative curvature. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 19 OF 34 (iv) Sketch the graph of f, showing all features identified above, as well as any axis intercepts. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 20 OF 34 2. (a) Evaluate {2 _x_2_dx }0 1 + x3 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 21 OF 34 (b) Evaluate jevxdx Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 22 OF 34 (c) Consider the area contained in the triangle in the first quadrant of the xy-plane enclosed by the x-axis, the y-axis, and the line y = mx + c which passes through (x, y) = (2, 4), where m < 0. What is the equation of the line that causes the triangle to contain the least area? Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 23 OF 34 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 24 OF 34 3. (a) Given that z = l + i is one solution, solve the equation z 3 + 3z 2 - 8z + 10 = 0. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 25 OF 34 (b) Let Ci and C2 be the lines in ~ 3 with the following parametric equations Ci : x = l + s, y = 2 + s, z = 3 + s, s E C2 : X = l + t, y = 2, z = 3 + 2t, t E Find the genral equation of the plane containing Ci and C2. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 26 OF 34 (c) Find the value(s) of the parameter a E for which the following system of linear equations has a unique solution. X + ay + Z 2 x + y + az 1 X + y + Z 1 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 27 OF 34 (d) ( i) Find the inverse of the matrix [! 1 -1 !01 !-11]. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 28 OF 34 ( ii) Use your answer to (i), or another valid method, to solve the system of equa- tions 4x + 2y + 3z =1 -x y z =2 -x z = -1 5x + y + z + 2w =3 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 29 OF 34 4. (a) Let M (i) [ 1 l Find the eigenvalues of M. (ii) Find an invertible matrix P such that p- 1 MP is a diagonal matrix. You do not need to find p- 1. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 30 OF 34 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 31 OF 34 (b) Let A be a 3 x 3 matrix and u be a nonzero vector in ~ 3 , and denote by v = Au. Show that if X = SU + tv, S, t E ~, is a line through the origin (rather than a plane through the origin), then u is an eigenvector of A. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 32 OF 34 (c) Suppose that A is a 3 x 3 matrix with eigenvalues -t, 0 and t for some positive real number t. Prove that A 3 is a scalar multiple of A. Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 33 OF 34 Downloaded by Yuki W ([email protected])  lOMoARcPSD|44658817 PRACTICE EXAM A SEMESTER 1 2024 PAGE 34 OF 34 Formula Sheet (n-/- -1) 9. J sec 2 xdx = tanx + C 2. J dx -;-=lnlxl+C 10. J cosec 2 x dx = - cot x + C 11. J sec x dx = ln Isec x + tan x I + C 4. J sinxdx = - cosx + C 12. J cosec x dx = lnlcosec x - cot xi + C 5. J cosxdx = sinx + C 13. J sinhxdx = coshx + C 6. J tanxdx = -lnlcosxl + C 14. J coshxdx = sinhx + C 7. J cotxdx = lnlsinxl + C 15. J tanhxdx = lncoshx + C 8. J 2 dx = -l tan _ 1 - + C a +x 2 a a (x) 16. J dx = sin- 1 (~) + C (lxl < a) a 17. J dx + a2 = sinh- 1 (~) + C = ln(x + a + a 2 ) + C' 18. j dx - a2 = cosh- 1 (~) +C = a - a 2 ) +C' (x > a) Linearity: J (>..J(x) + µg(x)) dx = >.. J f(x) dx +µ J g(x) dx Integration by substitution: J f(u(x)) ~: dx = J f(u) du Integration by parts: J f(x)g'(x) dx = f(x)g(x) - J J'(x)g(x) dx End of Extended Answer Section End of Examination Downloaded by Yuki W ([email protected])