Việc sở hữu đề thi Phương pháp tính sẽ giúp bạn nắm vững cấu trúc và nội dung thi, từ đó dễ dàng đạt điểm cao trong kì thi cuối kì ở Đại học Bách Khoa. Đề thi thường phản ánh chính xác những kiến thức trọng tâm mà giảng viên muốn nhấn mạnh, giúp bạn tập trung ôn luyện những phần quan trọng. Hơn nữa, khi làm quen với đề thi, bạn sẽ biết được các dạng bài tập thường xuất hiện và cách giải quyết chúng một cách hiệu quả. Ngoài ra, việc luyện tập với đề thi còn giúp bạn cải thiện kỹ năng quản lý thời gian, tránh được tình trạng lúng túng khi làm bài. Vì vậy, hãy mua và sử dụng đề thi Phương pháp tính một cách thông minh để tối ưu hóa kết quả học tập của mình.
Trang 1Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 0.0929 B 0.8589 C 0.0451 D 0.3857 E -0.05355 Using the Runge-Kutta 4, approximate the solution at x = −0.82.
A 0.3757 B 0.6493 C 0.2643 D -0.0165 E -0.0767
Trang 2dt = ty + 3.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (2.6519,2.3696) B (2.8809,2.489) C (3.2335,2.6452) D (2.6045,2.3918) E (2.4441,1.8272)
7 Using the modified Euler method, approximate the solution at t = 1.1.
x0x1x2x3x4y = f1(x)
y = f2(x)0.57
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Trang 313 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A None of others B -0.2223 C -0.2222 D -0.222 E -0.2221
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0019 B None of others C 0.0021 D 0.002 E 0.0018
Questions 15 to 18.Given the linear system
4x − y = 4
4x + 8y = 4 and the initial vector
x(0)y(0)
= 03
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
0.25y(n−1)+ 1−0.5x(n−1)+ 0.5
B None of others C x(n)y(n)
= 0.0336y(n−1)+ 10.8322x(n−1)+ 0.5
.D x(n)
=0.8322y(n−1)+ 30.0157x(n−1)0.5
E x(n)y(n)
= 0.0336y(n−1)+ 10.0157x(n−1)+ 0.5
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
Trang 4Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 52.9089 B 52.8413 C 52.9441 D 51.9941 E 52.94275 Using the Runge-Kutta 4, approximate the solution at x = −3.3.
A 29.6934 B 30.2014 C 29.7734 D 29.845 E 29.851
Trang 5dt = ty + 1.7where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
x0x1x2x3x4y = f1(x)
Trang 613 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.2221 B -0.2222 C -0.222 D -0.2223 E None of others
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0019 B 0.002 C 0.0021 D 0.0018 E None of others
Questions 15 to 18.Given the linear system
4x + y = 3
3x + 7y = 2 and the initial vector
x(0)y(0)
= −12
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
−0.25y(n−1)+ 0.75−0.4286x(n−1)+ 0.2857
B None of others C x(n)y(n)
= 0.6575y(n−1)+ 0.750.473x(n−1)+ 0.2857
.D x(n)
0.6575y(n−1)+ 0.750.2788x(n−1)+ 0.2857
E x(n)y(n)
= 0.2788y(n−1)+ 30.473x(n−1)0.2857
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
Trang 7Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 12.6367 B 12.9233 C 12.9147 D 13.1435 E 12.40275 Using the Runge-Kutta 4, approximate the solution at x = 2.26.
A 13.3802 B 13.2412 C 12.946 D 12.536 E 12.4454
Trang 8dt = ty + 3.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (2.7337,2.1622) B (3.1105,1.9974) C (2.4441, 1.8272) D (3.2163,2.5158)E (2.6141,2.0318)
7 Using the modified Euler method, approximate the solution at t = 1.1.
x0x1x2x3x4y = f1(x)
Trang 913 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.2858 B -0.2855 C -0.2857 D None of others E -0.2856
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0058 B 0.0059 C None of others D 0.0061 E 0.006
Questions 15 to 18.Given the linear system
4x + y = 4
2x + 6y = 2 and the initial vector
x(0)y(0)
= 03
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
0.2828y(n−1)+ 30.8862x(n−1)0.3333
B None of others C x(n)y(n)
0.5323y(n−1)+ 10.8862x(n−1)+ 0.3333
.D x(n)
−0.25y(n−1)+ 1−0.3333x(n−1)+ 0.3333
E x(n)y(n)
0.5323y(n−1)+ 10.2828x(n−1)+ 0.3333
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
2 + 1, where m is the last digit in the ID student For example, if ID student is 2022345,
Trang 10Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 126.4286 B 126.7178 C 126.2002 D 126.1628 E 126.70245 Using the Runge-Kutta 4, approximate the solution at x = 4.3.
A 59.1213 B 59.3857 C 58.6861 D 58.9255 E 58.8031
Trang 11dt = ty + 3.3where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (3.2276,2.724) B (2.8646,2.6766) C (2.5286,2.1188) D (3.1474,2.7308) E (2.4078,1.7936)
7 Using the modified Euler method, approximate the solution at t = 1.1.
A (1.9267,1.7022) B (1.1373,1.6764) C (1.0805, 0.9034) D (1.3673,1.1162)E (2.0615,1.5848)
0.55 0.6
x0x1x2x3x4y = f1(x)
y = f2(x)0.87
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Trang 1213 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.2856 B None of others C -0.2858 D -0.2855 E -0.2857
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.006 B None of others C 0.0059 D 0.0058 E 0.0061
Questions 15 to 18.Given the linear system
8x + 5y = 2
−3x + 7y = 1 and the initial vector
x(0)y(0)
= 01
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
0.5767y(n−1)+ 0.250.0101x(n−1)+ 0.1429
B x(n)y(n)
0.5767y(n−1)+ 0.250.6827x(n−1)+ 0.1429
C None ofothers D x(n)
0.6827y(n−1)+ 30.0101x(n−1)0.1429
E x(n)y(n)
= −0.625y(n−1)+ 0.250.4286x(n−1)+ 0.1429
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
2 + 1, where m is the last digit in the ID student For example, if ID student is 2022345,
Trang 13Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 85.0636 B 84.5524 C 84.772 D 84.804 E 84.51425 Using the Runge-Kutta 4, approximate the solution at x = 4.98.
A 96.0588 B 95.84 C 95.5252 D 95.245 E 95.915
Trang 14dt = ty + 2.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (2.081, 1.4909) B (2.579,1.8129) C (2.7006,2.4743) D (2.972,1.9277) E (2.3962,1.8827)
7 Using the modified Euler method, approximate the solution at t = 1.1.
A (1.9431,1.6159) B (1.5751,1.5627) C (0.9815, 0.8085) D (1.5397,1.1183)E (1.8599,1.1159)
0.375 0.3750.625
x0x1x2x3x4y = f1(x)
Trang 1513 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.25 B -0.2499 C None of others D -0.2498 E -0.2501
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A None of others B 0.0034 C 0.0033 D 0.0031 E 0.0032
Questions 15 to 18.Given the linear system
5x + 2y = 4
3x + 7y = 2 and the initial vector
x(0)y(0)
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
0.0023y(n−1)+ 0.80.9962x(n−1)+ 0.2857
B None of others C x(n)y(n)
0.606y(n−1)+ 30.9962x(n−1)0.2857
.D x(n)
0.0023y(n−1)+ 0.80.606x(n−1)+ 0.2857
E x(n)y(n)
−0.4y(n−1)+ 0.8−0.4286x(n−1)+ 0.2857
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
Trang 16Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 8.1736 B 7.8316 C 8.097 D 8.6718 E 8.51025 Using the Runge-Kutta 4, approximate the solution at x = −0.8.
A 5.7104 B 5.2144 C 5.5902 D 5.3484 E 6.1728
Trang 17dt = ty + 1.3where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
0.55 0.575
x0x1x2x3x4y = f1(x)
y = f2(x)0.78
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Trang 1813 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.1998 B -0.2001 C None of others D -0.1999 E -0.2
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0011 B 0.0012 C None of others D 0.0014 E 0.0013
Questions 15 to 18.Given the linear system
8x + 5y = 3
−3x + 7y = 1 and the initial vector
x(0)y(0)
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
0.269y(n−1)+ 0.3750.3567x(n−1)+ 0.1429
B x(n)y(n)
0.269y(n−1)+ 0.3750.6246x(n−1)+ 0.1429
C None ofothers D x(n)
0.6246y(n−1)+ 30.3567x(n−1)0.1429
E x(n)y(n)
= −0.625y(n−1)+ 0.3750.4286x(n−1)+ 0.1429
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
2 + 1, where m is the last digit in the ID student For example, if ID student is 2022345,
Trang 19Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A 38.0209 B 37.5007 C 37.5641 D 37.9267 E 38.22215 Using the Runge-Kutta 4, approximate the solution at x = 3.08.
A 17.5807 B 17.4309 C 17.4207 D 17.7349 E 18.2503
Trang 20dt = ty + 1.6where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (1.9891,2.1521) B (2.4125,2.1823) C (1.7905, 1.2219) D (2.6207,2.0327)E (2.3185,2.2127)
7 Using the modified Euler method, approximate the solution at t = 1.1.
A (1.6675,1.0967) B (1.7423,0.8403) C (1.4743,1.2619) D (0.8935, 0.7241)E (1.2155,1.0745)
x40.725 0.65
x0x1x2x3x4y = f1(x)
1.42 y = f2(x)0.6
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Trang 2113 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.1999 B None of others C -0.2 D -0.1998 E -0.2001
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0013 B 0.0012 C None of others D 0.0011 E 0.0014
Questions 15 to 18.Given the linear system
7x + 4y = 3
−2x + 6y = 3 and the initial vector
x(0)y(0)
= −10
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
=0.0493y(n−1)+ 0.42860.4377x(n−1)+ 0.5
B None of others C x(n)y(n)
=0.0493y(n−1)+ 0.42860.4433x(n−1)+ 0.5
.D x(n)
=−0.5714y(n−1)+ 0.42860.3333x(n−1)+ 0.5
E x(n)y(n)
=0.4433y(n−1)+ 30.4377x(n−1)0.5
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
2 + 1, where m is the last digit in the ID student For example, if ID student is 2022345,
Trang 22Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCMFaculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point).
Full name: Proctor 1: ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θdt2 + g
A -31.1724 B -32.0588 C -31.9988 D -31.2534 E -32.0965 Using the Runge-Kutta 4, approximate the solution at x = −0.12.
A -17.2064 B -17.3768 C -16.6828 D -17.5148 E -17.0452
Trang 23dt = ty + 3.1where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1.
6 Using the Euler method, approximate the solution at t = 1.3.
A (3.0544,2.017) B (2.3352, 1.7264) C (3.0094,1.8674) D (3.1278,2.565) E (2.6536,2.6118)
7 Using the modified Euler method, approximate the solution at t = 1.1.
A (1.3329,1.0433) B (1.8049,0.9889) C (1.0585, 0.8823) D (1.9801,1.0347)E (1.5971,1.4763)
x0x1x2x3x4y = f1(x)
Trang 2413 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.A -0.2221 B -0.222 C -0.2222 D -0.2223 E None of others
14 By the Newton method, find approximately the error of the solution x1 ∈ (−1, 0) (let x0 = 0).A 0.0021 B 0.0019 C None of others D 0.002 E 0.0018
Questions 15 to 18.Given the linear system
7x + 4y = 2
x + 5y = 2 and the initial vector
x(0)y(0)
= −3−3
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?A x(n)
=0.5178y(n−1)+ 0.28570.9178x(n−1)+ 0.4
B x(n)y(n)
=0.5178y(n−1)+ 0.28570.3618x(n−1)+ 0.4
C x(n)y(n)
=0.3618y(n−1)+ 3
D None of others E x(n)y(n)
=−0.5714y(n−1)+ 0.2857−0.2x(n−1)+ 0.4
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations).
2 + 1, where m is the last digit in the ID student For example, if ID student is 2022345,