Việc mua và ôn luyện đề thi Phương pháp tính là một chiến lược quan trọng để đạt điểm cao trong kỳ thi giữa kỳ ở Đại học Bách Khoa. Đề thi sẽ giúp bạn nắm vững cấu trúc và nội dung thi, từ đó làm quen với các dạng câu hỏi thường gặp và các vấn đề trọng tâm mà giảng viên có thể ra. Khi luyện tập với đề thi, bạn sẽ có cơ hội rèn luyện kỹ năng giải các bài tập cụ thể, cũng như nắm bắt những phương pháp giải nhanh và chính xác. Điều này không chỉ giúp củng cố kiến thức mà còn tăng cường sự tự tin khi bước vào phòng thi. Hơn nữa, việc luyện tập với đề thi sẽ giúp bạn cải thiện kỹ năng quản lý thời gian, tránh được tình trạng lúng túng và mất điểm oan. Vì vậy, hãy đầu tư vào việc mua đề thi Phương pháp tính và sử dụng chúng một cách hiệu quả để chuẩn bị tốt nhất cho kỳ thi giữa kỳ sắp tới.
Trang 1Lecturer: Date Approved by: Date
.
HCM University of TechnologyFaculty of Applied Science
Midterm test YearDate 2023-202417/10/2023 Semester 1Subject Numerical methods
- The question sheet has 20 questions.
- Documents are allow to used, except laptop, tablet, mobile phone.- Unless stated otherwise, round all answers to 4 decimal digits.
- Each wrong answer will get a penalty of 0.1 point.
Full name:
ID: Invigilator 1: Class: Invigilator 1: Suppose that we have to compute √3
21 Given that the exact number A belongs to [2, 3] Answer thequestions (1)-(8).
1 By the fixed point method, which sequence below converges to A?A none of them B xn =21xn−1
C xn =21xn−1xn−1+1
D xn =x 21n−1+1
E xn =
5 Evaluate the absolute error of the value obtained in (4) by using the general formula.A 0.0926 B 0.0514 C 0.1040 D 0.0193 E 0.0819
6 Using the bisection method, find the least number of steps n such that the approximate value xnof A has the accuracy within 10−3.
A n = 10 B n = 8 C n = 12 D n = 9 E n = 14
7 Using the Newton method and choosing x0 appropriately, compute an approximate value of A with2 number of iterations.
A 2.4104 B 2.8531 C 2.7591 D 2.1982 E 2.42918 Estimate the error of value found in question 7.
A 0.0010 B 0.0003 C 0.0059 D 0.0037 E 0.0070
Trang 2Midterm test - MT1009 2311 Semester I
From question (9) to (14), given the information: Let C =
11 Carry out 2 iterations to get the approximate solution X2 = (x1, x2, x3)T What is x2?A 0.2256 B 1.0437 C 0.1629 D 0.7987 E 0.0926
12 Compute the priori error of vector solution X2 in question 11, using ∞-norm.A 0.1089 B 0.2564 C 0.6517 D 0.1461 E 0.9317
13 With 2 iterations, one gets vector solution X2 = (x1, x2, x3)T Find x1.A 0.0434 B 0.7374 C 0.6690 D 0.8413 E 0.0871
14 Using the 1-norm, estimate the posteriori error of X2 found in question 13.A 0.3447 B 0.9862 C 0.0488 D 0.2310 E 0.4367
Questions (15)-(19) A function y = f (x) is interpolated by the natural cubic spline as follows:
S(x) =(
A Absolute error B Round-off error C Truncation error D Relative error E Algorithmicerror
THE END
Trang 3Midterm test - MT1009 2311 Semester I
Trang 6Lecturer(Date)Approved by(Date)
HCMUTFaculty of AS
Midterm test Semester/ Year 1 2022 - 2023
SubjectNumerical methodsSubject codeMT1009
Note: - This is an opened book exam, documents are allowed to used, EXCEPT laptop, smartphone.
Q 1.Given the matrix A=
Factor A=LU the Doolittle method Compute U11+ U22+ U33.
Q 2.Using the Newton method to find the intersection point of y = exand y = 3−1.2x on the interval [0; 1].Compute x2, knowing that x0is chosen by the Fourier condition.
D.All the other answers are wrong.E.0.8535.
Q 3.With the bisection method, perform 4 iterations(x3) to find approximately the common point between thegraphs y = ex and y = 3− 3.8x on the interval [0; 1].
D.All the other answers are wrong.E.0.4375.
Q 4.Given the matrix A=
Factor A=LU by the Doolittle method Compute U23.
E.All the other answers are wrong.
Q 5.Given the matrix A=
For which value of x below, A is positive definite?
Q 6.The dimensions of a tank whose shape is an inverted circular cone are measured and came up with thefollowing data: the height h = 8, the radius of base r = 6 The absolute errors of both h and r are δ Estimateδ so that the possible absolute error of volume of the tank less than 0.6429 if π = 3.14 ± 0.0016.
E.All the other answers are wrong.
Q 7.Given the system AX = B with A= 18.450.05−0.0714.12
x2+2.4 and the interval [2.3; 4] Taking x0=3.4, find the number of steps n(at least)such that |xn− xn−1| < 0.004.
Trang 7Q 10.Given the system AX = B with A= 33.440.03
−0.0322.4 , B= 1 3T
x2+2.8 By the fixed-point method, compute the approximate root x4, given thatthe previous root is x3=3.7.
Q 12.Given the equation x = 5
x2+3.3 and the interval [2.3; 4] Taking x0=3.4, estimate the error of x4by the fixedpoint method and the priori formula.
D.All the other answers are wrong.E.0.85523.Q 13.Given the system AX = B with A= 20.690.05
E.All the other answers are wrong.
Q 14.Given the matrix A=
Factor A = BBTwhere B is a lower triangular matrix by the Choleskymethod, find B32.
Q 15.Given the matrix A=0.98 1.632.585
Compute the condition number of A with 1-norm.
A.72.3513.B.72.3512.C.All the other answers are wrong.
Q 20.Given the equation x = 5
x2+2.5 and the interval [2.3; 4] Taking x0=2.8, using the priori formula, find thenumber of steps n(at least) such that ∆xn< 0.0027.
D.All the other answers are wrong.E.35.
THE END.
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Q 1 B.Q 2 C.Q 3 E.Q 4 A.
Q 5 D.Q 6 D.Q 7 D.Q 8 B.
Q 9 B.Q 10 D.Q 11 E.Q 12 C.
Q 13 D.Q 14 E.Q 15 D.Q 16 A.
Q 17 C.Q 18 D.Q 19 B.Q 20 C.
Trang 1/2- Đề 0
Trang 9Lecturer: Date Approved by: Date
HCM University of TechnologyFaculty of Applied Science
Midterm test Year 2022-2023 Semester 2
Date 7/3/2023Subject Numerical methodsSubject code MT1009
- The question sheet has 20 questions.
- Documents are allow to used, except laptop, tablet, mobile phone.- Round the answers to 4 decimal digits.
The centroid of an arc of a circle of radius r is located at ¯x = rsin α
α Determine α such that ¯x = r3
4 by solvingthe equation α = 4
3sin(α), knowing that α ∈ (1, 2) Answer the questions from 1 to 91 Use the fixed point method with 2 steps, choose α0 = 1.19.
A α2 ≈ 1.2645 B All other answers are wrong C α2 ≈ 1.3497 D α2≈ 1.2700 E α2 ≈ 1.26012 Taking α0 in question 1, estimate the absolute error of α2 with the fixed point method and the posteriori
3sin x, the sequence xn+1= g(xn), n ∈ N Which statement below is not correct?
A Start with some α0 ∈ (1, 2), we have |α2023 − α2022| < |α2020 − α2019| B g(x) is a contractivemapping C All others statements are correct D g(x) has unique fixed point on (1, 2) E Thesequence xn converges to some fixed point of g(x) on (1, 2) depending on the initial step α0.
5 Use the bisection method, calculate approximately α with 4 steps(α3).
A All other answers are wrong B α3 ≈ 1.3335 C α3 ≈ 1.3125 D α3≈ 1.4031 E α3 ≈ 1.36356 Let α3 be the value found in question 5, evaluate the absolute error of this value by using the general
A All other answers are wrong B n = 8 C n = 10 D n = 12 E n = 9
8 Use the Newton method with 2 number of iterations, choose α0 by the Fourier condition, we get theapproximate value α2 is :
A All other answers are wrong B α2 ≈ 1.4103 C α2 ≈ 1.3537 D α2≈ 1.3105 E α2 ≈ 1.36069 Estimate the error of α2 found in question 8.
A ∆x2≈ 0.0089 B ∆x2 ≈ 0.0049 C ∆x2≈ 0.0081 D All other answers are wrong E ∆x2 ≈0.0790
From question 10 to 11, using the information:Given the system AX = B, where A =
9 21 1521 98 70
, B = 0.94 0.12 0.73T.
Trang 10Midterm test - MT1009 2224 Semester II - 7/03/2023
10 Let C be the Cholesky matrix when we factor A = CCT, knowing that C33= 3 Find m.A m = 59 B m = 63 C All other answers are wrong D m = 61 E m = 5711 With m found in question 10, find S =
0.33 0.76 0.690.62 0.41 0.970.36 0.49 0.33
0.33 3.86 0.690.62 0.41 0.970.36 0.49 0.33
, B = 0.95 0.05T
Given that the initial vector isX0= 0.03 0.97T.
13 Evaluate the conditional number of matrix A, using the 1-norm.
A All other answers are wrong B k1 ≈ 3.9208 C k1 ≈ 3.2581 D k1 ≈ 3.6150 E k1≈ 3.5396From question 14 to 16: use the Jacobi method.
14 Carry out 2 iterations to get the approximate solution X2 = (x1, x2)T What is x2?
A x2 ≈ −0.1987 B x2 ≈ −0.4291 C x2 ≈ 0.2821 D x2 ≈ 0.1955 E All other answers arewrong.
15 Compute the priori error of vector solution X2 in question 14, using ∞-norm.
A ∆X2 ≈ 0.6604 B ∆X2 ≈ 0.0476 C ∆X2 ≈ 0.5715 D All other answers are wrong E ∆X2 ≈0.5906
16 How many iterations(at least) that we need to do to get the solution with accuracy within 10−6, using thepriori formula and ∞-norm?
A All other answers are wrong B n= 26 C n= 22 D n= 28 E n= 24From question 17 to 18, use the Gauss-Seidel method
17 With 2 iterations, one gets vector solution is X2 = (x1, x2)T Find x1.
A All other answers are wrong B x1 ≈ 1.9013 C x1 ≈ 1.6604 D x1≈ 2.5166 E x1 ≈ 2.375518 Use the 1-norm, estimate the posteriori error of X2 found in question 17.
A ∆X2 ≈ 0.7311 B ∆X2 ≈ 0.1378 C All other answers are wrong D ∆X2 ≈ 4.2360 E ∆X2 ≈0.2815
19 Let A, B be matrices size n Choose the correct statement.
A ∥A · B∥∞ = ∥A∥∞+ ∥B∥∞ B ∥A + B∥1 = ∥A∥1+ ∥B∥1 C ∥A∥∞ ≤ ∥A∥1 D ∥A · B∥1 ≤∥A∥1∥B∥1.
20 Suppose that we have a sequence of n approximate values a1< a2 < < an with the same absolute errorsδ = 0.083672 Estimate the absolute error of
nP
Trang 11Midterm test - MT1009 2224 Semester II - 7/03/2023
Trang 12Lecturer: Date Approved by: Date
HCM University of TechnologyFaculty of Applied Science
Midterm test Year 2022-2023 Semester 2
Date 07/03/2023Subject Numerical methodsSubject code MT1009
- The question sheet has 20 questions.
- Documents are allow to used, except laptop, tablet, mobile phone.- Round the answers to 4 decimal digits.
Let ¯x be the exact intersection point between the graphs y = x and y = cos√x + 1 Knowing that ¯x ∈ (0, 1).Answer the questions from 1 to 9
1 Use the fixed point method to approximate ¯x with 2 steps, choose x0 = 0.51.
A x2 ≈ 0.4034 B x2 ≈ 0.5294 C x2 ≈ 0.0606 D x2 ≈ 0.3178 E All other answers arewrong.
2 Taking x0 in question 1, estimate the absolute error of x2 with the fixed point method and the prioriformula.
A ∆x2≈ 0.3178 B ∆x2 ≈ 0.0534 C ∆x2≈ 0.7522 D All other answers are wrong E ∆x2 ≈0.5294
3 With the same x0 in question 1, how many iterations are necessary to get the accuracy within 10−3?(Usethe prior formula).
A All other answers are wrong B n = 10 C n = 7 D n = 17 E n = 6
4 Let g(x) = cos√x + 1, the sequence xn+1= g(xn), n ∈ N Which statement below is not correct?A The sequence xn converges to some fixed point on (0,1) of g(x) depending on the initial step x0.B g(x) is a contractive mapping C Start with some x0 ∈ (0, 1), we have |x2023−x2022| < |x2020−x2019|.D g(x) has unique fixed point on (0, 1) E All others statements are correct.
5 Use the bisection method, calculate an approximate value of ¯x with 4 steps(x3).A 0.2915 B 0.7104 C 0.3119 D 0.4375 E All other answers are wrong.
6 Let x3 be the value found in question 5, evaluate the absolute error of this value by using the generalformula.
A ∆x3≈ 0.0639 B All other answers are wrong C ∆x3 ≈ 0.0912 D ∆x3 ≈ 0.0255 E ∆x3 ≈0.0550
7 With the bisection method, find the least number of steps n such that the approximate value xn of ¯x hasthe accuracy within 10−3.
A n = 8 B n = 12 C n = 10 D n = 9 E All other answers are wrong.
8 Use the Newton method, choose x0 by the Fourier condition Compute an approximate value of ¯x with 2number of iterations.
A x2 ≈ 0.5847 B x2 ≈ 0.9481 C x2 ≈ 0.3842 D x2 ≈ 0.8383 E All other answers arewrong.
9 Estimate the error of x2 found in question 8.
A ∆x2≈ 0.0058 B All other answers are wrong C ∆x2 ≈ 0.0083 D ∆x2 ≈ 0.0001 E ∆x2 ≈0.0029
From question 10 to 11, using the information:Given the system AX = B, where A =
16 12 2012 45 27
, B = 0.13 0.67 0.57T.
Trang 13Midterm test - MT1009 2222 Semester II - 7/03/2023
10 Let C be the lower triangular matrix when we factor A = CCT by the Cholesky method, knowing thatC33= 3 Find m.
A m = 38 B m = 36 C m = 42 D All other answers are wrong E m = 40
11 With m found in question 10, find vector y such that Ly = B, where L is the matrix in the decompositionA = LU by Dolittle’s method.
A y = (0.2777, 0.7202, 0.3643)T B All other answers are wrong C y = (0.2998, 0.7423, 0.3864)TD y = (0.2777, 0.7201, 0.3643)T E y = (0.1300, 0.5725, 0.2167)T
12 Which matrix below is strictly diagonally dominant?A
3.46 0.91 0.550.03 −2.09 0.050.81 0.45 4.26
0.38 3.63 0.330.79 0.71 0.650.36 0.87 0.97
, B = 0.41 0.59T Given that the initial vector isX0= 0.31 0.69T.
13 Evaluate the conditional number of matrix A, using the 1-norm.
A All other answers are wrong B k1 ≈ 2.6795 C k1 ≈ 2.9159 D k1 ≈ 2.1845 E k1≈ 1.9207From question 14 to 16: use the Jacobi method.
14 Carry out 2 iterations to get the approximate solution X2 = (x1, x2)T What is x2?A All other answers are wrong B 1.4546 C 0.8600 D 0.6735 E 0.869315 Compute the priori error of vector solution X2 in question 14, using ∞-norm.
A ∆X2 ≈ 0.8023 B ∆X2 ≈ 0.0214 C ∆X2 ≈ 0.4242 D All other answers are wrong E ∆X2 ≈0.9924
16 How many iterations(at least) that we need to do to get the solution with accuracy within 10−6, using thepriori formula and ∞-norm?
A All other answers are wrong B n= 18 C n= 16 D n= 14 E n= 12From question 17 to 18, use the Gauss-Seidel method.
17 With 2 iterations, one gets vector solution is X2 = (x1, x2)T Find x1.
A x1 ≈ 0.7436 B All other answers are wrong C x1 ≈ 0.3871 D x1≈ 0.4603 E x1 ≈ 1.196118 Use the 1-norm, estimate the posteriori error of X2 found in question 17.
A ∆X2 ≈ 0.5910 B ∆X2 ≈ 0.0003 C All other answers are wrong D ∆X2 ≈ 0.1938 E ∆X2 ≈0.9102
19 Let A, B be matrices size n Choose the correct statement.
A ∥A · B∥1 ≤ ∥A∥1∥B∥1 B ∥A∥∞ ≤ ∥A∥1 C ∥A · B∥∞ = ∥A∥∞+ ∥B∥∞ D ∥A + B∥1 =∥A∥1+ ∥B∥1.
20 Suppose that we have a sequence of n approximate values a1< a2 < < an with the same absolute errorsδ = 0.043237 Estimate the absolute error of
nP
Trang 14Midterm test - MT1009 2222 Semester II - 7/03/2023