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 Technology
Faculty of Applied Science
Midterm test YearDate 2023-202417/10/2023 Semester 1 Subject Numerical methods
Subject code
MT1009
Note
- 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 the questions (1)-(8)
1 By the fixed point method, which sequence below converges to A?
A none of them B xn =21xn−1
x n−1 +1
13
C xn =21xn−1
x n−1 +1
12
D xn =x 21
n−1 +1
12
E xn =
21
x n−1
12
2 Starting at x0 = 2.0, compute an approximate value of A with 3 iterations
A 2.8754 B 2.8721 C 2.8755 D 2.8772 E 2.8817
3 Estimate the postepriori error of the value obtained in question (2)
A 1.4038 B 1.4050 C 1.3923 D 1.3990 E 1.4006
4 Use the bisection method to approximate A with 4 iterations
A 2.1016 B 2.7813 C 2.1088 D 2.8125 E 2.9595
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 xn
of 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 with
2 number of iterations
A 2.4104 B 2.8531 C 2.7591 D 2.1982 E 2.4291
8 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 =
0.5 10 0
be the matrix Cholesky
in the factorization A = CCT of a matrix A
9 Find the element U33 of the matrix U in the factorization A = LU
10 Evaluate the conditional number of matrix A, using the 1-norm
A 3.3787 B 3.3659 C 3.1937 D 3.5221 E 3.1381
Consider the system AX = B, with B = 2 4 3T and the initial vector X0 = 4 2 2T Answer questions (11)-(12) by using the Jacobi method and questions (13) to (14) by the Gauss-Seidel method
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) =
(
4 + b(x − 1) + c(x − 1)2+ d(x − 1)3 if 1 ≤ x ≤ 2
15 Find b
A 1.0000 B 11.0000 C 5.0000 D 4.0000 E 0.0000
16 Given that f′(0) = 0, f′(2) = 0, using the clamped cubic spline to approximate f (0.1)
A 0.7365 B 2.0110 C 0.4195 D 1.0705 E 1.1630
17 By using polynomial interpolation at 3 nodes 0, 1, 2, approximate f (0.1)
A 0.7111 B 2.0109 C 1.1009 D 1.9839 E 1.4800
18 Interpolating f (x) with the Newton formula, we get f (x) ≈ a0+ a1x + a2x(x − 1), what is a2?
A −2.0000 B −3.0000 C 1.0000 D 2.0000 E 8.0000
19 Using the least square method and the model y = a + bx + cx2, find c
A 2.0000 B 8.0000 C −2.0000 D 1.0000 E −3.0000
20 Given the description of a type of error: This error is a result of the finite precision of numerical representation in computers It occurs when numbers are rounded to fit within the available bits What is this error?
A Absolute error B Round-off error C Truncation error D Relative error E Algorithmic error
THE END
Trang 3Midterm test - MT1009 2311 Semester I
Key answers
Question Key
Trang 6Lecturer (Date) Approved by (Date)
HCMUT
Faculty of AS
Midterm test Semester/ Year 1 2022 - 2023
Subject Numerical methods Subject code MT1009
Note: - This is an opened book exam, documents are allowed to used, EXCEPT laptop, smartphone.
Q 1 Given the matrix A=
−9 18.7 −11.5 1.8 46.66 −13.22
Factor A=LU the Doolittle method Compute U 11 + U 22 + U 33
Q 2 Using the Newton method to find the intersection point of y = e x and y = 3−1.2x on the interval [0; 1] Compute x 2 , knowing that x 0 is chosen by the Fourier condition.
D All the other answers are wrong E 0.8535.
Q 3 With the bisection method, perform 4 iterations(x 3 ) to find approximately the common point between the graphs 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=
−5 −5.7 7.1
−4.5 0.87 11.69
−19 7.74 46.95
Factor A=LU by the Doolittle method Compute U 23
E All the other answers are wrong.
Q 5 Given the matrix A=
−8 22.3 5
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 the following data: the height h = 8, the radius of base r = 6 The absolute errors of both h and r are δ Estimate
δ 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= 18.45 0.05
−0.07 14.12
, B= 4 3 T
, and X (0) = 2 1 T
Using the Gauss-Seidel method, estimate the error ∆X(2) with the posteriori formula and 1-norm.
A All the other answers are wrong B ∆X(2) ≈0.1058 C ∆X(2) ≈0.2933.
D ∆X(2) ≈0.0001 E ∆X(2) ≈0.7305.
Q 8 Given the equation x = 5
x 2 +2.4 and the interval [2.3; 4] Taking x 0 =3.4, find the number of steps n(at least) such that |x n − x n−1 | < 0.004.
Q 9 Let xn be the approximate intersection point of y = e x and y = 3-3.8x on the interval [0; 1] by the Newton method Find the least number of neccessary steps to get the solution whose error is less than 0.0002.
D All the other answers are wrong E 5.
Trang 1/2- Đề 0
Trang 7Q 10 Given the system AX = B with A= 33.44 0.03
−0.03 22.4 , B= 1 3
T
, and X (0) = 4 1 T
Using the Jacobi method, compute the approximate solution X (2)
A 0.5309 0.4085 T
B 0.3322 0.6666 T
C All the other answers are wrong.
D 0.0298 0.134 T
E 0.5892 0.3791 T
.
Q 11 Given the equation x = 5
x 2 +2.8 By the fixed-point method, compute the approximate root x 4 , given that the previous root is x 3 =3.7.
Q 12 Given the equation x = 5
x 2 +3.3 and the interval [2.3; 4] Taking x 0 =3.4, estimate the error of x 4 by the fixed point method and the priori formula.
D All the other answers are wrong E 0.85523.
Q 13 Given the system AX = B with A= 20.69 0.05
−0.03 36.56
, B= 3 4 T
, and X (0) = 1 4 T
Using the Jacobi method, estimate the error ∆X(5) with the priori formula and ∞-norm.
A ∆X(5) ≈0.9131 B ∆X(5) ≈0.9511 C ∆X(5) ≈0.2569 D ∆X(5) ≈0.0001.
E All the other answers are wrong.
Q 14 Given the matrix A=
4 20 18
10 18 45
Factor A = BB T where B is a lower triangular matrix by the Cholesky method, find B 32
Q 15 Given the matrix A= 0.98 1.63
2.58 5
Compute the condition number of A with 1-norm.
A 72.3513 B 72.3512 C All the other answers are wrong.
Q 16 Given the system AX = B with A= 21.87 0.05
−0.05 31.45
, B= 1 1 T
, and X(0)= 4 1 T
Using the Gauss-Seidel’s method, compute the approximate solution X (2)
A 0.0457 0.0319 T
B 0.4377 0.9753 T
C 0.754 0.2494 T
.
D All the other answers are wrong E 0.9732 0.8171 T
.
Q 17 Let x n ∈ (0; 1) be the approximate intersection point between f (x) = e x and g(x) = 3− 3.1x by the bisection method Compute n(at least) so that ∆xn<0.0018.
Q 18 Given the system AX = B with A= 29.83 1.32
−1.2 25.76
, B= 3 3 T
, and X (0) = 1 4 T
Using the Jacobi’s method, the priori error and ∞−norm, find n(at least) such that ∆X(n) < 10−6.
Q 19 Using the Newton method to find the intersection point of y = e x and y = 3−2.5x on the interval [0; 1] Estimate the error of x 2
Q 20 Given the equation x = 5
x 2 +2.5 and the interval [2.3; 4] Taking x 0 =2.8, using the priori formula, find the number of steps n(at least) such that ∆ xn< 0.0027.
D All the other answers are wrong E 35.
THE END.
Trang 2/2- Đề 0
Trang 8Đề 0 ĐÁP ÁN
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 Technology
Faculty of Applied Science
Midterm test Year 2022-2023 Semester 2
Date 7/3/2023 Subject Numerical methods Subject code MT1009
Note
- 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 = r
3
4 by solving the equation α = 4
3sin(α), knowing that α ∈ (1, 2) Answer the questions from 1 to 9
1 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 ≈ 1.2601
2 Taking α0 in question 1, estimate the absolute error of α2 with the fixed point method and the posteriori formula
A ∆α2≈ 0.0557 B All other answers are wrong C ∆α2≈ 0.0772 D ∆α2 ≈ 0.0312 E ∆α2 ≈ 0.0574
3 With the same α0 in question 1, how many iterations are necessary to get the accuracy within 10−3?(Use the prior formula)
A n = 16 B n = 14 C n = 17 D n = 19 E All other answers are wrong
4 Let g(x) = 4
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 contractive mapping C All others statements are correct D g(x) has unique fixed point on (1, 2) E The sequence 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 ≈ 1.3635
6 Let α3 be the value found in question 5, evaluate the absolute error of this value by using the general formula
A ∆α3 ≈ 0.0629 B All other answers are wrong C ∆α3 ≈ 0.0391 D ∆α3 ≈ 0.0837
E ∆α3 ≈ 0.0102
7 With the bisection method, find the least number of steps n such that the approximate value αn has the accuracy within 10−3
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 the approximate value α2 is :
A All other answers are wrong B α2 ≈ 1.4103 C α2 ≈ 1.3537 D α2≈ 1.3105 E α2 ≈ 1.3606
9 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 15
21 98 70
, B = 0.94 0.12 0.73T
Trang 10
Midterm 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 = 57
11 With m found in question 10, find S =
3 P
i=1
Uii, where U is the upper triangular matrix in the decomposition
A = LU by Dolittle’s method
A All other answers are wrong B S= 67.0000 C S= 67.3983 D S= 67.8332 E S= 67.6465
12 Which matrix below is strictly diagonally dominant?
A
0.33 0.76 0.69
0.62 0.41 0.97
0.36 0.49 0.33
0.33 0.76 −4.31 0.62 0.41 0.97 0.36 0.49 0.33
3.59 0.75 0.84 0.32 −2.87 0.55 0.98 0.55 4.53
E
0.33 3.86 0.69
0.62 0.41 0.97
0.36 0.49 0.33
From question 13 to 18, using the information:
Given the system AX = B, where : A = 0.66 0.34
0.35 0.65
, B = 0.95 0.05T
Given that the initial vector is
X0= 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≈ 3.5396 From 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 are wrong
15 Compute the priori error of vector solution X2 in question 14, using ∞-norm
A ∆X 2 ≈ 0.6604 B ∆X 2 ≈ 0.0476 C ∆X 2 ≈ 0.5715 D All other answers are wrong E ∆X 2 ≈ 0.5906
16 How many iterations(at least) that we need to do to get the solution with accuracy within 10−6, using the priori formula and ∞-norm?
A All other answers are wrong B n= 26 C n= 22 D n= 28 E n= 24
From 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 ≈ 2.3755
18 Use the 1-norm, estimate the posteriori error of X2 found in question 17
A ∆X 2 ≈ 0.7311 B ∆X 2 ≈ 0.1378 C All other answers are wrong D ∆X 2 ≈ 4.2360 E ∆X 2 ≈ 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 errors
δ = 0.083672 Estimate the absolute error of
n P
i=1
ai when n = 10
A All other answers are wrong B Abs.error ≈ 0.1386 C Abs.error ≈ 0.5882 D Abs.error ≈ 0.8368 E Abs.error ≈ 0.3662
THE END
Trang 11Midterm test - MT1009 2224 Semester II - 7/03/2023
Answer key
Question Key
Trang 12Lecturer: Date Approved by: Date
HCM University of Technology
Faculty of Applied Science
Midterm test Year 2022-2023 Semester 2
Date 07/03/2023 Subject Numerical methods Subject code MT1009
Note
- 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 are wrong
2 Taking x0 in question 1, estimate the absolute error of x2 with the fixed point method and the priori formula
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?(Use the 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 general formula
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 has the 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 2 number of iterations
A x2 ≈ 0.5847 B x2 ≈ 0.9481 C x2 ≈ 0.3842 D x2 ≈ 0.8383 E All other answers are wrong
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 20
12 45 27
, B = 0.13 0.67 0.57T