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Test 1 Mathematics 2 Gtu

The document contains three tests (IT-A, CE-A, CE-B) with various mathematical problems. Topics include finding matrix inverses using the Gauss-Jordan method, solving systems of equations with Gauss elimination, and determining eigenvalues and eigenvectors. Additional problems involve differential equations and the application of the Cayley-Hamilton theorem.

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hetjoshi3812
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34 views2 pages

Test 1 Mathematics 2 Gtu

The document contains three tests (IT-A, CE-A, CE-B) with various mathematical problems. Topics include finding matrix inverses using the Gauss-Jordan method, solving systems of equations with Gauss elimination, and determining eigenvalues and eigenvectors. Additional problems involve differential equations and the application of the Cayley-Hamilton theorem.

Uploaded by

hetjoshi3812
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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TEST 1 (BE02000011)

IT - A
1 2 3
1 Find the inverse of a matrix 𝐴 = [2 5 3] using Gauss-Jordan method.
1 0 8
2 Solve the system of equations using Gauss elimination method:
𝑥 + 𝑦 + 2𝑧 = 8 , −𝑥 − 2𝑦 + 3𝑧 = 1 , 3𝑥 − 7𝑦 + 4𝑧 = 10
3 0 1 0
Find Eigen values and Eigen vectors of the matrix 𝐴 = [0 0 1]
1 −3 3
4 1 1 2
Use Cayley-Hamilton theorem to find 𝐴−1 given that 𝐴 = [2 2 4]
3 4 5
5 2 2 2
Solve: (𝑥 − 2𝑥𝑦 − 𝑦 )𝑑𝑥 − (𝑥 + 𝑦) 𝑑𝑦 = 0
6 𝑑𝑦
Solve: + 𝑦 tan(𝑥) = 4 sin2 (2𝑥)
𝑑𝑥

7 Solve: 𝑥 2 𝑦 ′′ − 2𝑥𝑦 ′ + 𝑦 = sin(log 𝑥)


8 Solve: 𝑦 ′′ + 𝑦 = cosec⁡(𝑥) using the method of variation of parameters.

TEST 1 (BE02000011)
CE - A
1 2 3
1 Find the inverse of a matrix 𝐴 = [0 1 4] using Gauss-Jordan method.
5 6 0
2 Solve the system of equations using Gauss elimination method:
𝑥 + 𝑦 + 𝑧 = 6 , 𝑥 + 2𝑦 + 3𝑧 = 14 , 2𝑥 + 4𝑦 + 7𝑧 = 30
3 4 0 1
Find Eigen values and Eigen vectors of the matrix 𝐴 = [−2 1 0]
−2 0 1
4 1 3 3
Use Cayley-Hamilton theorem to find 𝐴−1 given that 𝐴 = [1 4 3]
1 3 4
5 2 2 2
Solve: (𝑥 − 2𝑥𝑦 − 𝑦 )𝑑𝑥 − (𝑥 + 𝑦) 𝑑𝑦 = 0
6 𝑑𝑦
Solve: 𝑑𝑥 + 𝑦 cot(𝑥) = cos(𝑥)

7 Solve: 𝑥 2 𝑦 ′′ − 7𝑥𝑦 ′ + 12𝑦 = 𝑥 2


8 Solve: 𝑦 ′′ + 25𝑦 = sec⁡(5𝑥) using the method of variation of parameters.
TEST 1 (BE02000011)
CE - B
2 −1 2
1 Use Cayley-Hamilton theorem to find 𝐴−1 given that 𝐴 = [−1 2 −1]
1 −1 2
2 ′′
Solve: 𝑦 + 49𝑦 = cot⁡(7𝑥) using the method of variation of parameters.
3 1 3 3
Find the inverse of a matrix 𝐴 = [1 4 3] using Gauss-Jordan method.
1 3 4
4 Solve: (𝑒 𝑦 + 1) cos(𝑥) 𝑑𝑥 + 𝑒 𝑦 sin(𝑥) 𝑑𝑦 = 0
5 0 0 −2
Find Eigen values and Eigen vectors of the matrix 𝐴 = [1 2 1 ]
1 0 3
6 𝑑𝑦
Solve: 𝑑𝑥 − 𝑦 cot(𝑥) = 2𝑥 sin(𝑥)
7 Solve the system of equations using Gauss elimination method:
𝑥 + 𝑦 + 2𝑧 = 9, 3𝑥 + 6𝑦 − 5𝑧 = 0 , 2𝑥 + 4𝑦 − 3𝑧 = 1
8 Solve: 𝑥 2 𝑦 ′′ − 𝑥𝑦 ′ + 𝑦 = 𝑥 2

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