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Mathematics for Computer Engineering
Previous Year Question Paper with Solution
Engineering · Sem 3
Dec 2025 Examination
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Engineering Sem 3 Mathematics for Computer Engineering Dec 2025 Question Paper with Solutions
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First published on munotes.in on 5 November 2026.
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The question paper reproduced here is the paper as set by the University of Mumbai at the Dec 2025 examination.
Contents
The questions below are the paper as the University of Mumbai set it at the Dec 2025 examination, in the order it was set.
MarksPage
MarksPage
MarksPage
The Paper as Set
The questions in this volume are the questions asked at the Dec 2025 examination, reproduced as the University of Mumbai set them, in the order it set them. Nothing has been reworded, added or left out. Only the answers are ours. See the original question paper.
Duration 2 hours · Total marks 60 · 21 questions answered
How to use this volume
Solve the paper first, under exam conditions and against the clock. Then read the answers here and mark your own. Reading a solution before attempting the question feels productive and teaches very little, because recognising an answer is not the same as being able to write one.
Que. 1
Que. 1
Attempt any Five questions of the following · 15 Marks
a.If A is the matrix with rows (3, 2, 1), (0, -1, 4), (0, 0, -2), find the Eigen values of A⁻¹ + 6I.[3]
Answer
For full marks, cover: the eigenvalues of A read off the triangular matrix, the property that A⁻¹ has eigenvalues 1/λ, the property that adding kI adds k to each eigenvalue, and the three results.
Eigenvalues of A
A = [3 2 1 ; 0 -1 4 ; 0 0 -2]. It is an upper triangular matrix. The eigenvalues of a triangular matrix are its diagonal elements. So the eigenvalues of A are 3, -1 and -2.
Check: the characteristic equation is (3 - λ)(-1 - λ)(-2 - λ) = 0. The product of the eigenvalues, 3 × (-1) × (-2) = 6, equals |A| = 6. So A is non-singular and A⁻¹ exists.
Two properties used
- If λ is an eigenvalue of a non-singular matrix A, then 1/λ is an eigenvalue of A⁻¹. This follows from AX = λX, which gives A⁻¹X = (1/λ)X.
- If μ is an eigenvalue of a matrix B, then μ + k is an eigenvalue of B + kI. This follows from (B + kI)X = μX + kX = (μ + k)X.
Applying them
The eigenvalues of A⁻¹ are 1/3, 1/(-1) = -1 and 1/(-2) = -1/2.
The eigenvalues of A⁻¹ + 6I are therefore:
| Eigenvalue of A | Of A⁻¹ | Of A⁻¹ + 6I |
|---|---|---|
| 3 | 1/3 | 1/3 + 6 = 19/3 |
| -1 | -1 | -1 + 6 = 5 |
| -2 | -1/2 | -1/2 + 6 = 11/2 |
Check by direct work: A⁻¹ = [1/3 2/3 3/2 ; 0 -1 -2 ; 0 0 -1/2]. So A⁻¹ + 6I = [19/3 2/3 3/2 ; 0 5 -2 ; 0 0 11/2]. It is again triangular, with diagonal 19/3, 5 and 11/2.
Conclusion. The eigenvalues of A are 3, -1 and -2. Those of A⁻¹ are their reciprocals, and adding 6I adds 6 to each. So the eigenvalues of A⁻¹ + 6I are 19/3, 5 and 11/2.
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