Probability Engine · MTH163

Mathematics II (BSc CSIT, MTH163): the questions likely to come

84 analyzed questions from 7 past papers (2074-2081), grouped by syllabus unit — each with its probability, how often it's been asked, and where to study the answer.

7
Papers analyzed
2074-2081
84
Analyzed questions
across 7 syllabus units
5
Very likely units
high-probability topics
5
Units = 80% of marks
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U5 · Q1/22 · 208110 marks
Eigenvalues and Eigenvectors

Find the eigenvalues and eigenvectors of the given matrix and hence diagonalize it. Use the diagonal form to compute (A^4).

20%
Occasional to appearAppeared in 1 of the last 1 board papers
Seen in
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MODEL ANSWERU5 · 10 marks

Eigenvalues, Eigenvectors, Diagonalization and A4A^4

Since the paper does not print a specific matrix, take the standard representative

A=[2112].A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}.

The method below applies to any diagonalizable matrix.

Step 1: Characteristic equation

det⁡(A−λI)=∣2−λ112−λ∣=(2−λ)2−1=λ2−4λ+3=0.\det(A - \lambda I) = \begin{vmatrix} 2-\lambda & 1 \\ 1 & 2-\lambda \end{vmatrix} = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = 0.

So (λ−1)(λ−3)=0(\lambda - 1)(\lambda - 3) = 0, giving eigenvalues λ1=1, λ2=3\lambda_1 = 1,\ \lambda_2 = 3.

Step 2: Eigenvectors

For λ1=1\lambda_1 = 1: (A−I)X=0⇒[1111]X=0⇒x1+x2=0(A - I)X = 0 \Rightarrow \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}X = 0 \Rightarrow x_1 + x_2 = 0, eigenvector v1=[1−1]v_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}.

For λ2=3\lambda_2 = 3: (A−3I)X=0⇒[−111−1]X=0⇒x1=x2(A - 3I)X = 0 \Rightarrow \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}X = 0 \Rightarrow x_1 = x_2, eigenvector v2=[11]v_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}.

Step 3: Diagonalization

Let

P=[11−11],D=[1003],P−1=12[1−111].P = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}, \qquad D = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}, \qquad P^{-1} = \tfrac{1}{2}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}.

Then A=PDP−1A = PDP^{-1} (equivalently P−1AP=DP^{-1}AP = D).

Step 4: Compute A4A^4

Because A4=PD4P−1A^4 = P D^4 P^{-1} and D4=diag⁡(14,34)=diag⁡(1,81)D^4 = \operatorname{diag}(1^4, 3^4) = \operatorname{diag}(1, 81),

A4=[11−11][10081]⋅12[1−111]=12[181−181][1−111].A^4 = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 81 \end{bmatrix} \cdot \tfrac{1}{2}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} = \tfrac{1}{2}\begin{bmatrix} 1 & 81 \\ -1 & 81 \end{bmatrix}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}. A4=12[82808082]=[41404041].A^4 = \tfrac{1}{2}\begin{bmatrix} 82 & 80 \\ 80 & 82 \end{bmatrix} = \begin{bmatrix} 41 & 40 \\ 40 & 41 \end{bmatrix}.

Result: eigenvalues 1,31, 3; eigenvectors (1,−1)T, (1,1)T(1,-1)^T,\ (1,1)^T; A=PDP−1A = PDP^{-1}; and A4=[41404041]A^4 = \begin{bmatrix} 41 & 40 \\ 40 & 41 \end{bmatrix}.

AI-generated answer · unverifiedView in 2081 paper →
U5 · Question 1 of 22
Question Priority · U5ranked by appearance likelihood — study top-down

Eigenvalues and Eigenvectors

Analyzed next20%
1
★ TOP PICK

Find the eigenvalues and eigenvectors of the given matrix and hence diagonalize it. Use the diagonal form to compute (A^4).

10 marksSEEN IN
20%
2

Reduce the given quadratic form to canonical form by an orthogonal transformation and determine its nature (positive definite, negative definite, etc.).

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16%
3

Find the eigenvalues and eigenvectors of the given matrix and verify the Cayley-Hamilton theorem for it.

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14%
4

Diagonalize the given matrix A by finding a matrix P such that (P^{-1}AP) is diagonal. State the conditions under which a matrix is diagonalizable.

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13%
5

What is the algebraic multiplicity of an eigenvalue?

5 marksSEEN IN
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6

What is the canonical form of a quadratic form?

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7

Find the sum and product of eigenvalues of [[1,2],[3,4]] using trace and determinant.

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8

State and prove the Cayley-Hamilton theorem. Use it to find the inverse of a given 3x3 matrix.

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Define eigenvalues and eigenvectors. Find the eigenvalues and corresponding eigenvectors of a given 3x3 matrix.

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10

State the spectral theorem for symmetric matrices.

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11

Find the eigenvectors of [[3,0],[0,3]].

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12

State two properties of eigenvalues.

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13

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14

Define a quadratic form with an example.

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Find the eigenvalues of the identity matrix of order 3.

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16

What is the geometric multiplicity of an eigenvalue?

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17

Define a positive definite matrix.

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18

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19

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03The mock

Sit a probable paper

A full mock exam built from the most likely questions, mirroring the real paper's structure. Every slot is a real past question.

Most Probable Paper

Mirrors the real structure · 60 marks · based on 7 past papers

Section A: Long Answer QuestionsAttempt any TWO questions.
  1. 1.

    Find the eigenvalues and eigenvectors of the given matrix and hence diagonalize it. Use the diagonal form to compute (A^4).

    [10 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2081 paper →

    Asked once (2081); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  2. 2.

    Reduce the given quadratic form to canonical form by an orthogonal transformation and determine its nature (positive definite, negative definite, etc.).

    [10 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2079 paper →

    Asked once (2079); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  3. 3.

    Find the eigenvalues and eigenvectors of the given matrix and verify the Cayley-Hamilton theorem for it.

    [10 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2078 paper →

    Asked once (2078); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

Section B: Short Answer QuestionsAttempt any EIGHT questions.
  1. 1.

    What is the algebraic multiplicity of an eigenvalue?

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2081 paper →

    Asked once (2081); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  2. 2.

    What is the canonical form of a quadratic form?

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2081 paper →

    Asked once (2081); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  3. 3.

    Find the sum and product of eigenvalues of [[1,2],[3,4]] using trace and determinant.

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2081 paper →

    Asked once (2081); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  4. 4.

    State the spectral theorem for symmetric matrices.

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2080 paper →

    Asked once (2080); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  5. 5.

    Find the eigenvectors of [[3,0],[0,3]].

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2080 paper →

    Asked once (2080); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  6. 6.

    State two properties of eigenvalues.

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2080 paper →

    Asked once (2080); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  7. 7.

    What is a diagonalizable matrix?

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2080 paper →

    Asked once (2080); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  8. 8.

    Define a quadratic form with an example.

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2079 paper →

    Asked once (2079); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

  9. 9.

    Find the eigenvalues of the identity matrix of order 3.

    [5 marks]
    Eigenvalues and EigenvectorsVery likelyfrom 2079 paper →

    Asked once (2079); so far only in internal assessments, not the board; and its topic (Eigenvalues and Eigenvectors) appears in 100% of years.

04The receipts

Behind the numbers

The raw evidence the predictions are computed from: marks per unit per year, syllabus weights, trends, and coverage.

Show the heatmap, topic table and coverage analysis

The receipt: marks per unit, per year

Each row is a syllabus unit, each column an exam year, each cell the marks that unit earned that year. Click any cell to see the actual questions behind it.

Marks:nonefew → many
2074
2075
2077
2078
2079
2080
2081
Total
U5Eigenvalues and Eigenvectors
140
U4General Vector Spaces
125
U1Matrices and Determinants
80
U7Inner Product Spaces and Orthogonality
60
U6Linear Transformations
55
U2Solving Linear Systems
60
U3Euclidean Vector Spaces
5
#Syllabus unitProbabilityAppearedAvg marksSyllabus weightExam vs syllabusTrendQuestions
1U5Eigenvalues and EigenvectorsVery likely100%
2074207520772078207920802081
2013%6 lecture hrsOver-examinedexam 27% · syllabus 13%Steadynone repeat22 total
2U4General Vector SpacesVery likely100%
2074207520772078207920802081
17.922%10 lecture hrsBalancedexam 24% · syllabus 22%Steadynone repeat21 total
3U1Matrices and DeterminantsVery likely100%
2074207520772078207920802081
11.411%5 lecture hrsBalancedexam 15% · syllabus 11%Steadynone repeat15 total
4U7Inner Product Spaces and OrthogonalityVery likely86%
207520772078207920802081
109%4 lecture hrsBalancedexam 11% · syllabus 9%Steadynone repeat9 total
5U6Linear TransformationsVery likely86%
207520772078207920802081
9.213%6 lecture hrsBalancedexam 10% · syllabus 13%Steadynone repeat8 total
6U2Solving Linear SystemsLikely71%
20742077207820792081
1218%8 lecture hrsUnder-examinedexam 11% · syllabus 18%Steadynone repeat8 total
7U3Euclidean Vector SpacesOccasional14%
2078
513%6 lecture hrsUnder-examinedexam 1% · syllabus 13%Steadynone repeat1 total

Study smart, not hard

Drag the slider: studying the top 5 units in priority order covers ~88% of all observed marks.

  5. ~80% line

Lecture time vs exam marks

Where the exam pays more than the curriculum spends: ● lectures vs ● exam marks, as a share of the whole course. A long teal-leading bar = high-yield unit.

U5Eigenvalues and Eigenvectors
13% of lectures → 27% of markshigh yield
U4General Vector Spaces
22% of lectures → 24% of marks
U1Matrices and Determinants
11% of lectures → 15% of marks
U7Inner Product Spaces and Orthogonality
9% of lectures → 11% of marks
U6Linear Transformations
13% of lectures → 10% of marks
U2Solving Linear Systems
18% of lectures → 11% of markslow yield
U3Euclidean Vector Spaces
13% of lectures → 1% of markslow yield

Topics are the official MTH163 syllabus units. Predictions are data-driven probabilities computed from 7 past papers (2074-2081) by mapping each real question to its syllabus unit. They indicate what has historically been likely, not guaranteed questions. Always study the full syllabus.