Probability Engine · CSC207

Numerical Method (BSc CSIT, CSC207): the questions likely to come

20 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
20
Analyzed questions
across 6 syllabus units
4
Very likely units
high-probability topics
4
Units = 80% of marks
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Model answers for this subject are being written. Every question links to its original paper so you can study from the source meanwhile.
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U2 · Q1/5 · 208110 marks
Interpolation and Approximation

What is curve fitting? Explain the least squares method to fit a straight line y = a + bx to a given set of data points.

52%
Possible to appearAppeared in 3 of the last 3 board papers
Seen in
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MODEL ANSWERU2 · 10 marks

Curve Fitting

Curve fitting is the process of constructing a smooth curve (a mathematical function) that best represents the trend of a given set of discrete data points (xi,yi)(x_i,y_i). Unlike interpolation, the fitted curve need not pass through every point; it captures the overall relationship and is used for smoothing data, prediction, and approximating an underlying law. The most widely used technique is the method of least squares.

Least Squares Method for a Straight Line y=a+bxy = a + bx

Given nn data points (xi,yi)(x_i,y_i), we seek constants aa and bb so the line y=a+bxy=a+bx fits best. The error (residual) at each point is ei=yi−(a+bxi)e_i=y_i-(a+bx_i). The least squares principle minimizes the sum of squared residuals:

S(a,b)=∑i=1n(yi−a−bxi)2S(a,b)=\sum_{i=1}^{n}\bigl(y_i-a-bx_i\bigr)^2

For a minimum, set the partial derivatives to zero:

∂S∂a=−2∑(yi−a−bxi)=0\frac{\partial S}{\partial a}= -2\sum (y_i-a-bx_i)=0 ∂S∂b=−2∑xi(yi−a−bxi)=0\frac{\partial S}{\partial b}= -2\sum x_i(y_i-a-bx_i)=0

These give the normal equations:

∑yi=n a+b∑xi\sum y_i = n\,a + b\sum x_i ∑xiyi=a∑xi+b∑xi2\sum x_i y_i = a\sum x_i + b\sum x_i^2

Solving the two simultaneous equations gives:

b=n∑xiyi−∑xi∑yin∑xi2−(∑xi)2,a=∑yi−b∑xin=yˉ−bxˉb=\frac{n\sum x_i y_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2},\qquad a=\frac{\sum y_i - b\sum x_i}{n}=\bar{y}-b\bar{x}

Procedure

  1. Tabulate xi, yi, xi2, xiyix_i,\ y_i,\ x_i^2,\ x_i y_i and compute the column sums.
  2. Substitute ∑xi, ∑yi, ∑xi2, ∑xiyi\sum x_i,\ \sum y_i,\ \sum x_i^2,\ \sum x_i y_i into the formulas for aa and bb.
  3. Write the fitted line y=a+bxy=a+bx.

The resulting line is the best-fit line in the least squares sense, minimizing the total squared vertical distance between the data points and the line.

AI-generated answer · unverifiedView in 2081 paper →
U2 · Question 1 of 5
Question Priority · U2ranked by appearance likelihood — study top-down

Interpolation and Approximation

Analyzed next71%
1
★ TOP PICK

What is curve fitting? Explain the least squares method to fit a straight line y = a + bx to a given set of data points.

10 marksSEEN IN
52%
2

Explain Newton's divided difference interpolation formula.

5 marksSEEN IN
71%
3

Fit a second-degree parabola y = a + bx + cx^2 to a set of data using the least squares principle.

5 marksSEEN IN
71%
4

What is interpolation? Derive Newton's forward and backward difference interpolation formulae and explain when each is used.

10 marksSEEN IN
35%
5

State and explain Lagrange's interpolation formula with a suitable example.

5 marksSEEN IN
64%
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.

    Explain the Gauss elimination method with partial pivoting to solve a system of linear equations. Solve the system 2x + y + z = 10, 3x + 2y + 3z = 18, x + 4y + 9z = 16.

    [10 marks]
    Solution of Linear Algebraic EquationsVery likelyfrom 2080 paper →

    This question has recurred in 4 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Linear Algebraic Equations) appears in 100% of years.

  2. 2.

    Explain the fourth-order Runge-Kutta method for solving ordinary differential equations. Solve dy/dx = x + y, y(0) = 1 to find y(0.2) taking h = 0.1.

    [10 marks]
    Solution of Ordinary Differential EquationsVery likelyfrom 2081 paper →

    This question has recurred in 4 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Ordinary Differential Equations) appears in 86% of years.

  3. 3.

    What is curve fitting? Explain the least squares method to fit a straight line y = a + bx to a given set of data points.

    [10 marks]
    Interpolation and ApproximationVery likelyfrom 2081 paper →

    This question has recurred in 3 of 7 years; so far only in internal assessments, not the board; and its topic (Interpolation and Approximation) appears in 100% of years.

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

    Differentiate between the Gauss elimination and Gauss-Jordan methods.

    [5 marks]
    Solution of Linear Algebraic EquationsVery likelyfrom 2081 paper →

    This question has recurred in 6 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Linear Algebraic Equations) appears in 100% of years.

  2. 2.

    Explain numerical differentiation using forward and backward difference formulae.

    [5 marks]
    Numerical Differentiation and IntegrationVery likelyfrom 2081 paper →

    This question has recurred in 6 of 7 years; so far only in internal assessments, not the board; and its topic (Numerical Differentiation and Integration) appears in 100% of years.

  3. 3.

    Explain Euler's method to solve an ordinary differential equation with an example.

    [5 marks]
    Solution of Ordinary Differential EquationsVery likelyfrom 2081 paper →

    This question has recurred in 6 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Ordinary Differential Equations) appears in 86% of years.

  4. 4.

    Explain Newton's divided difference interpolation formula.

    [5 marks]
    Interpolation and ApproximationVery likelyfrom 2081 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic (Interpolation and Approximation) appears in 100% of years.

  5. 5.

    Fit a second-degree parabola y = a + bx + cx^2 to a set of data using the least squares principle.

    [5 marks]
    Interpolation and ApproximationVery likelyfrom 2081 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic (Interpolation and Approximation) appears in 100% of years.

  6. 6.

    State and explain Lagrange's interpolation formula with a suitable example.

    [5 marks]
    Interpolation and ApproximationVery likelyfrom 2080 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic (Interpolation and Approximation) appears in 100% of years.

  7. 7.

    Explain the power method for finding the largest eigenvalue of a matrix.

    [5 marks]
    Solution of Linear Algebraic EquationsVery likelyfrom 2081 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Linear Algebraic Equations) appears in 100% of years.

  8. 8.

    Explain the Gauss-Seidel iterative method to solve a system of linear equations.

    [5 marks]
    Solution of Linear Algebraic EquationsVery likelyfrom 2080 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic (Solution of Linear Algebraic Equations) appears in 100% of years.

  9. 9.

    Define absolute, relative and percentage errors. Explain the sources of errors in numerical computation.

    [5 marks]
    Solution of Nonlinear EquationsLikelyfrom 2081 paper →

    This question has recurred in 5 of 7 years; so far only in internal assessments, not the board; and its topic recurs in 5 of 7 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
U2Interpolation and Approximation
125
U4Solution of Linear Algebraic Equations
120
U1Solution of Nonlinear Equations
125
U3Numerical Differentiation and Integration
75
U5Solution of Ordinary Differential Equations
70
U6Numerical Solution of Partial Differential Equations
10
#Syllabus unitProbabilityAppearedAvg marksSyllabus weightExam vs syllabusTrendQuestions
1U2Interpolation and ApproximationVery likely100%
2074207520772078207920802081
17.922%10 lecture hrsBalancedexam 24% · syllabus 22%Steady5 recurring5 total
2U4Solution of Linear Algebraic EquationsVery likely100%
2074207520772078207920802081
17.120%9 lecture hrsBalancedexam 23% · syllabus 20%Steady4 recurring4 total
3U1Solution of Nonlinear EquationsLikely71%
20742077207820792081
2520%9 lecture hrsBalancedexam 24% · syllabus 20%Steady5 recurring5 total
4U3Numerical Differentiation and IntegrationVery likely100%
2074207520772078207920802081
10.720%9 lecture hrsUnder-examinedexam 14% · syllabus 20%Steady3 recurring3 total
5U5Solution of Ordinary Differential EquationsVery likely86%
207420752077207920802081
11.711%5 lecture hrsBalancedexam 13% · syllabus 11%Steady2 recurring2 total
6U6Numerical Solution of Partial Differential EquationsOccasional14%
2078
107%3 lecture hrsBalancedexam 2% · syllabus 7%Steadynone repeat1 total

Study smart, not hard

Drag the slider: studying the top 4 units in priority order covers ~85% of all observed marks.

  4. ~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.

U2Interpolation and Approximation
22% of lectures → 24% of marks
U4Solution of Linear Algebraic Equations
20% of lectures → 23% of marks
U1Solution of Nonlinear Equations
20% of lectures → 24% of marks
U3Numerical Differentiation and Integration
20% of lectures → 14% of markslow yield
U5Solution of Ordinary Differential Equations
11% of lectures → 13% of marks
U6Numerical Solution of Partial Differential Equations
7% of lectures → 2% of marks

Topics are the official CSC207 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.