About this previous year paper
Subject Code: KAI051
Subject Name: Mathematical Foundation of AI, ML, and Data Science
Course: B.Tech (Semester V)
Maximum Marks: 100
Time: 3 Hours
Exam Year: 2023–24
Sections: A, B, and C
SECTION A – Short Answer Questions (2 × 10 = 20 Marks)
Attempt all questions briefly.
Explain the reading and interpretation of bar graphs.
Using Chebyshev’s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean.
Options: (i) 11% (ii) 89% (iii) 90%
Discuss the need of sampling.
Explain the use of chi-square test in hypothesis testing.
Briefly explain Gibbs sampling.
Interpret the need of random number generators.
Discuss vector space.
Explain linear independence.
Differentiate between symmetric and anti-symmetric matrices.
Discuss eigenvalues and eigenvectors.
SECTION B – Medium-Length Questions (10 × 3 = 30 Marks)
Attempt any three of the following.
The table below shows the number of Maruti cars sold by five dealers — draw a bar graph:
| Dealer | Saya | Bagga Links | DD Motors | Bhasin Motors | Competent Motors |
|---|---|---|---|---|---|
| Cars Sold | 60 | 40 | 20 | 15 | 10 |
Discuss the Central Limit Theorem and its applications.
Explain the Metropolis–Hastings algorithm.
Explain and prove the Cauchy–Schwarz Inequality.
Explain Orthogonal Diagonalization with an example.
SECTION C – Long/Analytical Questions (10 × 5 = 50 Marks)
Q3. Probability & Random Variables
a. Three persons A, B, C have selection chances in ratio 1:2:4. Their probabilities of improving profits are 0.8, 0.5, and 0.3. If no change occurs, find the probability that C was appointed.
OR
b. Determine the mean and variance of a random variable XXX with:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| P(X) | 0.15 | 0.10 | 0.10 | 0.01 | 0.08 | 0.01 | 0.05 | 0.02 | 0.28 | 0.20 |
Q4. ANOVA & PCA
a. Calculate the ANOVA coefficient for:
| Plant | Number | Average Span | S |
|---|---|---|---|
| Hibiscus | 5 | 12 | 2 |
| Marigold | 5 | 16 | 1 |
| Rose | 5 | 20 | 4 |
OR
b. Discuss the steps of dimensionality reduction using PCA.
Q5. Distributions & Sampling
a. Explain the following joint distributions:
Discrete Joint Distribution
Continuous Joint Distribution
Multinomial Distribution
OR
b. Given age distribution at a convention, select a stratified sample (n = 80) proportionally:
| Age Group | 5–15 | 16–25 | 26–40 | 41–60 | 61+ |
|---|---|---|---|---|---|
| Number of People | 132 | 678 | 543 | 289 | 108 |
Q6. Vector Spaces
a. Explain the Gram–Schmidt Process for obtaining orthonormal bases.
OR
b. Explain how to find a basis of a vector space.
Q7. Linear Transformations & Diagonalization
a. Diagonalize the matrix
[300−349003]\begin{bmatrix} 3 & 0 & 0 \\ -3 & 4 & 9 \\ 0 & 0 & 3 \end{bmatrix}3−30040093
OR
b. Show that T:V2(R)→V2(R)T: V_2(\mathbb{R}) \rightarrow V_2(\mathbb{R})T:V2(R)→V2(R) defined by T(a,b)=(a+b,a)T(a, b) = (a + b, a)T(a,b)=(a+b,a) is a linear transformation.
Key Topics to Prepare
Statistics & Probability: Sampling, Chebyshev’s inequality, Central Limit Theorem, ANOVA.
Linear Algebra: Vector spaces, Linear independence, Basis, Gram–Schmidt, Eigenvalues, Diagonalization.
AI/ML Foundations: Gibbs sampling, Metropolis–Hastings, Random numbers, PCA, Joint distributions.
Theoretical Proofs: Cauchy–Schwarz inequality, Orthogonal diagonalization.
Study Tips
Practice derivations (Cauchy–Schwarz, Orthogonal Diagonalization).
Revise probability laws — Bayes, Conditional, Expectation, Variance.
Understand sampling techniques — stratified, random, Gibbs.
Learn PCA process: standardization → covariance matrix → eigen decomposition → projection.
Focus on matrix operations — orthogonality, linear transformations, eigenvalues.