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IIT M DIPLOMA AN EXAM QDD4 13 Apr 2025 - 13 Apr 2025

MLF End Term14 questions40 marks7 MSQ · 5 SA · 2 MCQDiploma / afternoon / QDD4

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Q1MCQ2 marks

Q2MCQ3 marks

Q3MSQ3 marksselect all that apply

Which of the following statements about singular value decomposition (SVD) is/are true?

Singular value decomposition is possible only for a square matrix.
Singular value decomposition is possible only for a symmetric matrix.
The singular values of a matrix are always non-negative.
The singular values of matrices A and AT are the same.

Q4MSQ3 marksselect all that apply

In the context of Linear Algebra, which of the following statements about Principal Component Analysis (PCA) is/are true?

The principal components of a matrix are given by the eigenvectors of itscovariance matrix.
The principal components of a matrix are given by the singular values of itscovariance matrix.
PCA finds the principal components by maximizing the variance along eachcomponent.
The principal components form a basis that is not necessarily orthogonal.

Q5MSQ3 marksselect all that apply

Which of the following options is/are true?

Q6MSQ3 marksselect all that apply

Which of the following options is/are true ?

Q7MSQ3 marksselect all that apply

Q8SA3 marks

Your numeric answer:

Q9SA3 marks

Your numeric answer:

Q10SA4 marks

Your numeric answer:

Q11SA2 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q12MSQ2 marksselect all that apply

Which of the following statements correctly represent the Karush-Kuhn-Tucker (KKT) conditions for the given optimization problem?

Q13MSQ3 marksselect all that apply

Which of the following statements are correct based on the KKT conditions and the solution to the optimization problem?

Q14SA3 marks

Your numeric answer:

Answer key

Q1 (MCQ, 2 marks): A
Q2 (MCQ, 3 marks): A
Q3 (MSQ, 3 marks): C, D
Q4 (MSQ, 3 marks): A, C
Q5 (MSQ, 3 marks): A, B
Q6 (MSQ, 3 marks): B, C
Q7 (MSQ, 3 marks): A, D
Q8 (SA, 3 marks): 12
Q9 (SA, 3 marks): 0.2
Q10 (SA, 4 marks): 46
Q11 (SA, 2 marks): -0.2
Q12 (MSQ, 2 marks): A, C
Q13 (MSQ, 3 marks): A, C
Q14 (SA, 3 marks): 0.25

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Key Concepts

A. Probability & Statistics

Exponential Distribution
- CDF: $F_X(x) = 1 - e^{-\lambda x}$ (for $\lambda = 1$ , $F_X(5) = 1 - e^{-5}$ ).
- PDF: $f_X(x) = \lambda e^{-\lambda x}$ .

Gaussian (Normal) Distribution
- Conditional Probability: For $z = Cy + \epsilon$ where $\epsilon \sim \mathcal{N}(0, \sigma^2 I_d)$ , $p(z \mid y) = \mathcal{N}(Cy, \sigma^2 I_d)$ .
- Linearity of Expectation: $E[3X_1 + 2X_2] = 3E[X_1] + 2E[X_2]$ .

Markov’s Inequality
- For non-negative $Y$ , $P(Y \geq a) \leq \frac{E[Y]}{a}$ .
- Example: $P(Y \geq 100) \leq \frac{20}{100} = 0.2$ .

B. Linear Algebra & Optimization

Singular Value Decomposition (SVD)
- Properties:
  - Applicable to any matrix (not just square/symmetric).
  - Singular values are non-negative.
  - Singular values of $A$ and $A^T$ are identical.

Principal Component Analysis (PCA)
- Key Ideas:
  - Principal components = eigenvectors of the covariance matrix.
  - Maximizes variance along each component.
  - Basis is orthogonal.

Convexity
- Convex Sets:
  - Linear constraints (e.g., $\sum a_i x_i = b$ ) define convex sets.
  - Lines in $\mathbb{R}^2$ are convex; unions of convex sets are not necessarily convex.
- Convex Functions:
  - $e^x$ , $e^{x^2+2}$ are convex.
  - Composition $f \circ g$ is not guaranteed convex.
- Optimization:
  - Convex cost functions (e.g., $f(x,y) = 2x^2 + xy + y^2$ ) have global minima.
  - KKT Conditions for constrained optimization:
    - $\nabla L = 0$ (stationarity).
    - $\beta (w - \delta) = 0$ (complementary slackness).
    - $\beta \geq 0$ (dual feasibility).

Gradient Descent
- Update Rule: $x_{k+1} = x_k - \eta \nabla f(x_k)$ .
- Example: For $f(x,y) = x^2 + 2y^2 - 6x + 8y$ , $\nabla f = (2x - 6, 4y + 8)$ . With $\eta = 0.2$ , $(x_1, y_1) = (1.8, -0.4)$ .

C. Calculus & Constrained Optimization

Lagrange Multipliers
- Lagrangian: $L = f(x) + \sum \lambda_i g_i(x)$ .
- KKT Conditions:
  - $\nabla L = 0$ , $\lambda_i g_i(x) = 0$ , $\lambda_i \geq 0$ (for inequalities).

Maximization on Constraints
- Example: Maximize $f(x,y,z) = 5x + 7y + z$ on $x^2 + y^2 = 25$ and $x + y - z = 4$ .
  - Use Lagrange multipliers or parameterize the cylinder.

2. Key Equations

Topic	Equation
Exponential CDF	$F_X(x) = 1 - e^{-\lambda x}$
Gaussian Conditional	$p(z \mid y) = \mathcal{N}(Cy, \sigma^2 I_d)$
Markov’s Inequality	$P(Y \geq a) \leq \frac{E[Y]}{a}$
SVD	$A = U \Sigma V^T$ (singular values $\sigma_i \geq 0$ )
PCA	Covariance matrix eigenvectors $=$ principal components
Convex Function	$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$
Gradient Descent	$x_{k+1} = x_k - \eta \nabla f(x_k)$
KKT Conditions	$\nabla L = 0$ , $\lambda_i g_i(x) = 0$ , $\lambda_i \geq 0$
Lagrange Multipliers	$L = f(x) + \sum \lambda_i g_i(x)$

Graph 2: Linear Algebra & Optimization

Graph 3: Calculus & Constrained Optimization

4. Exam Tips

Probability: Focus on CDF/PDF of exponential/Gaussian distributions and conditional probability.
Linear Algebra: SVD properties, PCA (eigenvectors vs. singular values), and convexity.
Optimization: KKT conditions, gradient descent updates, and Lagrange multipliers.
Calculus: Maximization/minimization on constraints (use substitution or Lagrange multipliers).