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IIT M FOUNDATION DIPLOMA FN EXAM QDF1 28 Apr 2024 - 28 Apr 2024

MLF End Term14 questions40 marks6 MSQ · 6 SA · 2 MCQFoundation / Diploma / forenoon / QDF1

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Comprehension passage - the questions below it refer to this

A person went to the market to buy hens. Old hens can be bought for ₹ 200.00 each but young hens can be bought for ₹ 500.00 each. The old hens lay 3 eggs per week and the young hens lay 5 eggs per week. Each egg costs Rs. 11. A hen costs ₹ 50.00 per week to feed. If the financial constraint is ₹ 8000 per week to buy hens and the capacity constraint is that the total number of hens bought cannot exceed 20 per week. The objective is to earn more profit. Let x represent the number of old hens and y represent the number of young hens. Use the above information to answer the given subquestions

Q1MCQ2 marks

Choose the correct Primal optimization problem from the following.

Q2MSQ2 marksselect all that apply

If (x*, y*) is the optimal solution of the primal, u1 and u2 are Lagrange multipliers then which of the following options is/are true?

Q3MSQ3 marksselect all that apply

Which of the following options is/are true?

The Given LPP is infeasible.
The Given LPP is feasible.
The optimum value is x = 0 and y = 16
The optimum value is x = 20 and y = 0

Q4SA3 marks

Your numeric answer:

Q5MSQ3 marksselect all that apply

Q6MSQ3 marksselect all that apply

Comprehension passage - the questions below it refer to this

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

Q7SA3 marks

Find the number of discontinuous points.

Your numeric answer:

Q8SA3 marks

Find the number of points where the function is not differentiable.

Your numeric answer:

Q9MSQ3 marksselect all that apply

Let A be a real symmetric positive definite matrix. Which of the following options is/are true?

The eigenvectors of A may or may not be orthogonal.
If A is similar to a diagonal matrix B, then B is also positive definite.
AT A is not positive definite.
AT A is not similar to a diagonal matrix.

Q10MSQ3 marksselect all that apply

Q11MCQ3 marks

Q12SA3 marks

Your numeric answer:

Q13SA3 marks

In a large town, it is estimated that 20% of students own a laptop. A random sample of 400 students is selected from the town. What is the variance of the number of students with the laptops in a given sample?

Your numeric answer:

Q14SA3 marks

A quality control manager at a factory measures the thickness of glass sheets produced. Five randomly selected sheets have the thicknesses(in mm) 10, 8, 11, 9, 12 respectively. Assuming the thickness follows a normal distribution and the variance is known, what is the maximum likelihood estimator (MLE) of the mean thickness, (μ), in mm of the glass sheets?

Your numeric answer:

Answer key

Q1 (MCQ, 2 marks): A
Q2 (MSQ, 2 marks): C
Q3 (MSQ, 3 marks): B, C
Q4 (SA, 3 marks): 4
Q5 (MSQ, 3 marks): A, C, D
Q6 (MSQ, 3 marks): B, D
Q7 (SA, 3 marks): 2
Q8 (SA, 3 marks): 2
Q9 (MSQ, 3 marks): A, B
Q10 (MSQ, 3 marks): B, C
Q11 (MCQ, 3 marks): A
Q12 (SA, 3 marks): 1
Q13 (SA, 3 marks): 64
Q14 (SA, 3 marks): 10

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. Linear Programming (LP) & Optimization

Key Concepts:

Primal Problem Formulation:
- Objective: Maximize/Minimize a linear function (e.g., profit).
- Constraints: Linear inequalities (e.g., budget, capacity).
- Non-negativity: Variables ≥ 0.
- Example (from Q2):
  - Maximize: $5y - 17x$ (profit function derived from egg sales minus costs).
  - Subject to:

x + y \leq 20 \quad \text{(capacity constraint)},

200x + 500y \leq 8000 \quad \text{(budget constraint)}.

Feasibility & Optimality:
- Feasible Region: Set of points satisfying all constraints.
- Optimal Solution: Vertex of the feasible region (use corner point method or simplex algorithm).
- Example (from Q4): Optimal solution is $x = 0, y = 16$ .

Lagrange Multipliers & KKT Conditions:
- Complementary Slackness: For active constraints, multipliers $u_i \neq 0$ ; else $u_i = 0$ .

u_1(x^* + y^* - 20) = 0, \quad u_2(200x^* + 500y^* - 8000) = 0.

Stationary Condition: Gradient of Lagrangian = 0.

\nabla f(x^*, y^*) + u_1 \nabla g_1(x^*, y^*) + u_2 \nabla g_2(x^*, y^*) = 0.

Equations:

Profit Function Derivation (Q2):
- Revenue: $11 \times (3x + 5y)$ (eggs sold).
- Cost: $50(x + y)$ (feeding) + $200x + 500y$ (purchase).
- Net Profit: $11(3x + 5y) - 50(x + y) - (200x + 500y) = -17x + 5y$ .

B. Convexity & Optimization

Key Concepts:

Convex Functions:
- Definition: $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$ for $\lambda \in [0,1]$ .
- Examples (from Q6):
  - $f(x, y) = 2x^2 + 3y^2$ (convex, as Hessian is positive semi-definite).
  - $g(x) = -(x + 1)^2$ (concave, not convex).

Convex Sets:
- Definition: For any $x, y$ in the set, $\lambda x + (1-\lambda)y$ is also in the set.
- Examples (from Q7):
  - Convex: $x^2 + y^2 < 1$ (interior of a circle), $\frac{x^2}{4} + \frac{y^2}{9} < 1$ (interior of an ellipse).
  - Non-convex: $x^2 + y^2 = 1$ (boundary), $x^2 + y^2 > 1$ (exterior).

Global vs. Local Minima:
- For convex functions, any local minimum is a global minimum.

Equations:

Hessian Matrix (for convexity check):

H_f = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}.

If $H_f$ is positive semi-definite, $f$ is convex.

C. Calculus & Function Analysis

Key Concepts:

Continuity & Differentiability:
- Discontinuity: Points where $\lim_{x \to c} f(x) \neq f(c)$ .
- Non-differentiability: Corners (e.g., $|x|$ at $x=0$ ) or discontinuities.
- Example (from Q8, Q9):
  - Function:

f(x) = \begin{cases} 2x^2 & x < 2 \\ (x-2)^3 & 2 \leq x < 4 \\ 4 & x \geq 4 \end{cases}.

Discontinuous at $x=4$ (left limit $\neq$ right limit).
Non-differentiable at $x=2, 4$ (sharp corners or jumps).

Optimization with Constraints:
- Method: Lagrange multipliers or substitution.
- Example (from Q5):
  - Maximize $f(x, y) = xy$ subject to $\frac{x^2}{8} + \frac{y^2}{2} = 2$ .
  - Solution: Use substitution $y = \sqrt{2(2 - \frac{x^2}{8})}$ , then find critical points.

Equations:

Lagrange Multipliers:

\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda g(x, y).

Solve $\nabla \mathcal{L} = 0$ .

D. Linear Algebra

Key Concepts:

Positive Definite Matrices:
- Definition: $x^T A x > 0$ for all $x \neq 0$ .
- Properties (from Q10):
  - Eigenvalues > 0.
  - Eigenvectors are orthogonal.
  - $A^T A$ is positive definite if $A$ is invertible.

Rank & PCA:
- Rank: Maximum number of linearly independent rows/columns.
- PCA (Principal Component Analysis):
  - For data matrix $A$ , compute covariance matrix $C = \frac{1}{n} A^T A$ .
  - If $d >> n$ , use $\frac{1}{n} A A^T$ (from Q11).

Equations:

Covariance Matrix:

C = \frac{1}{n} A^T A, \quad A = \begin{bmatrix} (x_1 - \overline{x})^T \\ \vdots \\ (x_n - \overline{x})^T \end{bmatrix}.

Eigendecomposition:

C = V \Lambda V^T, \quad \Lambda = \text{diag}(\lambda_1, \dots, \lambda_d).

E. Probability & Statistics

Key Concepts:

Exponential Distribution:
- PDF: $f(x) = \lambda e^{-\lambda x}$ .
- Transformation of Variables (from Q12):
  - Given $Y_1 = X_1 + X_2$ , $Y_2 = X_1 - X_2$ , find joint PDF using Jacobian:

f_{Y_1, Y_2}(y_1, y_2) = f_{X_1, X_2}(x_1, x_2) \cdot |J|,

where $J$ is the Jacobian determinant.

Expected Value:
- For independent exponentials $X \sim \text{Exp}(\lambda)$ , $E[X] = \frac{1}{\lambda}$ .
- Example (from Q13): $E[X] = 1$ for $\lambda = 1$ .

Binomial Distribution:
- Variance: $np(1-p)$ .
- Example (from Q14): $n = 400$ , $p = 0.2$ , variance = $400 \times 0.2 \times 0.8 = 64$ .

Maximum Likelihood Estimation (MLE):
- For normal distribution, MLE of mean $\mu$ is the sample mean.
- Example (from Q15): $\hat{\mu} = \frac{10 + 8 + 11 + 9 + 12}{5} = 10$ .

Equations:

Joint PDF Transformation:

f_{Y_1, Y_2}(y_1, y_2) = \frac{\lambda_1 \lambda_2}{2} e^{-\frac{\lambda_1 + \lambda_2}{2} y_1 - \frac{\lambda_1 - \lambda_2}{2} y_2}, \quad y_1 \pm y_2 \geq 0.

Binomial Variance:

\text{Var}(X) = np(1-p).

MLE for Normal Distribution:

\hat{\mu} = \frac{1}{n} \sum_{i=1}^n x_i.

IIT M FOUNDATION DIPLOMA FN EXAM QDF1 28 Apr 2024 - 28 Apr 2024

Answer key

Concepts this paper tests

1. Core Topics & Key Concepts

A. Linear Programming (LP) & Optimization

Key Concepts:

Equations:

B. Convexity & Optimization

Key Concepts:

Equations:

C. Calculus & Function Analysis

Key Concepts:

Equations:

D. Linear Algebra

Key Concepts:

Equations:

E. Probability & Statistics

Key Concepts:

Equations:

Graph 2: Convexity & Function Analysis

Graph 3: Probability & Statistics

Graph 4: Linear Algebra & PCA