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2024 July07: IIT M AN EXAM QDF4 - 07 Jul 2024

MLF Quiz 112 questions40 marks5 MCQ · 4 MSQ · 3 SAafternoon / QDF4
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Comprehension passage - the questions below it refer to this
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Q1SA3 marks
Find the value of α + 2β.
Q2MSQ4 marksselect all that apply
Which of the following options is/are true ?
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Comprehension passage - the questions below it refer to this
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Q3MSQ4 marksselect all that apply
Which of the following options is/are true?
  1. A is a symmetric matrix.
  2. A is a diagonal matrix.
  3. A is orthogonally diagonalizable.
  4. Eigenvalues of A are -1, -1 and 2.
Q4MSQ3 marksselect all that apply
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Q5MSQ4 marksselect all that apply
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Q6MCQ4 marks
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Q7MCQ4 marks
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Q8MCQ3 marks
Which of the following statements is true for a function f(x, y) that is differentiable at a point (a, b)?
  1. The directional derivative exists in every direction.
  2. The directional derivative is the same in every direction.
  3. The directional derivative can only be calculated along the axes.
  4. The directional derivative does not exist.
Q9MCQ3 marks
Which of the following functions cannot be approximated linearly around x = 0?
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Q10SA3 marks
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Q11SA3 marks
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Q12MCQ2 marks
Consider a dataset from an e-commerce company such as Flipkart that has details of customer behavior, which of the following problems can be best approached using clustering?
  1. Predicting the exact number of sales for each customer segment.
  2. Classifying customers into segments based on labeled customer data.
  3. Identifying inherent groupings in customer data without predefined labels.
  4. Reducing the number of features in the customer dataset.

Answer key

  1. Q1 (SA, 3 marks): 3
  2. Q2 (MSQ, 4 marks): A, C, D
  3. Q3 (MSQ, 4 marks): A, C, D
  4. Q4 (MSQ, 3 marks): A, C
  5. Q5 (MSQ, 4 marks): A, B, C
  6. Q6 (MCQ, 4 marks): A
  7. Q7 (MCQ, 4 marks): B
  8. Q8 (MCQ, 3 marks): A
  9. Q9 (MCQ, 3 marks): C
  10. Q10 (SA, 3 marks): 20.58 to 20.64
  11. Q11 (SA, 3 marks): 0.2
  12. Q12 (MCQ, 2 marks): C

Concepts this paper tests

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

Summary of Core Topics, Concepts, Principles, and Equations

Mathematics

  1. Matrices and Eigenvalues

    • Eigenvalues and Eigenvectors: For a matrix AA, if Av=λvAv = \lambda v, then λ\lambda is an eigenvalue and vv is the corresponding eigenvector.
    • Projection Matrix: P=aaTaTaP = \frac{a a^T}{a^T a}
    • Properties of Projection Matrix: Symmetric, Idempotent, Singular.
    • Rank and Null Space: Rank of a matrix, null space, column space.

  1. Differentiability and Continuity

    • Differentiability Conditions: For a function to be differentiable at a point, it must be continuous and have a well-defined derivative at that point.
    • Directional Derivatives: Existence and calculation in different directions.

  1. Linear Approximation

    • Linear Approximation: Approximating functions linearly around a point.
    • Gradient: ∇f=(∂f∂x,∂f∂y)\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)
    • Length of Gradient: ∥∇f∥=(∂f∂x)2+(∂f∂y)2\|\nabla f\| = \sqrt{\left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}

  1. Least Squares Method

    • Best-Fit Line: Finding the line y^=mx+c\hat{y} = mx + c that minimizes the sum of squared residuals.

Data Science

  1. Classification Models

    • Misclassification Rate: Calculating the rate of misclassified labels.
    • Step Function: u(z)={1,if z≥threshold0,otherwiseu(z) = \begin{cases} 1, & \text{if } z \geq \text{threshold} \\ 0, & \text{otherwise} \end{cases}

  1. Clustering

    • Clustering Techniques: Identifying inherent groupings in data without predefined labels.

Equations

  1. Eigenvalue Equation: Av=λvAv = \lambda v
  2. Projection Matrix: P=aaTaTaP = \frac{a a^T}{a^T a}
  3. Gradient: ∇f=(∂f∂x,∂f∂y)\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)
  4. Length of Gradient: ∥∇f∥=(∂f∂x)2+(∂f∂y)2\|\nabla f\| = \sqrt{\left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}
  5. Least Squares Method: Minimizing ∑(yi−y^i)2\sum (y_i - \hat{y}_i)^2
  6. Misclassification Rate: Number of misclassified labelsTotal number of labels\frac{\text{Number of misclassified labels}}{\text{Total number of labels}}