Home / Past papers / MLF Quiz 1 / this paper

IIT M FOUNDATION AN EXAM QDF3 23 Feb - 23 Feb 2025

MLF Quiz 114 questions40 marks8 MCQ · 4 MSQ · 2 SAFoundation / afternoon / QDF3

Elapsed 00:00

Q1MCQ3 marks

0.162
0.257
0.157
1.57

Q2MCQ3 marks

Eigenvectors corresponding to the two different eigenvalues of M areorthogonal.
The matrix M is not diagonalizable.
The eigenvalues of M are 4 and 1
The rank of M is 1.

Q3MCQ3 marks

Q4MCQ3 marks

Q5MSQ3 marksselect all that apply

Which of the following options is / are true ?

Q6MSQ3 marksselect all that apply

Q7MSQ3 marksselect all that apply

Which of the following statements is/are correct?

Q8MSQ2 marksselect all that apply

Q9SA2 marks

Your numeric answer:

Q10SA3 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q11MCQ3 marks

For the input x = [0.9, 5, 1], what will be the predicted output u(z)?

0
1
1.75
Cannot be determined

Q12MCQ3 marks

If the threshold in the step function is changed to z ≥ 1.0, what will happen to the misclassification rate?

Increase
Decrease
Remain same
Cannot be determined

Comprehension passage - the questions below it refer to this

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

Q13MCQ3 marks

If B is diagonalizable, what is the invertible matrix P such that B = PDP−1?

Q14MCQ3 marks

If B is diagonalizable, which of the following is a diagonal matrix D such that B = PDP−1? Use the matrix P obtained in the previous question.

Answer key

Q1 (MCQ, 3 marks): C
Q2 (MCQ, 3 marks): D
Q3 (MCQ, 3 marks): A
Q4 (MCQ, 3 marks): B
Q5 (MSQ, 3 marks): A, D
Q6 (MSQ, 3 marks): B, C
Q7 (MSQ, 3 marks): B, C, D
Q8 (MSQ, 2 marks): A, B, C
Q9 (SA, 2 marks): 15
Q10 (SA, 3 marks): 6.37 to 6.43
Q11 (MCQ, 3 marks): B
Q12 (MCQ, 3 marks): A
Q13 (MCQ, 3 marks): D
Q14 (MCQ, 3 marks): A

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

1. Linear Approximation

Concept: Linear approximation of a function around a point.
Equation: For a function $f(x, y)$ , the linear approximation $L(x, y)$ around a point $(a, b)$ is given by:

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

2. Matrix Properties

Concepts: Eigenvalues, eigenvectors, diagonalizability, rank, nullity.
Equations:
- Rank-Nullity Theorem: $\text{rank}(A) + \text{nullity}(A) = n$ for an $m \times n$ matrix $A$ .
- Eigenvalue equation: $A \mathbf{v} = \lambda \mathbf{v}$ .

3. Directional Derivative

Concept: Rate of change of a function in the direction of a unit vector.
Equation: The directional derivative of $f$ at $(x_0, y_0)$ in the direction of $\mathbf{u}$ is:

D_{\mathbf{u}} f(x_0, y_0) = \nabla f(x_0, y_0) \cdot \mathbf{u}

4. Least Squares Method

Concept: Fitting a polynomial to a dataset using the least sum of squares.
Equation: For a second-order polynomial $\theta_0 + \theta_1 x + \theta_2 x^2$ , the least squares method minimizes:

\min_{\theta_0, \theta_1, \theta_2} \sum_{i=1}^{n} \left( y_i - (\theta_0 + \theta_1 x_i + \theta_2 x_i^2) \right)^2

5. Vector Projection

Concept: Projection of one vector onto another.
Equation: The projection of $\mathbf{a}$ onto $\mathbf{b}$ is:

\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}

6. Continuity and Differentiability

Concepts: Continuity and differentiability of functions.
Examples: $f(x) = \ln(x)$ , $f(x) = e^x$ , $f(x) = \sqrt{x}$ , $f(x) = \frac{1}{x}$ .

7. Gradient and Length

Concept: Gradient of a function and its length.
Equation: The gradient of $f(x, y, z)$ is:

\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)

The length of the gradient is:

\|\nabla f\| = \sqrt{\left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 + \left( \frac{\partial f}{\partial z} \right)^2}

8. Classification Model

Concept: Using a linear combination and step function for classification.
Equation: Linear combination $z = w_1 x_1 + w_2 x_2 + w_3 x_3$ , and step function:

u(z) = \begin{cases} 1 & \text{if } z \geq \text{threshold} \\ 0 & \text{otherwise} \end{cases}

9. Diagonalization

Concept: Diagonalization of a matrix.
Equation: $B = PDP^{-1}$ , where $D$ is a diagonal matrix and $P$ is the matrix of eigenvectors.