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Machine Learning Foundations 15 Mar 26 - 15 Mar 2026

MLF Quiz 116 questions40 marks6 SA · 5 MCQ · 5 MSQFoundation

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Q1MCQ2 marks

Let the vectors

be orthogonal. Which of the following must be true?

Q2MCQ2 marks

Consider the function

Which of the following is true?

is continuous at , but not differentiable at .
is both continuous and differentiable at .
is neither continuous nor differentiable at .
is differentiable at , but not continuous at .

Q3MCQ3 marks

Consider the following matrices: question figure

Which of the following is true?

is diagonalizable, is not diagonalizable.
is not diagonalizable, is diagonalizable.
Both and are diagonalizable.
Neither nor is diagonalizable

Q4MCQ3 marks

Consider a dataset

for a regression problem where each element is of the form

, for

is the feature and

is the label of the

data-point.

Let

be two regression models such that:

Which of the following options is correct?

has the lower mean squared loss.
has the lower mean squared loss.
Both models have same mean squared loss.

Q5MCQ3 marks

Which of the following is a basis for the column space of the matrix

given below? question figure

Q6MSQ3 marksselect all that apply

The matrix question figure

has eigenvalues

and

. Which of the following options is/are correct?

Q7MSQ3 marksselect all that apply

Which of the following is/are true?

If and are matrices of the same size, then .
The rank of a matrix is equal to the number of non-zero rows in it.
If is a matrix which is obtained by permuting the rows of the identity matrix, then the rank of is .
If is a non-zero column vector in , then the rank of the matrix is .

Q8MSQ3 marksselect all that apply

Let

and

be two lines in

defined in parametric form: question figure

For example, every point on line

is of the form

. Which of the following is/are true?

and are parallel
and are not parallel
and intersect at a point
and do not intersect at any point

Q9MSQ3 marksselect all that apply

Which of the following is/are true ?

Every square matrix with real entries has at least one real eigenvalue.
A set of two eigenvectors corresponding to the same eigenvalue of a square matrix must be linearly dependent.
Given a square matrix of order , the set of all eigenvectors for an eigenvalue , along with the zero vector, forms a subspace of .
A square matrix is not invertible if and only if is one of its eigenvalues.

Q10MSQ2 marksselect all that apply

Suppose you are building a credit card fraud detection system and have two approaches: (i) Manually writing rules such as "If transaction amount is above Rs. 50,000 and the location is overseas, mark it as fraud.” (ii) Training a model using millions of past transactions labeled as fraud or not fraud. Which of the following options is/are correct?

Approach (i) is not machine learning.
Approach (ii) is machine learning.
Both approaches are machine learning.
Neither approach is machine learning.

Q11SA2 marks

Consider the function

given by

. If the best linear approximation to

is given as

, find

. Your answer should be an integer.

Your numeric answer:

Q12SA3 marks

Find the directional derivative of the function

given by

at the point

in the direction of the vector question figure

. Your answer should be an integer.

Your numeric answer:

Comprehension passage - the questions below it refer to this

Let

be the projection matrix that projects vectors in

onto the line

. Let

be the matrix that projects vectors in

onto the nullspace of

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

Q13SA3 marks

Find

. Your answer should be an integer.

Your numeric answer:

Q14SA1 mark

Find

. Your answer should be an integer.

Your numeric answer:

Comprehension passage - the questions below it refer to this

Consider the matrix

given below: question figure

Based on the above data, answer the given subquestions.

Q15SA2 marks

is the smallest eigenvalue of

, enter the value of

. Your answer should be an integer.

Your numeric answer:

Q16SA2 marks

is an eigenvector of

corresponding to

, find

. Enter your answer correct to two decimal places.

Your numeric answer:

Answer key

Q1 (MCQ, 2 marks): A
Q2 (MCQ, 2 marks): A
Q3 (MCQ, 3 marks): B
Q4 (MCQ, 3 marks): B
Q5 (MCQ, 3 marks): A
Q6 (MSQ, 3 marks): B, C
Q7 (MSQ, 3 marks): C, D
Q8 (MSQ, 3 marks): B, D
Q9 (MSQ, 3 marks): C, D
Q10 (MSQ, 2 marks): A, B
Q11 (SA, 2 marks): 11
Q12 (SA, 3 marks): 3
Q13 (SA, 3 marks): 16
Q14 (SA, 1 marks): 4
Q15 (SA, 2 marks): -3
Q16 (SA, 2 marks): 1.24 to 1.26

Concepts this paper tests

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

Here is a structured summary of the core topics, concepts, principles, and equations essential for the exam, followed by Mermaid syntax knowledge graphs.

Core Topics and Concepts

1. Linear Algebra

##### Vectors and Orthogonality

Orthogonal Vectors: Two vectors are orthogonal if their dot product is zero.
- Equation: $\mathbf{u} \cdot \mathbf{v} = 0$

##### Matrices

Diagonalizable Matrices: A matrix $A$ is diagonalizable if it can be written as $A = PDP^{-1}$ , where $D$ is a diagonal matrix.
Eigenvalues and Eigenvectors:
- Eigenvalue equation: $A\mathbf{v} = \lambda\mathbf{v}$
- Eigenvectors corresponding to the same eigenvalue form a subspace.
Rank of a Matrix: The maximum number of linearly independent row/column vectors.
Projection Matrix: Projects vectors onto a subspace. For a line defined by vector $\mathbf{a}$ , the projection matrix $P$ is:

P = \frac{\mathbf{a}\mathbf{a}^T}{\mathbf{a}^T\mathbf{a}}

##### Basis for Column Space

The column space of a matrix is spanned by its linearly independent columns.

2. Calculus

##### Continuity and Differentiability

Continuity at a Point: A function $f(x)$ is continuous at $x = a$ if:

\lim_{x \to a} f(x) = f(a)

Differentiability at a Point: A function $f(x)$ is differentiable at $x = a$ if the derivative $f'(a)$ exists.

##### Directional Derivative

The directional derivative of $f(x, y)$ at $(a, b)$ in the direction of $\mathbf{u} = (u_1, u_2)$ is:

D_{\mathbf{u}}f(a, b) = \nabla f(a, b) \cdot \mathbf{u}

##### Best Linear Approximation

The best linear approximation of $f(x, y)$ at $(a, b)$ is:

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

3. Machine Learning

##### Regression Models

Mean Squared Loss: Measures the average squared difference between predicted and actual values.

\text{MSE} = \frac{1}{n}\sum_{i=1}^n (y_i - \hat{y}_i)^2

##### Machine Learning Approaches

Rule-Based Systems: Not considered machine learning.
Model Training: Training a model on labeled data is machine learning.

4. Geometry

##### Lines in Parametric Form

Two lines are parallel if their direction vectors are scalar multiples of each other.
Two lines intersect if they are not parallel and satisfy a common point equation.

Key Equations

Orthogonality:

\mathbf{u} \cdot \mathbf{v} = 0

Eigenvalue Equation:

A\mathbf{v} = \lambda\mathbf{v}

Projection Matrix:

P = \frac{\mathbf{a}\mathbf{a}^T}{\mathbf{a}^T\mathbf{a}}

Directional Derivative:

D_{\mathbf{u}}f(a, b) = \nabla f(a, b) \cdot \mathbf{u}

Mean Squared Loss:

\text{MSE} = \frac{1}{n}\sum_{i=1}^n (y_i - \hat{y}_i)^2

Graph 1: Linear Algebra Concepts

Graph 2: Calculus Concepts

Graph 3: Machine Learning Concepts

Graph 4: Geometry Concepts

This summary and the knowledge graphs should help you understand the relationships between the core topics and prepare effectively for the exam.