Machine Learning Foundations End Term: 21 December 2025 (September 2025 term)

Consider the function defined as . For what values of is convex

A real estate company wants to create a model to predict house prices based on the size of the house. They collect data from three recent sales. There are two features: size (hundreds of sq ft) and number of bedrooms. The data-matrix and the price vector (in hundreds of thousands of rupees) are:

The objective is to find the weight vector that minimizes the sum of squared errors given by the loss function:

If is the value that minimizes , then which of the following is ?

Consider

Which of the following is/are true

Let . Find the directional derivative of at the point in the direction of unit vector along .

An ML team wants to predict whether an email is spam or not by using some classification models. Therefore, the team defines three binary features where, = Presence of suspicious keywords (1 = present, 0 = absent)• = Whether the sender is unknown (1 = unknown, 0 = known)• = Whether the email contains a link (1 = yes, 0 = no)• The team collects the following dataset from previously classified emails:

The model makes a prediction using the rule : and

Compute the average loss on this dataset. Enter your answer correct to one decimal place.

Consider the convex hull of the set of points . Which of the following points belongs to the convex hull?

Let be a continuous random variable with the probability density function:

What is the value of

If is a square invertible matrix, which of the following is/are true?

is a matrix whose eigenvalues are . Consider the set as defined below:

Find .

An analyst monitors the number of defective items produced daily. Let denote number of defective items produced in a day. Using Markov's inequality, the analyst finds an upper bound of on the probability that at least items are defective. Calculate the mean of .

Using the method of Lagrange multipliers, find a point on the circle centered at with radius that is closest to the point . If this distance is , enter as your answer. Your answer should be an integer.

Consider two datasets in , and , each having three points as shown below. The figure is drawn to scale. PCA is run on both these datasets separately.

and are the covariance matrices of the two datasets respectively. If is the eigenvector corresponding to the largest eigenvalue of and is the eigenvector corresponding to the smallest eigenvalue of , find the absolute value of . Both eigenvectors have unit norm. Your answer should be an integer.

A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.

Find the waiting time such that 95% of customers are served within minutes.

The time (in minutes) a bank employee spends assisting a customer follows an exponential distribution with parameter . The time spent (in minutes) from a random sample of 5 customers are as follows: Find the maximum likelihood estimate of for the given sample. Enter the answer correct to two decimal places.

A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.

What is the probability that a randomly chosen customer waits between and minutes

Anil throws a dart onto a circular board. Let a random variable denote the distance from the center to the point where the dart hits the board. Suppose the PDF of is

Calculate the value of

. Enter the answer correct to two decimal places.

A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.

Using Chebyshev's inequality, find an upper bound for , where represents the mean of the random variable . Enter your answer correct to two decimal places.

Using the method of Lagrange multipliers, find a point on the circle centered at with radius that is closest to the point . If this distance is , enter as your answer. Your answer should be an integer.

Consider the following constrained optimization problem:

Answer the sub-questions assuming that we take a very small step in the direction of interest.

At the point which of the following is/are descent directions for the objective function

Consider the following constrained optimization problem:

If

corresponds to the global minimum, find

. Enter your answer correct to one decimal place.

Consider the following constrained optimization problem:

Answer the sub-questions assuming that we take a very small step in the direction of interest.

At the point which of the following is/are feasible directions

If and , where

Based on the above data, answer the given subquestions.

What is the distribution of

Consider the following constrained optimization problem:

Answer the sub-questions assuming that we take a very small step in the direction of interest.

Is a potential candidate at which the problem could have a local minimum

If and , where

Based on the above data, answer the given subquestions.

What is the distribution of

If and , where

Based on the above data, answer the given subquestions.

What is the distribution of

If and , where

Based on the above data, answer the given subquestions.

Find the covariance matrix of .

If and , where

Based on the above data, answer the given subquestions.

What is the distribution of

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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

Find the updated point after the first iteration of the gradient descent algorithm.

If and , where

Based on the above data, answer the given subquestions.

Find the covariance matrix of .

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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

Compute the value of after the second iteration, i.e., .

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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

Find the updated point after the first iteration of the gradient descent algorithm.

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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

Compute the value of after the second iteration, i.e., .

A cafe records the time (in minutes) a customer waits for a freshly made sandwich. The waiting time follows an exponential distribution with mean 5 minutes. Based on the above data, answer the given subquestions.

Consider the following constrained optimization problem:

Answer the sub-questions assuming that we take a very small step in the direction of interest.

If and , where

Based on the above data, answer the given subquestions.

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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

If and , where

Based on the above data, answer the given subquestions.

A company aims to minimize its production cost, represented by the following quadratic function: The company applies the gradient descent algorithm to minimize this function, starting with an initial guess of and using a step size (learning rate) of

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