MLF End Term: 1 September 2024, Set QDF3 (May 2024 term)

A company produces two types of products P1 and P2. The cost price per unit of P1 and P2 are **₹**4 and **₹**3, respectively. The production process requires two types of resources: labor hours and machine hours. Each unit of P1 requires 2 labor hours and 1 machine hour, while each unit of P2 requires 1 labor hour and 3 machine hours. The company has constraints on the availability of labor and machine hours, which are 80 and 90 hours, respectively. Use the above information to answer the given sub-questions.

Choose the correct Primal optimization problem from the following.

A company produces two types of products P1 and P2. The cost price per unit of P1 and P2 are **₹**4 and **₹**3, respectively. The production process requires two types of resources: labor hours and machine hours. Each unit of P1 requires 2 labor hours and 1 machine hour, while each unit of P2 requires 1 labor hour and 3 machine hours. The company has constraints on the availability of labor and machine hours, which are 80 and 90 hours, respectively. Use the above information to answer the given sub-questions.

Find the minimum cost price of the products.

Which of the following options is/are true?

A gardener has 200 meters of fencing and wants to enclose a rectangular garden that borders a straight wall. The gardener does not need to fence along the wall. Find the maximum possible enclosed area of the garden.

Suppose a random variable X has a mean μ of 70 and a standard deviation σ of 8. Using Chebyshev’s inequality, determine the maximum probability that X will deviate from the mean by more than 24 units. Enter the answer correct to two decimal places.

Consider the following input data points:

Suppose we fit a linear model $f(\mathrm{x}) = x_1 + 3x_2 - x_3$, where $\mathrm{x} = (x_1, x_2, x_3)$. Compute the value of the loss function $L$ for this dataset which is defined as $L = \dfrac{1}{n} \sum\limits_{i=1}^{n} (f(\mathrm{x}^i) - y^i)^2$.

Enter the answer correct to one decimal place.

Let $X$ and $Y$ be two independent random variables, where $X \sim \mathrm{Normal}(-1, 11)$ and $Y \sim \mathrm{Normal}(1, 9)$. Define $U = 2X - 3Y$. Find the value of $P(U < 2)$. Enter the answer correct to three decimal places.

Hint: Use the following values of $F_Z$ if required. $F_Z$ stands for the CDF of the standard normal.

• $F_Z(0.62) = 0.7343$
• $F_Z(-0.62) = 0.2656$
• $F_Z(1.25) = 0.8947$
• $F_Z(-1.25) = 0.1052$

Consider a square matrix A of order 3 such that trace(A) = 3 and det(A) = 2. If 1 is an eigenvalue of A, then which of the following options is/are true?

Which of the following is/are true about PCA?

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.