Singular matrix

Decomposable matrix

Adjuzate of a matrix

 Solve the following differential equation

given Y(0) = 4

5. If Z = f(x,y) = xy

Find the maximum value for f(x,y) if x and y are constrained to sum to 1 (That is, x+y = 1). Solve the problem in two ways: by substitution and by using the Lagrangian multiplier method.

 Let Z = f(x,y) = 3x³-5y²- 225x + 70y + 23.

(i) Find the stationary points of z.

(ii) Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point.

An individual consumer consumes two commodities X₁ & X₂. The utility function is 

The price of commodity one is P₁ = Rs.3.00, the price of commodity two is P₂ = Rs.4.00, the individual's income per period is Rs.108. Determine the utility maximizing level of X₁ & X₂ and derive the demand curves for the two commodities.

 Consider the following two matrices
A =

B =  

(i) Find the rank of 'A' and 'B'

(ii) Show that (AB)⁻¹ = B⁻¹ A⁻¹

(iii) Show that (A⁻¹)⁻¹ = A

(iv) Show that (B⁻¹)⁻¹ = B

Fit a simple linear regression model on the data given in Question 2 for non-smokers only and obtain the followings:

(i) Diagonal of the hat matrix. Also, check the leverage points, if any.

(ii) Cook's Distances. Also, verify the influence points, if any.

A logistics company wants to predict the delivery efficiency score of its drivers based on two primary factors: years of driving experience and average hours of sleep per night. The data for 10 drivers are recorded as follows:

Based on the image provided, here is the data transcribed into a table format:

To investigate the effect of Age (in years) and Smoking Status (Group 1: Non-Smokers, Group 2: Smokers) on Systolic Blood Pressure (in mmHg), the following data were collected:

The txt from the table in the image is provided below:

(i) Fit a multiple regression model.

(ii) Obtain the individual regression models for Non-Smokers and Smokers separately based on the main model.

(iii) Test the significance of the overall fitted regression model at the 5% level of significance.

To investigate the effect of Age (in years) and Smoking Status (Group 1: Non-Smokers, Group 2: Smokers) on Systolic Blood Pressure (in mmHg), the following data were collected:

State whether the following statements are true or false and also give the reason in support of your answer.

(i) The model Y = BX + & is a linear model.

(ii) If total and residual sum of squares for a dataset are 124 and 15.5, respectively, then the coefficient of determination is 0.875.

(iii) We use Cook's distances to identify the autocorrelation of the error terms.

(iv) If the coefficient of determination for the auxiliary regression model of the explanatory variable X₁ is 0.8, the VIF for X₁ will be 5.

(v) We use baseline category logit models when we have ordered categories of a response variable.

Write a Python program using NumPy to
(i) Create following two matrices P and Q with following elements.
$
(ii) Compute the (a) Addition, (b) Multiplication, (c) Determinant and (d) Inverse of these matrices.
(iii) Combine the given matrices row-wise and determine the transpose of the combined matrix.
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What are different ways to create a NumPy array? Explain any two with examples.

Describe any two Python libraries used for data visualization.

Describe the while loop with an example program.

 Explain how file handling works in Python for text files. Mention different file modes, and a short description of read(), readline() and write().

Describe the key features of Spyder.

Write a Python program to accept a list of tuples (product_name, price) for 5 items from the user, then sort the list by price in descending order using a function.

Find and correct the error in the following code.

import numpy as np

arr = np.array[4, 5, 6]

print(arr)

Write the output of the following Python code.

nums = [10, 20, 30, 40, 50]

print(nums[-2])