Use the penalty (Big M) method to solve the following LP problem:
Minimise Z = 5x₁+ 3x₂
Subject to the constraints
2x₁+ 4x₂ ≤ 12
2x₁+ 2x₂ = 10
5x₁+ 2x₂ ≥ 10
x₁, x₂ ≥ 0.

A firm wants to know whether there is any linear relationship between the sales (X) and its yearly revenue (Y). The records for 10 years were examined and the following results were obtained:  ∑ X =  265, ∑Y = 27.73, ∑SS_x = 285.6, ∑ SS_Y = 6.978 and SS_XY = 57.456.
(a) Fit a regression line taking Y as the dependent variable and X as the independent variable.
(b) Test whether the sales have any effect on revenue at 5% level of significance.
(c) Comment on the goodness of fit of the regression line.

Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C₀, C₁, C₂, C₃ and C₄.
(b) Calculate auto-correlation coefficients r₁, r₂, r₃ and r₄.
(c) Plot the correlogram.

Using the graphical method to minimise the time required to process Job 1 and Job 2 on five machines A, B, C, D and E, find the minimum elapsed times an idle times to complete both jobs.

Four professors are capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below:

The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:

State whether the following statements are True or False. Give reasons in support of your answers.
(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns
(destinations) is feasible if number of possible allocations are 8.
(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.
(c) If the basic solutions for a system of equations are (- 2, 0, 1), (0, 1, 3), (- 2, 3, 0), then only (0, 1, 3) is feasible.
(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.
(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.