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Examine whether the following statement pattern is a tautology, a contradiction or a contingency.
(p ∧ ~ q) → (~ p ∧ ~ q)
Concept: Statement Patterns and Logical Equivalence
Write the converse, inverse, and contrapositive of the following statement.
"If it snows, then they do not drive the car"
Concept: Statement Patterns and Logical Equivalence
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Concept: Increasing and Decreasing Functions
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Concept: Increasing and Decreasing Functions
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Concept: Increasing and Decreasing Functions
Find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as Ec = (0.0003) I2 + (0.075) I ; When I = 1000.
Concept: Application of Derivatives to Economics
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
Concept: Methods of Integration: Integration Using Partial Fractions
Find the area of the region bounded by the parabola y2 = 4x and the line x = 3.
Concept: Area Under Simple Curves
If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`
Concept: Application of Differential Equations
Obtain the differential equation by eliminating arbitrary constants from the following equation:
y = Ae3x + Be–3x
Concept: Formation of Differential Equation by Eliminating Arbitary Constant
The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.
| Year | 1962 | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 |
| Production (million barrels) |
0 | 0 | 1 | 1 | 2 | 3 | 4 | 5 |
| Year | 1970 | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 | |
| Production (million barrels) |
6 | 7 | 8 | 9 | 8 | 9 | 10 |
- Obtain trend values for the above data using 5-yearly moving averages.
- Plot the original time series and trend values obtained above on the same graph.
Concept: Measurement of Secular Trend
The optimum value of the objective function of LPP occurs at the center of the feasible region.
Concept: Introduction of Linear Programming
Four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost matrix is given below:
| Machines | Places | ||||
| A | B | C | D | E | |
| M1 | 4 | 6 | 10 | 5 | 6 |
| M2 | 7 | 4 | – | 5 | 4 |
| M3 | – | 6 | 9 | 6 | 2 |
| M4 | 9 | 3 | 7 | 2 | 3 |
Find the optimal assignment schedule.
Concept: Special Cases of Assignment Problem
Find the sequence that minimizes the total elapsed time to complete the following jobs in the order AB. Find the total elapsed time and idle times for both the machines.
| Job | I | II | III | IV | V | VI | VII |
| Machine A | 7 | 16 | 19 | 10 | 14 | 15 | 5 |
| Machine B | 12 | 14 | 14 | 10 | 16 | 5 | 7 |
Concept: Types of Sequencing Problem
Use quantifiers to convert the following open sentences defined on N, into a true statement.
n2 ≥ 1
Concept: Quantifier and Quantified Statements in Logic
Which of the following is not a statement?
Concept: Truth Value of Statement
Write the converse, inverse, contrapositive of the following statement.
If a man is bachelor, then he is happy.
Concept: Statement Patterns and Logical Equivalence
The false statement in the following is ______.
Concept: Statement Patterns and Logical Equivalence
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Concept: Derivatives of Implicit Functions
If y = x log x, then `(d^2y)/dx^2`= ______.
Concept: The Concept of Derivative >> Derivatives of Logarithmic Functions
