Show that the equations 3x − 2y = 6, 6x − 4y = 10 are inconsistent

Show that the equations x + y + z = 6, x + 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

Evaluate \(\int { \frac { 1 }{ \sqrt { x+2 } -\sqrt { x-2 } } } dx\)

Evaluate ∫(2sin x − 5cos x)dx

Evaluate \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 3 } }{ \sin x } \) dx

Evaluate the following integrals:
ഽ\(\frac { dx }{ { e }^{ x }+6+{ 5e }^{ -x } } \)

If f'(x) = 8x³ -2x², f(2) = 1, find f(x)

Using Integration, find the area of the region bounded the line 2y + x = 8, the x axis and the lines x = 2, x = 4.

The marginal cost function of manufacturing x units of a commodity is 3x² - 2x + 8. If there is no fixed cost, find the total cost function?

Find the differential equation of the following
xy = c²

Write down the order and degree of the following differential equations.
\(\left( \frac { dy }{ dx } \right) ^{ 2 }-7\frac { d^{ 3 }y }{ { dx }^{ 3 } } +y\frac { { d }^{ 2 }y }{ dx^{ 2 } } +4\frac { dy }{ dx } \)- log x = 0

Solve: 3\(\frac { { d }^{ 2 }y }{ dx^{ 2 } } -5\frac { dy }{ dx } \)+ 2y = 0

Using Lagrange’s interpolation formula find a polynomial which passes through the points (0, –12), (1, 0), (3, 6) and (4,12).

Define random variable.

What do you understand by Mathematical expectation?

Determine whether the following is a probability distribution of a random variable X.

X	0	1	2
P(X)	0.6	0.1	0.2

In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses \(\frac{1}{2}\) mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random?

Mention the properties of binomial distribution.

What is an estimator?

The average score on a nationally administered aptitude test was 76 and the corresponding standard deviation was 8. In order to evaluate a state’s education system, the scores of 100 of the state’s students were randomly selected. These students had an average score of 72. Test at a significance level of 0.05 if there is a significant difference between the state scores and the national scores.

Write note on Fisher’s price index number.

What do you mean by process control?

Calculate the seasonal indices by the method of simple average for the following data.

Year	I quarter	II quarter	III quarter	IV quarter
1985	68	62	61	63
1986	65	58	66	61
1987	68	63	63	67

What is the Assignment problem?

For the given pay-off matrix, find the optimal decision under the minimax principle.