Find the rank of each of the following matrices.
\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 1 & -1 \\ 3 & -6 \end{matrix} \right) \)

Evaluate \(\int { \left( { x }^{ 3 }+7 \right) \left( x-4 \right) dx } \)

Integrate the following with respect to x.
\(\sqrt { 1-\sin2x } \)

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

he cost of over haul of an engine is ₹10,000 The operating cost per hour is at the rate of 2x − 240 where the engine has run x km. Find out the total cost if the engine run for 300 hours after overhaul.

Solve \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -4\frac { dy }{ dx } +5y\) = 0

Find the order and degree of the following differential equation
\(\frac { { d }^{ 2 }y }{ { dx }^{ 3 } } -3{ \left( \frac { dy }{ dx } \right) }^{ 6 }+2y={ x }^{ 2 }\)

Evaluate \({ \Delta }^{ 2 }\left( \frac { 1 }{ x } \right) \) by taking ‘1’ as the interval of differencing.

Prove that
E ∇ = Δ =  ∇E

The discrete random variable X has the probability function

X	1	2	3	4
P(X=x)	k	2k	3k	4k

Show that k = 0.1.

Describe what is meant by a random variable.

Write the conditions for which the poisson distribution is a limiting case of binomial distribution.

Write down the conditions in which the Normal distribution is a limiting case of binomial distribution.

Define parameter.

Define alternative hypothesis.

Explain cyclic variations.

The following table gives the number of small-scale units registered with the Directorate of Industries between 1985 and 1991. Show the growth on a trend line by the free hand method.

Years	1985	1986	1987	1988	1989	1990	1991	1992
No. of units (in‘000)	10	22	36	62	55	40	34	50

What is the Assignment problem?

What is the difference between Assignment Problem and Transportation Problem?

Find the rank of the matrix \(\left( \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right) \)

\(\int \frac{e^{5 \log _{e} x}-e^{+\log _{1} x}}{e^{3 \log _{e} x}-e^{2 \log _{1} x}} d x\)

Find the area under the curve y = \(4 x^{2}-8 x+6\) bounded by the y-axis, x-axis and the ordinate x = 2

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

Calculate a forward difference table for the following data

x	20	30	40	50
y	51	43	34	24

 

If \(\mathrm{F}(x)=\frac{1}{\pi}\left(\frac{\pi}{2}+\tan ^{-1} x\right)-\infty<x<\infty\) distribution function of a continuous variable X, find \(\mathrm{P}(0 \leq x \leq 1)\)

It is given that 3% of the electric bulbs manufactured by a company are defective find the probability that a simple of 100 bulbs will contain
(i) no defective
(ii) exactly one defective (e^-3 = 0.0498)

State any 2 demerits of simple random sampling.

Calculate the 3-yearlymoving averages of the production figures (in tonnes) for the following data.

Year	1973	1974	1975	1976	1977	1978	1979	1980	1981	1982	1983	1984	1985	1986	1987
Production	15	21	30	36	42	46	50	56	63	70	74	82	90	95	102

Consider the following pay-off (profit) matrix action, states

Action	States
	B1	B2
A1	8	6
A2	9	2
A3	6	4

Determine the best action using maximin principle.