Find the rank of the matrix \(\left( \begin{matrix} 0 & -1 & 5 \\ 2 & 4 & -6 \\ 1 & 1 & 5 \end{matrix} \right) \)

The total cost of 11 pencils and 3 erasers is Rs. 64 and the total cost of 8 pencils and 3 erasers is Rs. 49. Find the cost of each pencil and each eraser by Cramer’s rule.

Find the rank of the matrix A =\(\left( \begin{matrix} 4 & 5 & 2 \\ 3 & 2 & 1 \\ 4 & 4 & 8 \end{matrix}\begin{matrix} 2 \\ 6 \\ 0 \end{matrix} \right) \)

 Evaluate \(\int \frac{a x^{2}+b x+c}{\sqrt{x}} d x\)

Evaluate \(\int { \frac { 7x-1 }{ { x }^{ 2 }-5x+6 } dx } \)

Evaluate  \(\int { } \)cos³ x dx

Integrate the following with respect to x.
xⁿ log x

Integrate the following with respect to x.
\(\frac { { x }^{ e-1 }+{ e }^{ x-1 } }{ { x }^{ e }+{ e }^{ x } } \)

Evaluate ഽ\(\frac { { x }^{ 3 }dx }{ \sqrt { x^{ 8 }+1 } } \) 

Integrate the following with respect to x
\(\frac { { x }^{ 3 } }{ \sqrt { { x }^{ 8 }-1 } } \)

Evaluate \(\int _{ 0 }^{ 1 }{ [{ e }^{ a \log x }+{ e }^{ x \log a }] } dx\)

Evaluate the following: f(x) = \(\begin{cases} cx, \\ 0, \end{cases}\begin{matrix} 0 < x < 1 \\ \text{otherwise} \end{matrix}\)

Evaluate the following integrals:
ഽ(x +1)² log x dx

In year 2000 world gold production was 2547 metric tons and it was growing exponentially at the rate of 0.6% per year. If the growth continues at this rate, how many tons of gold will be produced from 2000 to 2013? [e^0.078 = 1.0811)

A manufacture’s marginal revenue function is given by MR = 275 − x − 0.3x². Find the increase in the manufactures total revenue if the production is increased from 10 to 20 units.

Find the differential equation corresponding to y = ae^4x + be^−x where a, b are arbitrary constants.

Solve x \(\frac{dy}{dx}\) + 2y = x⁴

Evaluate ∆(log ax).

A second degree polynomial passes though the point (1,-1) (2,-1) (3,1) (4,5). Find the polynomial.

Let X be a discrete random variable with the following p.m.f
\(p(x) = \begin{cases}0.3 & \text { for } x =3 \\ 0.2, & \text { for } x = 5 \\ 0.3, & \text { for } x = 8 \\ 0.2, & \text { for} x = 10 \\ 0, & \text { otherwise } \\ \end{cases}\)
Find and plot the c.d.f. of X.

The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function.\(f(x)= \begin{cases}3e^{-3x} & x > 0 \\ 0, & \text { otherwise }\end{cases}\)
Find the expected life of the piece of equipment.

The probability that a student get the degree is 0.4 Determine the probability that out of 5 students
(i) one will be graduate
(ii) atleast one will be graduate

Assume that a drug causes a serious side effect at a rate of three patients per one hundred. What is the probability that atleast one person will have side effects in a random sample of ten patients taking the drug?

Find the sample size for the given standard deviation 10 and the standard error with respect of sample mean is 3.

Calculate three-yearly moving averages of number of students studying in a higher secondary school in a particular village from the following data.

Year	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004
Number of students	332	317	357	392	402	405	410	427	435	438

An Enquiry was made into the budgets of the middle class families in a city gave the following information.

Expenditure	Food	Rent	Clothing	Fuel	Rice
Price(2010)	150	50	100	20	60
Price(2011)	174	60	125	25	90
Weights	35	15	20	10	20

What changes in the cost of living have taken place in the middle class families of a city?

Solve the following assignment problem.

Obtain an initial basic feasible solution to the following transportation problem by using least- cost method.

A farmer wants to decide which of the three crops he should plant on his 100-acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high medium and low. His estimated profit for each is shown in the table.

Rainfall	Estimated Conditional Profit(Rs.)
	crop A	crop B	crop C
High	8000	3500	5000
Medium	4500	4500	5000
Low	2000	5000	4000

If the farmer wishes to plant only crop, decide which should be his best crop using
(i) Maximin
(ii) Minimax