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

At marina two types of games viz., Horse riding and Quad Bikes riding are available on hourly rent. Keren and Benita spent Rs. 780 and Rs. 560 during the month of May.

Name	Number of hours		Total amount spent (in Rs)
	Horse Riding	Quad Bike Riding
Keren	3	4	780
Benita	2	3	560

Find the hourly charges for the two games (rides). (Use determinant method).

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 2 & -1 & 1 \\ 3 & 1 & -5 \\ 1 & 1 & 1 \end{matrix} \right) \)

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

Evaluate \(\int \frac{x^{3}+5 x^{2}-9}{x+2} d x\)

Integrate the following with respect to x.
\(\frac { { e }^{ 3x }+{ e }^{ 5x } }{ { e }^{ x }+{ e }^{ -x } } \)

Evaluate \(\int { } \)x³e^xdx

Evaluate \(\int { \frac { dx }{ x\left( { x }^{ 3 }+1 \right) } } \)

Evaluate ഽ \(\frac { dx }{ 2+x-{ x }^{ 2 } } \)

Integrate the following with respect to x
\(\frac { 1 }{ \sqrt { { x }^{ 2 }-3x+2 } } \)

Evaluate the following:
\(\int _{ 1 }^{ 4 }{ f(x) } dx\) where f(x) = \(\begin{cases} 4x+3, \\ 3x+5, \end{cases}\begin{matrix} 1 & \le & x \\ 2 & \le & x \end{matrix}\begin{matrix} \le & 2 \\ \le & 4 \end{matrix}\)

Evaluate the following integrals:
\(\int _{ 0 }^{ 1 }{ \sqrt { x(x-1) } } \) dx

The marginal cost function MC = 2 + 5e^x Find C if C (0)=100

The marginal cost function of a product is given by \(\frac { dC }{ dx } \) = 100 −10x + 0.1x² where x is the output. Obtain the total and the average cost function of the firm under the assumption, that its fixed cost is Rs. 500.

Calculate consumer’s surplus if the demand function p = 122 − 5x − 2x² and x = 6

Find the differential equation of the following
y = cx + c − c³

Solve sec²x tan y dx + sec²y tan x dy = 0

Find the order and degree of the following differential equations.
\(\frac{d^{3} y}{d x^{3}}+3\left(\frac{d y}{d x}\right)^{3}+2 \frac{d y}{d x}=0\)

By constructing a difference table and using the second order differences as constant, find the sixth term of the series 8,12,19,29,42…

Using graphic method, find the value of y when x = 48 from the following data:

x	40	50	60	70
y	6.2	7.2	9.1	12

If you toss a fair coin three times, the outcome of an experiment consider as random variable which counts the number of heads on the upturned faces. Find out the probability mass function and check the properties of the probability mass function.

Consider a random variable X with probability density function \(f(x)= \begin{cases}4x^3 & \text { if } 0< x < 1 \\ 0, & \text { otherwise }\end{cases}\)
Find E(X) and V(X).

A person tosses a coin and is to receive Rs. 4 for a head and is to pay Rs. 2 for a tail. Find the expectation and variance of his gains.

Suppose A and B are two equally strong table tennis players. Which of the following two events is more probable:
(a) A beats B exactly in 3 games out of 4 or
(b) A beats B exactly in 5 games out of 8 ?

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?

If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.

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

A sample of 100 items, draw from a universe with mean value 4 and S.D 3, has a mean value 63.5. Is the difference in the mean significant at 0.05 level of significance?

Explain the method of fitting a straight line.

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?

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

Here O_i and D_j denote i^th origin and j^th destination respectively.

A business man has three alternatives open to him each of which can be followed by any of the four possible events. The conditional pay offs for each action - event combination are given below:

Alternative	Pay – offs (Conditional events)
	A	B	C	D
X	8	0	-10	6
Y	-4	12	18	-2
A3	14	6	0	8

Determine which alternative should the businessman choose, if he adopts the maximin principle.

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