Find the rank of the matrix \(\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right] \)

Solve: x + 2y = 3 and 2x + 4y = 6 using rank method.

For what value of x, the matrix
\(A=\left| \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right| \) is singular?

Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)

Evaluate ∫ tan²x dx

Evaluate \(\int { \frac { 2+3cosx }{ { sin }^{ 2 }x } } dx\)

If \(\int _{ 0 }^{ a }{ { 3x }^{ 2 } } dx=8\) find the value of a

Find the area of the region bounded by the parabola x² = 4y, y = 2, y = 4 and the y-axis.

Find the area under the curve y = 4x - x² included between x = 0, x = 3 and the X-axis.

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?

If the marginal revenue for a commodity is MR = 9 - 6x² + 2x, find the total revenue function.

The marginal cost at a production level of x units is given by C '(x) = 85 +\(\frac{375}{x^2}\). Find the cost of producing 10 in elemental units after 15 units have been produced?

The marginal cost function is MC = \(\frac{100}{x}\). Find the cost function C(x) if C(16) = 100.

Find the consumer's surplus for the demand function p = 25 - x -x² when P_o = 19

Find the producer's surplus for the supply function p = x² + x + 3 when x_o = 4

Find the differential equation for y = mx + \(\frac { a }{ m } \) where m is arbitrary constant.

Form the differential equation of family of rectangular hyperbolas whose asymptotes are the Co-ordinate axes.

Solve: x dy +y dx = 0

Solve: \(\frac { dy }{ dx } \)+ ay = e^x (where a ≠ -1)

The change in the cost of ordering and holding C as quantity q is given by \(\frac { dC }{ dq } =a-\frac { c }{ q } \) where a is a Constanst. Find C as a function of q.

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

Solve: (D²-6D+25)y = 0

Find the missing term from the following data

x	1	2	3	4
f(x)	100	-	126	157

Find the second order backward differences of f(x).

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

An unbiased die is rolled. If the random variable X is defined as
X(w) = {1, the outcome w is an even number    
{0, if the outcome w is an odd number
Find the probability distribution of X.

Two eggs are drawn at random without replacement from a bag containing two bad eggs and eight good eggs. Find the probability of getting two bad eggs?

Verify whether \(f(x)=\begin{cases} \frac { 2x }{ 9 } ,\quad 0\le x\le \\ 0,\quad elsewhere \end{cases}\) is a probability density function

A continuous random variable. X has the p.d.f. defined by \(f(x)=\left\{\begin{array}{l} C e^{-a x}, \quad 0<x<\infty \\ 0, \quad \text { elsewhere } \end{array}\right.\) Find the value of C if a> 0

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?

In a gambling game a man wins Rs. 10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

Find the mean for the probability density function \(f(x)=\begin{cases} \frac { 1 }{ 24 } ,-12\le x\le 12 \\ 0,\quad otherwise \end{cases}\)

If the mean of the binomial distribution with 9 trial is 6, then find the variance.

If the mean of the binomial distribution is 20 and standard deviation is 4, then find the number of events.

Suppose X is a binomial variate X ~ B (5, p) and P(X = 2) = P(X = 3), then find p.

If 10 coins are tossed, find the probability that exactly 5 heads appears.

In a packet of 50 pens, 10 are defective, 10 pens are selected at random. What is the probability that atleast one is defective.

The random variable X has the normal distribution f(x) = \(C{ e }^{ -\left( \frac { x-100 }{ 50 } \right) ^{ 2 } }\), then find the value of C.

The probability of the happening of an event X is 0.002 in an experiment. If an experiment is reported 1000 times, find the probability that the event X happens exactly twice? (e^-2 = 0.1353)

Out of 1000 T.V. viewers, 320 watched a particular programme. Calculate the standard error.

Out of 1500 school students, a sample of 150 selected to test the accuracy of solving a problem in B.M. and of them 10 did a mistake. Calculate the standard error of sample proportion.

The income distribution of the population of a village has a mean of Rs. 6000 and a variance of Rs. 32,400. Could a sample of 64 persons with a mean income of Rs. 5950 belong to this population. (Test at 1% level of significance).

Using the method ofleast squares, fit a straight line trend for Σx = 10, Σy = 16.9, Σx² = 30, Σxy = 47.4 and n = 7.

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

Construct the cost of living index for 2003 on the basis of 2000 from the following data using family budget method.

Item	Price(Rs.)		Weights
Food	2000	2003	30
Rent	200	280	30
Clothing	150	120	20
Fuel & lighting	50	100	10
Miscellaneous	100	200	20

The following data shows the value of sample mean (\(\bar{X}\)) and the range R for 10 samples of size 5 each. Calculate the control limits for : mean chart and range chart.

Sample No.	1	2	3	4	5	6	7	8	9	10
Mean \(\bar{X}\)	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0
Range	7	4	8	5	7	4	8	4	7	9

(Given for n = 5, A₂ = .577, D₃ = 0, D₄ = 2.115)

Obtain the initial solution for the following problem using north-west corner rule.

Determine an initial basic feasible solution to the following transportation problem using feast cost method.

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

The following is the pay-off matrix (in rupees) for three strategies and three states of nature. Select a strategy using maximin principle.