Find k, if the equations x + y + z = 7,  x + 2y + 3z = 18,  y + kz = 6 are inconsistent

Show that the following system of equations have unique solution:
x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6 by rank method.

The price of 3 Business Mathematics books, 2 Accountancy books and one Commerce book is Rs. 840. The price of 2 Business Mathematics books, one Accountancy book and one Commerce book is Rs. 570. The price of one Business Mathematics book, one Accountancy book and 2 Commerce books is Rs. 630. Find the cost of each book by using Cramer’s rule.

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 45% of those who already subscribe will subscribe again while 30% of those who do not now subscribe will subscribe. On the last letter, it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

Find k if the equations x + y + z = 1, 3x − y − z = 4, x+ 5y + 5z = k are inconsistent.

Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

Integrate the following with respect x
\(\frac { 3x+2 }{ \left( x-2 \right) \left( x-3 \right) } \)

Evaluate \(\int\left[\frac{1}{\log x}-\frac{1}{(\log x)^{2}}\right] d x\)

Evaluate \(\int _{ 2 }^{ 3 }{ \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } } dx\)

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 1 }{ (x+1)(x+2) } } dx\)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sin x }{ \sin x+\cos x } } \) dx

Evaluate the integral as the limit of a sum: \(\int _{ 1 }^{ 2 }{ (2x+1) } dx\)

Evaluate the following integrals as the limit of the sum:
\(\int _{ 0 }^{ 1 }{ { x }^{ 2 } } dx\)

Using integration find the area of the circle whose center is at the origin and the radius is a units.

A firm has the marginal revenue function given by MR = \(\frac { a }{ { (x+b) }^{ 2 } } \) - c where x is the output and a, b, c are constants. Show that the demand function is given by \(x=\frac { a }{ b(p+c) } -b\).

Elasticity of a function \(\frac{Ey}{Ex}\) is given by \(\frac{Ey}{Ex}\) = \(​​\frac { -7x }{ (1-2x)(2+3x) } \). Find the function when x = 2, y = \(\frac{3}{8}\)

A firm’s marginal revenue function is MR = 20e^-x/10 \(\left( 1-\frac { x }{ 10 } \right) \). Find the corresponding demand function.

Find the consumer’s surplus and producer’s surplus for the demand function p_d = 25 − 3x and supply function p_s = 5 + 2x.

Solve 3e^xtan ydx +(1 + e^x)sec²ydy = 0 given y(0) = \(\frac { \pi }{ 4 } \)

Solve the differential equation \(\frac { dy }{ dx } =\frac { x-y }{ x+y } \)

Solve the following homogeneous differential equations.
\((x-y)\frac { dy }{ dx } =x+3y\).

Solve the following:
\(\frac { dy }{ dx } \)+ ytan x = cos³ x

Suppose that \({ Q }_{ d }=30-5P+2\frac { dp }{ dt } +\frac { { d }^{ 2 }P }{ { dt }^{ 2 } } \) and Q_s = 6 + 3P. Find the equilibrium price for market clearance.

Evaluate \(\Delta \)\(\left[ \frac { 5x+12 }{ { x }^{ 2 }+5x+6 } \right] \) by taking ‘1’ as the interval of differencing.

Using Lagrange’s interpolation formula find y(10) from the following table:

x	5	6	9	11
y	12	13	14	16

Using interpolation estimate the output of a factory in 1986 from the following data

Year	1974	1978	1982	1990
Output in 1000 tones	25	60	80	170

Find f(0.5) if f(−1) = 202, f (0)= 175, f(1) = 82 and f(2) = 55

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
\(f(x)=\left\{\begin{array}{l} Ax,for \ 0≤x10 \\ A(20−x),for \ 10 ≤x< 20 \\ 0,\quad \quad \quad otherwise \end{array}\right.\)
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function\(F(x)\begin{cases} 0,\quad ​​​​\text{for}\quad x<0 \\ \frac { 1 }{ 2 } ,\quad ​​​​\text{for}\quad 0\le x<1 \\ 0,\quad ​​​​\text{for}\quad 1\le x<2\quad \\ \frac { 1 }{ 4 } ,\quad ​​​​\text{for}\quad 2\le x<4 \\ 0,\quad ​​​​\text{for}\quad x\ge 4 \end{cases}\)
(a) Is the distribution function continuous? If so, give its probability density function?
(b) What is the probability that a person will have to wait
(i) more than 3 minutes,
(ii) less than 3 minutes and
(iii) between 1 and 3 minutes?

The probability density function of a continuous random variable X is
\(f(x)=\left\{\begin{array}{l} a+b x^{2}, 0 \leq x \leq 1 \\ 0, \text { otherwise } \end{array}\right.\)
where a and b are some constants. Find
(i) a and b if E(X)\(\frac{3}{5}\)
(ii) Var(X).

What is the probability that a standard normal variate Z will be
(i) greater than 1.09
(ii) less than -1.65
(iii) lying between -1.00 and 1.96
(iv) lying between 1.25 and 2.75

900 light bulbs with a mean life of 125 days are installed in a new factory. Their length of life is normally distributed with a standard deviation of 18 days. What is the expected number of bulbs expire in less than 95 days?

An experiment succeeds twice as often as it fails, what is the probability that in next five trials there will be
(i) three successes and
(ii) at least three successes

Time taken by a construction company to construct a flyover is a normal variate with mean 400 labour days and standard deviation of 100 labour days. If the company promises to construct the flyover in 450 days or less and agree to pay a penalty of Rs. 10,000 for each labour day spent in excess of 450. What is the probability that
(i) the company pays a penalty of atleast Rs. 2,00,000?
(ii) the company takes at most 500 days to complete the flyover?

A sample of 100 measurements at breaking strength of cotton thread gave a mean of 7.4 and a standard deviation of 1.2 gms. Find 95% confidence limits for the mean breaking strength of cotton thread.

The mean breaking strength of cables supplied by a manufacturer is 1,800 with a standard deviation 100. By a new technique in the manufacturing process it is claimed that the breaking strength of the cables has increased. In order to test this claim a sample of 50 cables is tested. It is found that the mean breaking strength is 1,850. Can you support the claim at 0.01 level of significance.

Calculate the seasonal index for the monthly sales of a product using the method of simple averages.

Months	Jan	Feb	Mar	Apr	May	June	July	Aug	Sep	Oct	Nov	Dec
Year
2001	15	41	25	31	29	47	41	19	35	38	40	30
2002	20	21	27	19	17	25	29	31	35	39	30	44
2003	18	16	20	28	24	25	30	34	30	38	37	39

Calculate Fisher’s price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
Rice	10	5	11	6
Wheat	12	6	13	4
Rent	14	8	15	7
Fuel	16	9	17	8
Transport	18	7	19	5
Miscellaneous	20	4	21	3

Calculate the cost of living index number by consumer price index number for the year 2016 with respect to base year 2011 of the following data
\(\begin{array}{|c|c|c|c|} \hline & {\text { Price }} & \\ \begin{array}{c} \text { Commodities } \\ \text { } \end{array} & \begin{array}{c} \text { Base } \\ \text { year } \end{array} & \begin{array}{c} \text { Current } \\ \text { year } \end{array} & \text { Quantity } \\ \hline \text { Rice } & 32 & 48 & 25 \\ \hline \text { Sugar } & 25 & 42 & 10 \\ \hline \text { Oil } & 54 & 85 & 6 \\ \hline \text { Coffee } & 250 & 460 & 1 \\ \hline \text { Tea } & 175 & 275 & 2 \\ \hline \end{array}\)

The annual production of a commodity is given as follows :
\(\begin{array}{|c|c|} \hline \text { Year } & \text { Production (in tones) } \\ \hline 1995 & 155 \\ \hline 1996 & 162 \\ \hline 1997 & 171 \\ \hline 1998 & 182 \\ \hline 1999 & 158 \\ \hline 2000 & 180 \\ \hline 2001 & 178 \\ \hline \end{array}\)
Fit a straight line trend by the method of least squares.

Calculate price index number for 2005 by
(a) Laspeyre’s
(b) Paasche’s method

Commodity	1995		2005
	Price	Quantity	Price	Quantity
A	5	60	15	70
B	4	20	8	35
C	3	15	6	20

Calculate Fisher’s index number to the following data. Also show that it satisfies Time Reversal Test.

Commodity	Price in Rupees per unit		Number of units
	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)
Food	40	12	65	14
Fuel	72	14	78	20
Clothing	36	10	36	15
Wheat	20	6	42	4
Others	46	8	52	6

Ten samples each of size five are drawn at regular intervals from a manufacturing process. The sample means ( \(\overset{-}{X}\) ) and their ranges (R ) are given below:

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset {-}{X}\)	49	45	48	53	39	47	46	39	51	45
R	7	5	7	9	5	8	8	6	7	6

Calculate the control limits in respect of \(\overset {-}{X}\) chart. (Given A₂ = 0.58, D₃ = and D₄ = 2.115)  Comment on the state of control.

In a certain bottling industry the quality control inspector recorded the weight of each of the 5 bottles selected at random during each hour of four hours in the morning.

Time	Weights in ml
8:00 AM	43	41	42	43	41
9:00 AM	40	39	40	39	44
10:00 AM	42	42	43	38	40
11:00 AM	39	43	40	39	42

From the following data, calculate the control limits for the mean and range chart.

Sample No.	1	2	3	4	5	6	7	8	9	10
Sample Observations	50	51	50	48	46	55	45	50	47	56
	55	50	53	53	50	51	48	56	53	53
	52	53	48	50	44	56	53	54	49	55
	49	50	52	51	48	47	48	53	52	54
	54	46	47	53	47	51	51	57	54	52

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below :

How should the tasks be allocated to subordinates so as to minimize the total man-hours?

A natural truck-rental service has a surplus of one truck in each of the cities 1,2,3,4,5 and 6 and a deficit of one truck in each of the cities 7,8,9,10,11 and 12. The distance(in kilometers) between the cities with a surplus and the cities with a deficit are displayed below:

How should the truck be dispersed so as to minimize the total distance travelled?