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

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

Integrate \(\int{\sqrt{1-sin2x}dx }\)

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

A company receives a shipment of 200 cars every 30 days. From experience it is known that the inventory on hand is related to the number of days. Since the last shipment, I x( )= − 200 0 2. x . Find the daily holding cost for maintaining inventory for 30 days if the daily holding cost is ₹3.5

The marginal revenue ‘y’ of output ‘q’ is given by the equation \(\frac { dy }{ dq } =\frac { { q }^{ 2 }+{ 3 }y^{ 2 } }{ 2qy } \). Find  the total Revenue function when output is 1 unit and Revenue is Rs. 5.

(D² − 3D + 2)y = e^3x which shall vanish for x = 0 and for x = log 2

The following data are taken from the steam table

Temperature C0	140	150	160	170	180
Pressure kg f / cm2	3.685	4.854	6.302	8.076	10.225

Find the pressure at temperature t = 175⁰

The area A of circle of diameter ‘d’ is given for the following values

D	80	85	90	95	100
A	5026	5674	6362	7088	7854

Find the approximate values for the areas of circles of diameter 82 and 91 respectively 

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 marks obtained in a certain exam follow normal distribution with mean 45 and SD 10. If 1,300 students appeared at the examination, calculate the number of students scoring
(i) less than 35 marks and
(ii) more than 65 marks.

The annual salaries of employees in a large company are approximately normally distributed with a mean of Dallor. 50,000 and a standard deviation of Dallor.20,000.
(a) What percent of people earn less than Dallor.40,000?
(b) What percent of people earn between Dallor.45,000 and Dallor.65,000?
(c) What percent of people earn more than Dallor.70,00

Using the following random number table,

Tippet’s random number table
2952	6641	3992	9792	7969	5911	3170	5624
4167	9524	1545	1396	7203	5356	1300	2693
2670	7483	3408	2762	3563	1089	6913	7991
0560	5246	1112	6107	6008	8125	4233	8776
2754	9143	1405	9025	7002	6111	8816	6446

Draw a sample of 10 children with their height from the population of 8,585 children as classified here under.

Height (cm)	105	107	109	111	113	115	117	119	121	123	125
Number of children	2	4	14	41	83	169	394	669	990	1223	1329
Height(cm)	127	129	131	133	135	137	139	141	143	145
No. of children	1230	1063	646	392	202	79	32	16	5	2

The average score on a nationally administered aptitude test was 76 and the corresponding standard deviation was 8. In order to evaluate a state’s education system, the scores of 100 of the state’s students were randomly selected. These students had an average score of 72. Test at a significance level of 0.05 if there is a significant difference between the state scores and the national scores.

The following table shows the number of salesmen working for a certain concern:

Year	1992	1993	1994	1995	1996
No. of salesmen	46	48	42	56	52

Use the method of least squares to fit a straight line and estimate the number of salesmen in 1997.

A machine is set to deliver packets of a given weight. Ten samples of size five each were recorded. Below are given relevant data:

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset {-}{X}\)	15	17	15	18	17	14	18	15	17	16
R	7	7	4	9	8	7	12	4	11	5

Calculate the control limits for mean chart and the range chart and then comment on the state of control. (conversion factors for n = 5, A₂ = 0.58, D₃ = 0 and D₄ = 2.115

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.

Find the optimal solution for the assignment problem with the following cost matrix.

Find k if the equation x + 2y 3 = -2, 3x - y - 2z = 1 and 2x + 3y 5z = k are consistent

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

Find the area of the region \(\left\{(x, y) ; x^{2} \leq y \leq|x|\right\} .\)

The net profit p and quantity x satisfy the differential equation \(\frac{d p}{d x}=\frac{2 p^{3}-x^{3}}{3 x p^{2}}\). Find the relationship between net profit and demand given that p = 20, when x = 10.

From the data find the number of students whose height is between 80 cm and 90 cm.

Height in cm's	40-60	60-80	80-100	100-120	120-140
No. of students y	240	120	100	70	50

Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the value X has the following form, where k is some unknown constant \(p(X=x)= \begin{cases}0.1 & \text { if } x=0 \\ k x & \text { if } x=1 \text { or } 2 \\ k(5-x) & \text { if } x=3 \text { or } 4 \\ 0 & \text { otherwise }\end{cases}\)
(i) Find the value of k 
(ii) What is the probability that you study atleast 2 hours? 
(iii) Exactly 2 hours
(iv) At most 2 hours

What is the probability that Z
(a) lies between 0 and 1.83
(b) is greater than 1.54
(c) is greater than -0.86
(d) lies between 0.43 and 1.12
(e) is less than 0.77

A sample of 400 students is found to have a mean height of 171.38 cms. Can it reasonably be regarded as a sample from a large population with mean height of 171.17 cms and standard deviation of 3.3 cms (Test at 5% level)

Calculate Fisher's ideal index from the following data and verify that it satisfies both time reversal and factor reversal test

Commodity	Price		Quantity
	1985	1986	1985	1986
A	8	20	50	60
B	2	6	15	10
C	1	2	20	25
D	2	5	10	8
E	1	5	40	30

Solve the following assignment problem.

1. In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides the information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity

   Commodity Variety	Variety			Total weight
   	I	II	III
   A	1	2	3	11
   B	2	4	5	21
   C	3	5	6	27

   Find the weights assigned to the three varieties by using Cramer’s Rule.