Show that the equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.

Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

The price of three commodities X, Y and Z are x, y and z respectively Mr. Anand purchases 6 units of Z and sells 2 units of X and 3 units of Y. Mr. Amar purchases a unit of Y and sells 3 units of X and 2units of Z. Mr. Amit purchases a unit of X and sells 3 units of Y and a unit of Z. In the process they earn Rs. 5,000/-, Rs. 2,000/- and Rs. 5,500/- respectively. Find the prices per unit of three commodities by rank method.

An automobile company uses three types of Steel S₁, S₂ and S₃ for providing three different types of Cars C₁, C₂ and C₃. Steel requirement R (in tonnes) for each type of car and total available steel of all the three types are summarized in the following table.

Types of Steel	Types of Car			Total Steel available
	C1	C2	C3
S1	3	2	4	28
S2	1	1	2	13
S3	2	2	2	14

Determine the number of Cars of each type which can be produced by Cramer’s rule.

Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

Solve the equations x + 2y + z = 7, 2x − y + 2z = 4, x + y − 2z = −1 by using Cramer’s rule

Solve the following equation by using Cramer’s rule
x + 4y + 3z = 2, 2x−6y + 6z = −3, 5x− 2y + 3z = −5

Integrate the following with respect to x.
\(\frac { { 4x }^{ 2 }+2x+6 }{ { \left( x+1 \right) }^{ 2 }\left( x-3 \right) } \)

Integrate the following with respect to x.
e^x (1+ x) log(xe^x)

Evaluate \(\int _{ -1 }^{ 1 }{ { ({ x }^{ 3 }+{ 3x }^{ 2 }) }^{ 3 } } \) (x² +  2x)dx

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

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

Evaluate the following integrals as the limit of the sum:
\(\int _{ 0 }^{ 1 }{ (x+4) } \)dx

Evaluate the following integrals:
\(\int _{ 0 }^{ 3 }{ \frac { xdx }{ \sqrt { x+1 } +\sqrt { 5x+1 } } } \)

Calculate the area bounded by the parabola y² = 4ax and its latus rectum.

The marginal cost C'(x) and marginal revenue R'(x) are given by C'(x) = 50 + \(\frac{x}{50}\) and R'(x) = 60. The fixed cost is Rs. 200. Determine the maximum profit

When the Elasticity function is \(\frac { x }{ x-2 } \). Find the function when x = 6 and y = 16.

The marginal cost of production of a firm is given by C'(x) = 5 + 0.13x, the marginal revenue is given by R'(x) = 18 and the fixed cost is Rs. 120. Find the profit function.

The demand equation for a product is p_d = 20 − 5x and the supply equation is p_s = 4x + 8. Determine the consumer’s surplus and producer’s surplus under market equilibrium.

Solve : x - y \(\frac { dx }{ dy } =a\left( { x }^{ 2 }+\frac { dx }{ dy } \right) \)

If the marginal cost of producing x shoes is given by (3xy + y²)dx + (x² + xy)dy = 0 and the total cost of producing a pair of shoes is given by Rs. 12. Then find the total cost  function.

Solve the following homogeneous differential equations.
\(x\frac { dy }{ dx } -y=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

Solve (x² + 1)\(\frac { dy }{ dx } \) + 2xy = 4x² 

Solve the following:
\(\frac{d y}{d x}+\frac{3 x^{2}}{1+x^{3}} y=\frac{1+x^{2}}{1+x^{3}}\)

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

Solve the following differential equations (3D² + D − 14)y = 13e^2x

Solve (x² + y²)dx + 2xy dy = 0

Using Newton’s formula for interpolation estimate the population for the year 1905 from the table:

Year	1891	1901	1911	1921	1931
Population	98.752	1,32,285	1,68,076	1,95,690	2,46,050

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

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

Using Lagrange’s interpolation formula find a polynomial which passes through the points (0, –12), (1, 0), (3, 6) and (4,12).

The distribution of a continuous random variable X in range (–3, 3) is given by p.d.f.
\(f(x)=\left\{\begin{array}{l} \frac{1}{16}(3+x)^{2},-3 \leq x \leq-1 \\ \frac{1}{16}\left(6-2 x^{2}\right),-1 \leq x \leq 1 \\ \frac{1}{16}(3-x)^{2}, 1 \leq x \leq 3 \end{array}\right.\)
Verify that the area under the curve is unity.

Determine the mean and variance of the random variable X having the following probability distribution.

X=x	1	2	3	4	5	6	7	8	9	10
P(x)	0.15	0.10	0.10	0.01	0.08	0.01	0.05	0.02	0.28	0.20

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).

If the average rain falls on 9 days in every thirty days, find the probability that rain will fall on atleast two days of a given week.

If X is a normal variate with mean 30 and SD 5. Find the probabilities that 
(i) 26 ≤ X ≤ 40
(ii) X > 45

A bank manager has observed that the length of time the customers have to wait for being attended by the teller is normally distributed with mean time of 5 minutes and standard deviation of 0.6 minutes. Find the probability that a customer has to wait
(i) for less than 6 minutes
(ii) between 3.5 and 6.5 minutes

Out of 750 families with 4 children each, how many families would be expected to have
(i) atleast one boy
(ii) atmost 2 girls
(iii) and children of both sexes? Assume equal probabilities for boys and girls.

A car hiring firm has two cars. The demand for cars on each day is distributed as a Poisson variate, with mean 1.5. Calculate the proportion of days on which
(i) Neither car is used
(ii) Some demand is refused

In a distribution 30% of the items are under 50 and 10% are over 86. Find the mean and standard deviation of the distribution.

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?

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

An auto company decided to introduce a new six cylinder car whose mean petrol consumption is claimed to be lower than that of the existing auto engine. It was found that the mean petrol consumption for the 50 cars was 10 km per litre with a standard deviation of 3.5 km per litre. Test at 5% level of significance, whether the claim of the new car petrol consumption is 9.5 km per litre on the average is acceptable.

Explain in detail about simple random sampling with a suitable example.

A sample of 100 students are drawn from a school. The mean weight and variance of the sample are 67.45 kg and 9 kg. respectively. Find
(a) 95% and
(b) 99% confidence intervals for estimating the mean weight of the students.

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

Construct Fisher’s price index number and prove that it satisfies both Time Reversal Test and Factor Reversal Test for data following data.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
Rice	40	5	48	4
Wheat	45	2	42	3
Rent	90	4	95	6
Fuel	85	3	80	2
Transport	50	5	65	8
Miscellaneous	65	1	72	3

Construct the cost of living index number for 2011 on the basis of 2007 from the given data using family budget method.

Commodities	Price		Weights
	2007	2011
A	350	400	40
B	175	250	35
C	100	115	15
D	75	105	20
E	60	80	25

The sales of a commodity in tones varied from January 2010 to December 2010 as follows:

In year 2010	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Sales (in tones)	280	240	270	300	280	290	210	200	230	200	230	210

Fit a trend line by the method of semi-average.

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

Construct the cost of living Index number for 2015 on the basis of 2012 from the following data using family budget method.

Commodity	Price		Weight
	2012	2015
Rice	250	280	10
Wheat	70	280	5
Corn	150	170	6
Oil	25	35	4
Dhal	85	90	3

Construc \(\overset {-}{X}\) and R charts for the following data:

Sample Number	Observations
1	32	36	42
2	28	32	40
3	39	52	28
4	50	42	31
5	42	45	34
6	50	29	21
7	44	52	35
8	22	35	44

( Given for n = 3, A₂ = 0.58,D₃ = 0 and D₄ = 2.115)

Fit a straight line trend by the method of least squares to the following data.

Year	1980	1981	1982	1983	1984	1985	1986	1987
Sales	50.3	52.7	49.3	57.3	56.8	60.7	62.1	58.7

The following data gives the average life(in hours) and range of 12 samples of 5 lamps each. The data are

Sample No	1	2	3	4	5	6
Sample Mean	1080	1390	1460	1380	1230	1370
Sample Range	410	670	180	320	690	450
Sample No	7	8	9	10	11	12
Sample Mean	1310	1630	1580	1510	1270	1200
Sample Range	380	350	270	660	440	310

Construct control charts for mean and range. Comment on the control limits.

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?