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

A total of Rs. 8,500 was invested in three interest earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was Rs. 380 and the amount, invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule).

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

Integrate the following with respect to x. 
\(\frac { { 3x }^{ 2 }-2x+5 }{ { \left( x-1 \right) }\left( x^{ 2 }+5 \right) } \)

Evaluate \(\int _{ 1 }^{ e }{ \log x } \) dx

Evaluate the following integrals as the limit of the sum:
\(\int _{ 1 }^{ 3 }{ xdx } \)

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

The price of a machine is Rs. 5,00,000 with an estimated life of 12 years. The estimated salvage value is Rs. 30,000. The machine can be rented at Rs. 72,000 per year. The present value of the rental payment is calculated at 9% interest rate. Find out whether it is advisable to rent the machine.(e^−1.08 = 0.3396).

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.

The normal lines to a given curve at each point(x,y) on the curve pass through the point (1, 0). The curve passes through the point (1, 2). Formulate the differential equation representing the problem and hence find the equation of the curve.

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

Solve \(\frac { dy }{ dx } \) −3ycot x = sin 2x given that y = 2 when x = \(\frac { \pi }{ 2 } \)

\(\frac { dy }{ dx } +\frac { y }{ x } ={ xe }^{ x }\)

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

Solve \(\frac { dy }{ dx } +ycosx+x=2cosx\).

The population of a certain town is as follows

Year : x	1941	1951	1961	1971	1981	1991
Population in lakhs:y	20	24	29	36	46	51

Using appropriate interpolation formula, estimate the population during the period 1946.

Find the missing entries from the following

x	0	1	2	3	4	5
y = f(x)	0	-	8	15	-	35

Find f(2.8) from the following table.

x	0	1	2	3
f(x)	1	2	11	34

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

A continuous random variable X has p.d.f
f(x) = 5x⁴, 0\(\le\)x\(\le\)1 
Find a₁ and a₂ such that
i) P[X\(\le\)a₁] = P[X>a₁]   
ii) P[X>a₂] = 0.05

A continuous random variable X has the following distribution function:
\(f(x)=\left\{\begin{array}{l} 0 , \text{if} \ x \leq1 \\ k(x-1)^4, \text{if} \ 1< x \leq 3 \\ 1, \text{if} \ x > 3 \end{array}\right.\)
Find (i) k and (ii) the probability density function.

Determine the mean and variance of a discrete random variable, given its distribution as follows.

X = x	1	2	3	4	5	6
Fx(x)	\(\frac{1}{6}\)	\(\frac{2}{6}\)	\(\frac{3}{6}\)	\(\frac{4}{6}\)	\(\frac{5}{6}\)	1

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(X<0)

If the probability that an individual suffers a bad reaction from injection of a given serum is 0.001, determines the probability that out of 2,000 individuals
(a) exactly 3, and
(b) more than 2 individuals will suffer a bad reaction.

If 5% of the items produced turn out to be defective, then find out the probability that out of 20 items selected at random there are
(i) exactly three defectives
(ii) atleast two defectives
(iii) exactly 4 defectives
(iv) find the mean and variance

The average number of customers, who appear in a counter of a certain bank per minute is two. Find the probability that during a given minute
(i) No customer appears
(ii) three or more customers appear 

A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain
(a) no more than 2 rejects?
(b) at least 2 rejects?

A machine produces a component of a product with a standard deviation of 1.6 cm in length. A random sample of 64 componentsvwas selected from the output and this sample has a mean length of 90 cm. The customer will reject the part if it is either less than 88 cm or more than 92 cm. Does the 95% confidence interval for the true mean length of all the components produced ensure acceptance by the customer?

A random sample of 60 observations was drawn from a large population and its standard deviation was found to be 2.5. Calculate the suitable standard error that this sample is taken from a population with standard deviation 3?

Given below are the data relating to the sales of a product in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

Year	1995	1996	1997	1998	1999	2000	2001	2002
Sales	6.7	5.3	4.3	6.1	5.6	7.9	5.8	6.1

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

Commodities	Price		Quandity
	2003	2009	2003	2009
Rice	10	13	4	6
Wheat	125	18	7	8
Rent	25	29	5	9
Fuel	11	14	8	10
Miscellaneous	14	17	6	7

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 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.

Using Fisher’s Ideal Formula, compute price index number for 1999 with 1996 as base year, given the following:

Year	Commodity: A		Commodity: B		Commodity: C
	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)
1996	5	10	8	6	6	3
1999	4	12	7	7	5	4

A quality control inspector has taken ten samples of size four packets each from a potato chips company. The contents of the sample are given below, Calculate the control limits for mean and range chart.

Sample Number	Observations
	1	2	3	4
1	12.5	12.3	12.6	12.7
2	12.8	12.4	12.4	12.8
3	12.1	12.6	12.5	12.4
4	12.2	12.6	12.5	12.3
5	12.4	12.5	12.5	12.5
6	12.3	12.4	12.6	12.6
7	12.6	12.7	12.5	12.8
8	12.4	12.3	12.6	12.5
9	12.6	12.5	12.3	12.6
10	12.1	12.7	12.5	12.8

(Given for n = 5,  A₂ = 0.58, D₃  = 0 and D₄ = 2.115)

Using the following data, construct Fisher’s Ideal Index Number and Show that it satisfies Factor Reversal Test and Time Reversal Test?

Commodities	Price		Quantity
	Base Year	Current year	Base Year	Current year
Wheat	6	10	50	56
Ghee	2	2	100	120
Firewood	4	6	60	60
Sugar	10	12	30	24
Cloth	8	12	40	36

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

Determine an initial basic feasible solution to the following transportation problem by using North West Corner rule