Find k, if the equations x + 2y − 3z = −2, 3x − y − 2z = 1, 2x + 3y − 5z = k are consistent.

Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them 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.

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.

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

For what values of k, the system of equations kx+ y+z = 1, x+ ky+z= 1, x+ y+kz = 1 have
(I) Unique solution
(ii) More than one solution
(iii) no solution

Evaluate \(\int { { \left( \log x \right) }^{ 2 } } dx\)

Evaluate \(\int _{ 0 }^{ 1 }{ [{ e }^{ a \log x }+{ e }^{ x \log a }] } dx\)

Evaluate ഽx. log (1 + x) dx

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).

In year 2000 world gold production was 2547 metric tons and it was growing exponentially at the rate of 0.6% per year. If the growth continues at this rate, how many tons of gold will be produced from 2000 to 2013? [e^0.078 = 1.0811)

The marginal revenue function (in thousands of rupees) of a commodity is 7+^e-0.05x where x is the number of units sold. Find the total revenue from the sale of 100 units (e^-5 = 0.0067)

Find the particular solution of the differential equation x² dy + y(x + y)dx = 0 given  that x = 1, y = 1

If \(\frac { dy }{ dx } \) + 2y tan x = sin x and if y = 0 when x = \(\frac{\pi}{3}\) express y in terms of x.

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

Derive the mean and variance of binomial distribution.

Explain in detail about non-sampling error.

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

Compute
(i) Laspeyre's
(ii) Paasche's 
(iii) Fisher's price index number for 2000 from the following data.

Commodity	Price		Quantity
	1990	2000	1990	2000
A	2	4	8	6
B	5	6	10	5
C	4	5	14	10
D	2	2	19	13