Show that the equations are inconsistent x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 

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.

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.

An amount of Rs. 5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is Rs. 358/-. If the income from first two investments is Rs. 70/- more than the income from the third, then find the amount of investment in each bond by rank method.

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?

Find k if the equations 2x + 3y − z = 5, 3x − y + 4z = 2, x + 7y − 6z = k are consistent.

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

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

If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).

Sketch the graph \(y=\left| x+3 \right| \) and evaluate \(\int _{ -6 }^{ 0 }{ \left| x+3 \right| } \) dx.

Find the area bounded by the curve y = x² and the line y = 4

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

The Marginal revenue for a commodity is MR=\(\frac { { e }^{ x } }{ 100 } +x+{ x }^{ 2 }\), find the revenue function.

Solve cos² x \(\frac{dy}{dx}\) + y = tan x

Solve the following differential equations (D²+D−6)y=e^3x + e^−3x

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.

Find a polynomial of degree two which takes the values

x	0	1	2	3	4	5	6	7
y	1	2	4	7	11	16	22	29

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

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?

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.

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as \( f(x)\begin{cases} { Ae }^{ -x/5 },\quad \text{for}\quad x\ge 0 \\ 0 \quad ,\quad \text{otherwise }\end{cases}\)
(a) Find the value of A that makes fix) a p.d.f,
(b) What is the probability that the number of minutes that person will talk over the phone is
(i) more than 10 minutes
(ii) less than 5 minutes and
(iii) between 5 and 10 minutes.

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 distribution of the discrete random variables X and Y are given below

X	0	1	2	3
P(X)	\(\frac{1}{5}\)	\(\frac{2}{5}\)	\(\frac{1}{5}\)	\(\frac{1}{5}\)

Y	0	1	2	3
P(Y)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)	\(\frac{1}{10}\)

Prove that E(Y²) = 2E(X).

Measurements of the weights of a random sample of 200 ball bearings made by certain machine during one week showed a mean of 0.824 newtons and a S.D. of 0.042 newton's. Find
a) 95% and
b) 99% confidence limits for the mean weight of all the ball bearings.

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