Show that the equations x + y = 5, 2x + y = 8 are consistent and solve them.

A total of Rs. 8,600 was invested in two accounts. One account earned \(4\frac { 3 }{ 4 } %\)% annual interest and the other earned \(6\frac { 1 }{ 2 } %\)% annual interest. If the total interest for one year was Rs. 431.25, how much was invested in each account? (Use determinant method).

Find the rank of the matrix A =\(\left( \begin{matrix} -2 & 1 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{matrix}\begin{matrix} 4 \\ 2 \\ 7 \end{matrix} \right) \)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ -2 & 4 & -4 \end{matrix} \right) \)

Show that the equations x + 2y = 3, y - z = 2, x + y + z = 1 are consistent and have infinite sets of solution.

If \(A=\left( \begin{matrix} 2 & 4 \\ 4 & 3 \end{matrix} \right) ,X=\left( \begin{matrix} n \\ 1 \end{matrix} \right) B=\left( \begin{matrix} 8 \\ 11 \end{matrix} \right) \) and AX = B then find n.

Using second fundamental theorem, evaluate the following:
\(\int _{ 1 }^{ e }{ \frac { dx }{ x(1{ +logx) }^{ 3 } } } \)

If f' (x) = 3x² - \(\frac { 2 }{ { x }^{ 3 } } \) and f(1) = 0, find f(x)

The demand and supply function of a commodity are p_d = 18− 2x − x² and p_s = 2x − 3 . Find the consumer’s surplus and producer’s surplus at equilibrium price.

Find the area bounded by one arc of the curve y = sin ax and the x-axis.

Solve: (x²-yx²)dy + (y²+xy²)dx = 0

Find y when x = 0.2 given that

x	0	1	2	3	4
y	176	185	194	202	212

A player tosses two unbiased coins. He wins Rs. 5 if two heads appear, Rs. 2 if one head appear and Rs.1 if no head appear. Find the expected amount to win.

If the probability of success is 0.09, how many trials are needed to have a probability of atleast one success as 1/3 or more ?

Obtain K, μ and σ2 of of the normal distribution whose probability distribution function is f(x) = \(K{ e }^{ -2x^{ 2 }+4x-2 }\), -∞

A random sample of marks in mathematics secured by 50 students out of 200 students showed a mean of 75 and a standard deviation of 10. Find the 95% confidence limits for the estimate of their mean marks.

Write a brief note on seasonal variations

Compute the consumer price index for 2015 on the basis of 2014 from the following data.

Commodities	Quantities	Prices in 2015	Prices in 2016
A	6	5.75	6.00
B	6	5.00	8.00
C	1	6.00	9.00
D	6	8.00	10.00
E	4	2.00	1.50
F	1	20.00	15.00

Determine an initial basic feasible solution to the following transportation problem using North West corner rule.

Find the initial basic feasible solution for the following transportation problem by Vogel's approximation method.