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.

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.

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

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).

80% of students who do maths work during one study period, will do the maths work at the next study period. 30% of students who do english work during one study period, will do the english work at the next study period. Initially there were 60 students do maths work and 40 students do english work.
Calculate,
(i) The transition probability matrix
(ii) The number of students who do maths work, english work for the next subsequent 2 study periods.

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

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

Evaluate \(\int { \frac { { 3x }^{ 2 }+6x+1 }{ \left( x+3 \right) \left( { x }^{ 2 }+1 \right) } } dx\)

Integrate the following with respect x
\(\frac { 3x+2 }{ \left( x-2 \right) \left( x-3 \right) } \)

Evaluate \(\int _{ -1 }^{ 1 }{ x\sqrt { x+1 } } dx\)

Using integration find the area of the region bounded between the line x = 4 and the parabola y² = 16x.

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

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 slope of the tangent to a curve at any point (x, y) on it is given by (y³−2yx²)dx + (2xy²−x³)dy = 0 and the curve passes through (1, 2). Find the equation of the curve.

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

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

Calculate the value of y when x = 7.5 from the table given below

x	1	2	3	4	5	6	7	8
y	1	8	27	64	125	216	343	512

Using Lagrange’s interpolation formula find y(10) from the following table:

x	5	6	9	11
y	12	13	14	16

From the following table obtain a polynomial of degree y in x

x	1	2	3	4	5
y	1	-1	1	-1	1

Suppose the life in hours of a radio tube has the probability density function
\(f(x)=\left\{\begin{array}{l} e^{-\frac{x}{100}}, \text { when } x \geq 100 \\ 0, \quad \text { when } x<100 \end{array}\right.\)
Find the mean of the life of a radio tube.

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(0\(\le\)X\(\le\)10)

Derive the mean and variance of binomial distribution.

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

Calculate the cost of living index number for the following data.

Commodities	Quantity 2005	Price
		2005	2010
A	10	7	9
B	12	6	8
C	17	10	15
D	19	14	16
E	15	12	17