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

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 automobile company uses three types of Steel S₁, S₂ and S₃ for providing three different types of Cars C₁, C₂ and C₃. Steel requirement R (in tonnes) for each type of car and total available steel of all the three types are summarized in the following table.

Determine the number of Cars of each type which can be produced by Cramer’s rule.

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train this year.
(i) What percent of commuters will be using the transit system after one year?
(ii) What percent of commuters will be using the transit system in the long run?

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

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

Evaluate the following using properties of definite integrals:
\(\int _{ 0 }^{ 1 }{ \frac { x }{ ({ 1-x) }^{ \frac { 3 }{ 4 } } } dx } \)

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

A firm has the marginal revenue function given by MR = \(\frac { a }{ { (x+b) }^{ 2 } } \) - c where x is the output and a, b, c are constants. Show that the demand function is given by \(x=\frac { a }{ b(p+c) } -b\).

The elasticity of demand with respect to price p for a commodity is \(\eta _{ d }=\frac { p+2{ p }^{ 2 } }{ 100-p-{ p }^{ 2 } } \).Find demand function where price is Rs. 5 and the demand is 70.

The demand equation for a products is x = \(\sqrt { 100-p } \) and the supply equation is x = \(\frac{p}{2}\) -10. Determine the consumer’s surplus and producer’s surplus, under market equilibrium.

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.

If the marginal cost of producing x shoes is given by (3xy + y²)dx + (x² + xy)dy = 0 and the total cost of producing a pair of shoes is given by Rs. 12. Then find the total cost  function.

Solve the following:
\(x\frac { dy }{ dx } +2y={ x }^{ 4 }\)

From the following table find the number of students who obtained marks less than 45.

Using interpolation estimate the output of a factory in 1986 from the following data

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

The distribution of a continuous random variable X in range (–3, 3) is given by p.d.f.
\(f(x)=\left\{\begin{array}{l} \frac{1}{16}(3+x)^{2},-3 \leq x \leq-1 \\ \frac{1}{16}\left(6-2 x^{2}\right),-1 \leq x \leq 1 \\ \frac{1}{16}(3-x)^{2}, 1 \leq x \leq 3 \end{array}\right.\)
Verify that the area under the curve is unity.

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function\(F(x)\begin{cases} 0,\quad ​​​​\text{for}\quad x<0 \\ \frac { 1 }{ 2 } ,\quad ​​​​\text{for}\quad 0\le x<1 \\ 0,\quad ​​​​\text{for}\quad 1\le x<2\quad \\ \frac { 1 }{ 4 } ,\quad ​​​​\text{for}\quad 2\le x<4 \\ 0,\quad ​​​​\text{for}\quad x\ge 4 \end{cases}\)
(a) Is the distribution function continuous? If so, give its probability density function?
(b) What is the probability that a person will have to wait
(i) more than 3 minutes,
(ii) less than 3 minutes and
(iii) between 1 and 3 minutes?

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(|X|\(\le\)2)

A bank manager has observed that the length of time the customers have to wait for being attended by the teller is normally distributed with mean time of 5 minutes and standard deviation of 0.6 minutes. Find the probability that a customer has to wait
(i) for less than 6 minutes
(ii) between 3.5 and 6.5 minutes

A sample of 125 dry battery cells tested to find the length of life produced the following resultd with mean 12 and SD 3 hours. Assuming that the data to be normal distributed , what percentage of battery cells are expected to have life
(i) more than 13 hours
(ii) less than 5 hours
(iii) between 9 and 14 hours

Derive the mean and variance of binomial distribution.