HSC Commerce Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

A car hire company has one car at each of five depots a, b, c, d and e. A customer in each of the fine towers A, B, C, D and E requires a car. The distance (in miles) between the depots (origins) and the towers(destinations) where the customers are given in the following distance matrix.

How should the cars be assigned to the customers so as to minimize the distance travelled?

A natural truck-rental service has a surplus of one truck in each of the cities 1, 2, 3, 4, 5 and 6 and a deficit of one truck in each of the cities 7, 8, 9, 10, 11 and 12. The distance(in kilometers) between the cities with a surplus and the cities with a deficit are displayed below:

How should the truck be dispersed so as to minimize the total distance travelled?

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, 45% of those who already subscribe will subscribe again while 30% 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?

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 next year.

What percent of commuters will be using the transit system year after the next year?

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 next year.

What percent of commuters will be using the transit system in the long run?

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?

Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again where as 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?

Choose the correct alternative:

If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.4&0.6\\0.2&0.8\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then at equilibriuium A is equal to

Choose the correct alternative:

If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.7&0.3\\0.6&x\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then the value of x is

Choose the correct alternative:

In a transition probability matrix, all the entries are greater than or equal to

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?

Solve the following differential equation:

`("d"^2y)/("d"x^2) - 6 ("d"y)/("d"x) + 8y = 0`

Solve the following differential equation:

`("d"^2y)/("d"x^2) - 4("d"y)/("d"x) + 4y = 0`

Solve the following differential equation:

(D² + 2D + 3)y = 0

Solve the following differential equation:

`("d"^2y)/("d"x^2) - 2"k" ("d"y)/("d"x) + "k"^2y = 0`

Solve the following differential equation:

(D² – 2D – 15)y = 0 given that `("d"y)/("d"x)` = 0 and `("d"^2y)/("d"x^2)` = 2 when x = 0

Solve the following differential equation:

(4D² + 4D – 3)y = e^2x

Solve the following differential equation:

`("d"^2y)/("d"x^2) + 16y = 0`

Solve the following differential equation:

(D² – 3D + 2)y = e^3x which shall vanish for x = 0 and for x = log 2

Solve the following differential equation:

(D² + D – 2)y = e^3x + e^–3x