HSC Commerce कक्षा १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

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 

Solve the following differential equation:

(D² – 10D + 25) y = 4e^5x + 5

Solve the following differential equation:

`(4"D"^2 + 16"D" + 15)y = 4"e"^((-3)/2x)`

Solve the following differential equation:

(3D² + D – 14)y – 13e^2x

Solve the following differential equation:

Suppose that the quantity demanded Q_d = `13 - 6"p" + 2 "dp"/"dt" + ("d"^2"p")/("dt"^2)` and quantity supplied Q_s = `- 3 + 2"p"` where p is the price. Find the equilibrium price for market clearance