HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`

Solve the following differential equation `("d"y)/("d"x)` = x²y + y

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Solve the following differential equation

`y log y ("d"x)/("d"y) + x` = log y

Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0

y = `a + b/x`

`(dy)/(dx) = square`

`(d^2y)/(dx^2) = square`

Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`

= `x square + 2 square`

= `square`

Hence y = `a + b/x` is solution of `square`

Solve the following differential equation 

sec² x tan y dx + sec² y tan x dy = 0

Solution: sec² x tan y dx + sec² y tan x dy = 0

∴ `(sec^2x)/tanx  "d"x + square` = 0

Integrating, we get

`square + int (sec^2y)/tany  "d"y` = log c

Each of these integral is of the type

`int ("f'"(x))/("f"(x))  "d"x` = log |f(x)| + log c

∴ the general solution is

`square + log |tan y|` = log c

∴ log |tan x . tan y| = log c

`square`

This is the general solution.

Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)

Solution: `("d"y)/("d"x)` = cos(x + y)    ......(1)

Put `square`

∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`

∴ (1) becomes `"dv"/("d"x) - 1` = cos v

∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

Integrating, we get

`int 1/(1 + cos "v")  "d"v = int  "d"x`

∴ `int 1/(2cos^2 ("v"/2))  "dv" = int  "d"x`

∴ `1/2 int square  "dv" = int  "d"x`

∴ `1/2* (tan("v"/2))/(1/2)` = x + c

∴ `square` = x + c

Find the particular solution of the following differential equation

`("d"y)/("d"x)` = e^2y cos x, when x = `pi/6`, y = 0.

Solution: The given D.E. is `("d"y)/("d"x)` = e^2y cos x

∴ `1/"e"^(2y)  "d"y` = cos x dx

Integrating, we get

`int square  "d"y` = cos x dx

∴ `("e"^(-2y))/(-2)` = sin x + c₁

∴ e^–2y = – 2sin x – 2c₁

∴ `square` = c, where c = – 2c₁ 

This is general solution.

When x = `pi/6`, y = 0, we have

`"e"^0 + 2sin  pi/6` = c

∴ c = `square`

∴ particular solution is `square`

Choose the correct alternative:

A salesman receives 3% commission on the sales up to ₹ 50,000 and 4% commission on the sales over ₹ 50,000. His total income on the sale of ₹ 2,00,000 is ______.

Choose the correct alternative:

The present worth of ₹ 11,660 due 9 months hence is ₹ 11,000. The True discount is ______

An agent who gives guarantee to his principal that the party will pay the sale price of goods is called ______

The buyer is legally allowed ______ days grace period

When transactions like sale, purchase, auction etc. are done through some middlemen, such middlemen are called ______

State whether the following statement is True or False:

The trade discount is first calculated on the catalogue (list) price

State whether the following statement is True or False: 

A factor is an agent who is given the possession of goods and enters a contract for sale in his/her own name

Ananya gets salary of ₹ 15,000 per month and commission at 8% on the sales over ₹ 50,000. If she gets ₹ 17,400 in a certain month, Find the sales made by her in that month

An agent sold a car and charged 3% commission on sale value. If the owner of the car received ₹ 48,500, find the sale value of the car. If the agent charged 2% from the buyer, find his total remuneration

Three cars were sold through an agent for ₹ 2,40,000, ₹ 2,22,000 and ₹ 2,25,000 respectively. The rates of commission were 17.5% on the first, 12.5% on the second. If the agent overall received 14% commission on the total sales, find the rate of commission paid on the third car.

Solution: Total selling Price of three cars = 2,40,000 + 2,22,000 + 2,25,000

= `square`

Commision on total sale = 14%

= `14/100 xx square`

Selling price of First car = ₹ 2,40,000

Rate of commission = 17.5% 

= `17.5/100 xx 2,40,000 = square`

∴ Commission on first car = ₹ `square`

Selling price of Second car = ₹ 2,22,000

Rate of commission = 12.5%

= `12.5/100 xx 2,22,000 = square`

∴ Commission on second car = ₹ `square`

Selling price of third car = ₹ 2,25,000

Let the rate of commission be x

Commission on third car = `x/100 xx 2,25,000`

∴ Commission on third car = Total commission − (commission on first car + commission on second car)

∴ `x/100 xx 2,25,000 = square - {square + square}`

∴ x = `square`

State whether the following statement is True or False:

Regression analysis is used for measuring the degree of the relationship between the variables

State whether the following statement is True or False: 

The variable used for predicting the response is called the independent variable