HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

y = ae^2x + be^-3x is a solution of D.E. `(d^2y)/dx^2 + dy/dx + by = 0`

Mr. Pavan is paid a fixed weekly salary plus commission based on percentage of sales made by him. If on the sale of ₹68,000 and ₹73,000 in two successive weeks, he received in all ₹9880 and ₹10, 180. Complete the following activity to find his weekly salary and the rate of commission paid to him.

Solution: Income of Mr. Pavan = Weekly salary + Commission on sales

Salary + Commission on ₹68,000 = ₹9880      ... (1)

Salary+ Commission on ₹73,000 = ₹10,180     ... (2)

Subtracting (1) from (2), we get

Commission on ₹5000 = ₹ `square`

∴ the rate of commission = `square/5000 xx 100` = 6%

Commission on ₹68,000 at 6% = ₹`68000 xx6/100 = square`

From (1) and (3), we get Salary = ₹(9880 - 4080) = `square`

Hence, fixed weekly salary is ₹5800 and the rate of commission is 6%.

Find the equations of tangent and normal to the curve y = 6 - x² where the normal is parallel to the line x - 4y + 3 = 0

Complete the following activity to calculate, Laspeyre's and Paasche's Price Index Number for the following data :

Solution:

Laspeyre's Price Index Number:

P₀₁(L) = `(sum"p"_1"q"_0)/(sum"p"_0"q"_0) xx 100 = square/660xx100`

∴ P₀₁(L) = `square`

Paasche 's Price Index Number:

P₀₁(P) = `(sum"p"_1"q"_1)/(sum"p"_0"q"_1) xx 100=(620)/(square) xx 100`

∴ P₀₁(P) = `square`

A plant manager has four subordinates and four tasks to perform. The subordinates differ in efficiency and task differ in their intrinsic difficulty. Estimates of the time subordinate would take to perform tasks are given in the following table:

Complete the following activity to allocate tasks to subordinates to minimize total time.

Solution:

Step I: Subtract the smallest element of each row from every element of that row:

Step II: Since all column minimums are zero, no need to subtract anything from columns.

Step III: Draw the minimum number of lines to cover all zeros.

Since minimum number of lines = order of matrix, optimal solution has been reached

Optimal assignment is A →`square`  B →`square`

C →IV  D →`square`

Total minimum time = `square` hours.

Evaluate the following.

`int x sqrt(1+x^2) dx`

If X ∼ P(m) with P(X = 1) = P(X = 2), then find the mean and P(X = 2).

Given e^–2 = 0.1353

Solution: Since P(X = 1) = P(X = 2)

∴ `("e"^square"m"^1)/(1!) = ("e"^"-m""m"^2)/square`

∴ m = `square`

∴ P(X = 2) = `("e"^-2. "m"^2)/(2!)` = `square`

Evaluate `int 1 / (x (x - 1)) dx`

If f'(x) = 4x³ - 3x² + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

Evaluate.

`intx^3/sqrt(1+x^4) dx`

Evaluate.

`int x^3e^(x^2)dx`

Evaluate `int1/(x(x - 1))  dx`

Find `dy/dx if y=cos^-1(sqrt(x))`

Price P for demand D is given as P = 183 +120D - 3D² Find D for which the price is increasing

find dy/dx if `y=tan^-1((6x)/(1-5x^2))`

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

The price P for demand D is given as P = 183 + 120 D – 3D².
Find D for which the price is increasing.

If x = cos2 θ and y = cot θ then find `dy/dx  at  θ=pi/4`