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

If ∑ p₀q₀ = 120, ∑ p₀q₁ = 160, ∑ p₁q₁ = 140, ∑ p₁q_o = 200, find Laspeyre’s, Paasche’s, Dorbish-Bowley’s and Marshall-Edgeworth’s Price Index Numbers.

Five wagons are available at stations 1, 2, 3, 4 and 5. These are required at 5 stations I, II, III, IV and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?

In a town, 10 accidents take place in the span of 50 days. Assuming that the number of accidents follows Poisson distribution, find the probability that there will be 3 or more accidents on a day.

(Given that e^-0.2 = 0.8187)

In the following table, Laspeyre's and Paasche's Price Index Numbers are equal. Complete the following activity to find x :

Solution: P₀₁(L) = P₀₁(P)

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

`(20 + 5x)/square xx 100 = square/14 xx 100`

∴ x = `square`

A job production unit has four jobs P, Q, R, S which can be manufactured on each of the four machines I, II, III and IV. The processing cost of each job for each machine is given in the following table :

Complete the following activity to find the optimal assignment to minimize the total processing cost.

Solution:

Step 1: Subtract the smallest element in each row from every element of it. New assignment matrix is obtained as follows :

Step 2: Subtract the smallest element in each column from every element of it. New assignment matrix is obtained as above, because each column in it contains one zero.

Step 3: Draw minimum number of vertical and horizontal lines to cover all zeros:

Step 4: From step 3, as the minimum number of straight lines required to cover all zeros in the assignment matrix equals the number of rows/columns. Optimal solution has reached.

Examine the rows one by one starting with the first row with exactly one zero is found. Mark the zero by enclosing it in (`square`), indicating assignment of the job. Cross all the zeros in the same column. This step is shown in the following table :

Step 5: It is observed that all the zeros are assigned and each row and each column contains exactly one assignment. Hence, the optimal (minimum) assignment schedule is :

Hence, total (minimum) processing cost = 25 + 21 + 19 + 34 = ₹`square`

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.

The differential equation of (x - a)² + y² = a² is ______ 

Find `dy/dx` if, y = `e^(5 x^2 - 2x + 4)`

If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`

Evaluate the following.

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

Find the rate of change of demand (x) of acommodity with respect to its price (y) if

`y = 12 + 10x + 25x^2`

The position vector of points A and B are `6bar(a) + 2bar(b) and bar(a) -3bar(b)` . If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is 3`bar(a)- bar(b)`.

Find `dy/dx` if, y = `e^(5x^2-2x+4)`

Solve the following:

If y = `root5((3x^2 +8x+5)^4`,find `dy/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`

Find `dy/dx` if, y = `e^(5x^2 -2x + 4)`