Mathematics and Statistics Official 2024-2025 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If p : He is intelligent

q : He is strong

Then, symbolic form of statement "It is wrong that, he is intelligent or strong" is:

− p ∨ − q

− (p ∧ q)

− (p ∨ q)

p ∨ − q

`int(x + 1/x)^3 dx` = ______.

`1/4(x + 1/x)^4 + c`

`x^4/4 + (3x^2)/2 + 3log x - 1/(2x^2) + c`

`x^4/4 + (3x^2)/2 + 3log x + 1/x^2 + c`

`(x - x^(-1))^3 + c`

`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))  dx` = ______.

`7/2`

`5/2`

7

2

The area of the region bounded by the line y = 4 and the curve y = x² is ______. 

`32/3` square units

0 square unit

1 square unit

32 square units

`64/3` square units

`16/3` square units

64 square units

The order and degree of the differential equation `((d^2y)/(dx^2))^2 + ((dy)/(dx))^2 = a^x` are ______ respectively.

1, 1

1, 2

2, 2

2, 1

The integrating factor of the differential equation `(dy)/(dx) + y/x = x^3 −3` is ______.

log x

e^x

`1/x`

x

If A is a matrix and K is a constant, then (KA)^T = KA^T

True

False

`int log x  dx = x log x + x + c`

True

False

The differential equation obtained by eliminating arbitrary constants from bx + ay = ab is `(d^2y)/dx^2` = 0.

True

False

The average revenue R_A is 50 and elasticity of demand η is 5, the marginal revenue R_M is ______.

`inte^x(1/x - 1/x^2) dx` = ______ + c.

If f'(x) = x² + 5 and f(0) = −1 then f(x) = ______.

Write the converse, inverse, and contrapositive of the statement "If a triangle is equilateral, then it is equiangular."

Find x, y, z, if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, - 2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`

Evaluate the following.

`int 1/(x(x^6 + 1))` dx 

Solve the following equation by the method of inversion.

2x – y + z = 1,
x + 2y + 3z = 8,
3x + y – 4z = 1

Find MPC, MPS, APC and APS, if the expenditure E_c of a person with income I is given as E_c = (0.0003) I² + (0.075) I ; When I = 1000.

Evaluate:

`int_1^2 1/(x^2 + 6x + 5)  dx`

Find `dy/dx`if, y = `(x)^x + (a^x)`.

Find the area of the region bounded by the parabola y² = 25x and the line x = 5.

Obtain the differential equation by eliminating arbitrary constants from the following equation:

y = Ae^3x + Be^–3x

Using the truth table, verify.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

If x = `(4t)/(1 + t^2),  y = 3((1 - t^2)/(1 + t^2))` then show that `dy/dx = (-9x)/(4y)`.

Divide the number 84 into two parts such that the product of one part and square of the other is maximum. 

Solution:

Let one part be x then the other part will be 84 − x.

∴ f(x) = `square`

∴ f'(x) = 168x − 3x²

For extreme values f'(x) = 0

168x − 3x² = 0

∴ 3x(56 − x) = 0

∴ x = `square or square`

f'(x) = 168 − 6x

If x = 0, f'(0) = 168 − 6(0) = 168 > 0

∴ function attains maximum at x = 0

If x = 56, f'(56) = `square` < 0

∴ function attains maximum at x = 56

∴ Two parts of 84 are `square and square`

Solve the following differential equation (x² – yx²) dy + (y² + xy²) dx = 0.

Separating the variables, the given equation can be written as:

`square  dy + square  dx = 0` 

∴`(y^(-2) - 1/y)dy + (x^(-2) + 1/x)dx = 0`

`square  dy - 1/y  dy + x^(-2) dx + square  dx = 0`

Integrating, we get

`inty^(-2) dy - int1/y dy + int x^(-2)dx + int 1/x dx = 0`

 ∴ `y^(-1)/(-1) - square + x^(-1)/(-1) + square = c`

`-1/y - 1/x + log x - log y = c`

`log x - log y = square + c`

is the required solution.

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

Auctioneer

Del credere agent

Factor

Broker

In an ordinary annuity, payments or receipts occur at ______. 

Beginning of each period

End of each period

Mid of each period

Quarterly basis

Moving averages are useful in identifying ______. 

Seasonal component

Irregular component

Trend component

Cyclical component

If P₀₁(L) = 90 and P₀₁(P) = 40, then P₀₁(D – B) is ______.

65

50

25

130

The objective of an assignment problem is to assign ______. 

Number of jobs to equal number of persons at maximum cost.

Number of jobs to equal number of persons at minimum cost.

Only the maximize cost.

Only to minimize cost.

The expected value of the sum of two numbers obtained when two fair dice are rolled is ______.

5

6

7

8

Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).

Cyclic variation can occur several times in a year.

True

False

Cost of living index number is used in calculating purchasing power of money.

True

False

The amount paid to the holder of the bill after deducting banker's discount is known as ______.

The simplest method of measuring trend of time series is ______.

Quantity Index Number by Weighted Aggregate Method is given by ______.

Compute the appropriate regression equation for the following data:

X is the independent variable and Y is the dependent variable.

The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs ₹ 20 per kg and sand costs of ₹ 6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the L.P.P. for the cost to be minimum.

Find the mean of the number of heads in three tosses of a fair coin.

Obtain the trend value for the following data using 4-yearly centered moving averages:

Find the sequence that minimizes the total elapsed time to complete the following jobs in the order AB. Find the total elapsed time and idle times for both the machines.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that

1. all the five cards are spades?
2. only 3 cards are spades?
3. none is a spade?

A house valued at ₹ 8,00,000 is insured at 75% of its value. If the rate of premium is 0.80%, find the premium paid by the owner of the house. If agent’s commission is 9% of the premium, find agent’s commission.

Solve the following L.P.P. by graphical method:

Maximize: Z = 4x + 6y

Subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.

Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:

1. no defect
2. at least one defect
   Use e⁻¹ = 0.3678

The equations of two regression lines are 10x − 4y = 80 and 10y − 9x = − 40 Find:

1. `bar x and bar y`
2. b_YX and b_XY
3. If var (Y) = 36, obtain var (X)
4. r

Find x if the cost of living index is 150.

A bill of ₹ 18,000 was discounted for ₹ 17,568 at a bank on 25th October 2017. If the rate of interest was 12% p.a., what is the legal due date?

Solution:

Given SD = ₹18,000, CV = ₹17,568

r = 12% p.a.

Now, BD = `square`

= 18,000 − 17,568

= ₹ 432

Also, BD = `square`

∴ `432 = (18,000 xx n xx 12)/100`

`n = (432 xx 100)/(18,000 xx 12)`

`n = 1/5` years = `square` days

The period for which the discount is deducted is 73 days, which is counted from the date of discounting, i.e., 25^th October 2017:

Hence, legal due date is `square`

Solve the following assignment problem to maximization:

Step - I

Subtract the Smallest element of each row from every element of that row

Step - II

Subtract the smallest element of each column from every element of that column:

Step - III

Draw minimum number of lines covering all zeros.

Step - 1V

The smallest uncovered element is 1, which is to be subtracted from all uncovered elements and add it to all elements which lie at the intersection of two lines:

Step - V

Draw minimum number of lines that are required to cover all zeros:

Here minimum number of Lines = order of matrix.

Step - VI

Find smallest uncovered element (1). Subtract this number from all uncovered elements and add it to all elements which lie at the intersection of two lines:

Now minimium number of lines = order of matrix.

The optimal assignment can be made.

Optimal solution is

1 → I

2 → IV

3 → `square`

4 → `square`

5 → V

Minimum value = `square`