Mathematics and Statistics Official 2023-2024 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Which of the following sentences is a statement in logic?

He is a good actor.

Did you eat lunch yet?

Every real number is a complex number.

Bring the motor car here.

If y = 2x² + log 2 + 5, then `dy/dx` = ______.

x

4x

2x + log 2

−4x

If x = 2at² , y = 4at, then `dy/dx = ?`

`- 1/(2at^2)`

`1/(2at^3)`

`1/t`

`1/(4at^3)`

The equation of the tangent to the curve y = x² + 4x + 1 at P(−1, −2) is ______.

2x − y = 0

x + 2y + 5 = 0

2x + 4 = 3y

5x + y = 1

`int_(-2)^3 dx/(x + 5)` = ______.

`-log(8/3)`

`log(8/3)`

`log(3/8)`

`-log(3/8)`

`3 log (3/8)`

`-2 log (3/8)`

The order and degree of the differential equation `(d^2x)/(dt^2) + (dx/dt)^2 + 8 = 0` are ______.

order = 2, degree = 2

order = 1, degree = 2

order = 1, degree = 1

order = 2, degree = 1

Every identity matrix is a scalar matrix.

True

False

The rate of change of demand (x) of a commodity w.r.t. its price (y) is `dy/dx`.

True

False

The integrating factor of the differential equation `dy/dx - y = x` is e^−x.

True

False

If y = x log x, then `(d^2y)/dx^2`= ______.

If the marginal revenue R_m = 40 and elasticity of demand η is 5, then the average revenue R_A will be ______.

Area of the region bounded by y = x⁴, x = 1, x = 5 and the X-axis is _______.

Write the converse of the following statement:

‘If the train reaches on time, then I can catch the connecting flight.’

Write the inverse of the following statement:

‘If the train reaches on time, then I can catch the connecting flight.’

Write the contrapositive of the following statement:

‘If the train reaches on time, then I can catch the connecting flight.’

If A = `[(1, 2), (-1, -2)]`, B = `[(2, a), (-1, b)]` and if (A + B)² = A² + B², find value of a and b.

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

Evaluate the following.

`int x/(4x^4 - 20x^2 - 3) dx`

Evaluate:

`int_1^3 log x dx`

In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

A metal wire of  36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.

Evaluate:

`int (2x + 1)/(x(x - 1)(x - 4)) dx`.

Find the area of the region bounded by y² = 25x and the line x = 4.

Using the truth table, verify

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q.

Solve the following equations by method of inversion:

x + y + z = 1, x – y + z = 2 and x + y – z = 3

The cost C for producing x articles is given as C = x³ − 16x² + 47x. For what values of x, the average cost is decreasing?

Solution:

Given C = x³ − 16x² + 47x

Average cost `C_A = C/x`

∴ `C_A = square`

Differentiating w.r.t. x, we get

`d/dx (C_A) = square`

We know that C_A is decreasing, if

`d/dx (C_A) square 0`

∴ 2x − 16 < 0

∴ 2x < 16

∴ x < `square`

∴ average cost is decreasing for x ∈ (0, 8).

Solve the differential equation: `y - x dy/dx = 0`.

Solution: given equation is `y - x dy/dx = 0`

Separating the variables, we get

`dx/square = dy/square`

Integrating, we get

`int dx/square = int dy/square + c`

∴ log x = `square + c`

∴ log x − log y = log c₁, where c = log c₁

∴ `log (x/y) = log c_1`

∴ `x/square = c_1`

Hence, the required solution is x = c₁y.

The date on which the period of the bill expires is called ______.

Legal Due Date

Days of grace

The Nominal Due date

Date of Drawing

grace date

A person insured a property of ₹ 4,00,000. The rate of premium is ₹ 35 per thousand p.a. The amount of annual premium is ______.

₹ 14,000

₹ 24,000

₹ 34,000

₹ 15,000

Paasche’s Price Index Number is given by ______.

`(sump_0q_0)/(sump_1q_0) xx 100`

`(sump_0q_1)/(sump_1q_1) xx 100`

`(sump_1q_0)/(sump_0q_0) xx 100`

`(sump_1q_1)/(sump_0q_1) xx 100`

If jobs I, II, III have processing times as 8, 6, 5 on machine M₁ and 8, 3, 4 on machine M₂ in the order M₁-M₂, then the optimal sequence is ______.

I, II, III

I, III, II

II, I, III

III, II, I

If E(X) = 4 and X follows Poisson’s distribution, then V(X) = ______.

2

−2

4

−4

Three coins are tossed simultaneously. X is the number of heads. Then the expected value of X is ______.

1

1.5

1.9

1.017

In the regression of Y on X, X is the independent variable and Y is the dependent variable.

True

False

The region represented by the inequalities x ≤ 0, y ≤ 0 lies in the first quadrant.

True

False

In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.

True

False

If an agent charges 12% commission on the sales of ₹ 48,000, then his total commission is ₹ ______.

The optimal value of the objective function is attained at the ______ points of the feasible region.

Given p.d.f. of a continuous random variable X is

`f(x) = x/8`, for 0 < x < 4

= 0, otherwise.

Then P(1 < X < 2) = ...........

Find the rate of interest compounded annually if an immediate annuity of ₹ 20,000 per year amounts to ₹ 41,000 in 2 years.

Find the Value Index Number using the Simple Aggregate Method in the following example.

Five jobs must pass through a lathe and a surface grinder, in that order. The processing times in hours are shown below. Determine the optimal sequence of the jobs. Also, find the total elapsed time:

A bill was drawn on 14^th April for ₹ 7000 and was discounted on 6^th July at 5% p.a. The banker paid ₹ 6930 for the bill. Find the period of the bill.

The following table gives the production of steel (in millions of tons) for years 1976 to 1986:

Fit a trend line to the above data by the method of least squares.

Solve the following LPP by graphical method:

Maximize: z = 7x + 11y, subject to:

3x + 4y ≤ 24, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0.

For bivariate data. `bar x = 53`, `bar y = 28`, b_yx = −1.2, b_xy = −0.3. Find the correlation coefficient between x and y.

For a bivariate data, `bar x = 53`, `bar y = 28`, b_yx = −1.5 and b_xy = −0.2. Estimate y when x = 50.

Given that ∑p₀q₀ = 220, ∑p₀q₁ = 380, ∑p₁q₁ = 350 and Marshall-Edgeworth’s Price Index Number is 150, find Laspeyre’s Price Index Number.

The following data gives the production of bleaching powder (in ’000 tons) for the years 1962 to 1972:

Obtain the trend values for the above data using 5-yearly moving averages.

Four new machines M₁, M₂, M₃ and M₄ are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M₂ cannot be placed at C and M₃ cannot be placed at A. The cost matrix is given below:

Find the optimal assignment schedule.

There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

The equations of the two regression lines are 3x + 2y − 26 = 0 and 6x + y − 31 = 0. Obtain the correlation coefficient between x and y.

Solution: To find correlation coefficient, we have to find the regression coefficients b_yx and b_xy.

Let 3x + 2y = 26 be equation of the line of regression of y on x.

This gives y = `square` + x + 13

∴ b_yx = `-3/2`

Now, consider 6x + y = 31 as equation of the line of regression of x on y.

This can be written as x = `square y + 31/6`

∴ b_yx = `-1/6`

Now, `r^2 = square = 0.25`

As both b_yx and b_xy are negative,

∴ r = `square`

The probability distribution of X is as follows:

Find:

1. k
2. P(X < 2)
3. P(1 ≤ X < 4)
4. F(2)

Solution: The table gives a probability distribution.

∴ ∑p_i = 1       

∴ 0.1 + k + 2k + 2k + k = 1

1. k = `square`
2. P(X < 2) = P(X = 0) + P(X = 1) = `square`
3. P(1 ≤ X < 4) = P(1) + P(2) + P(3) = `square`
4. F(2) = P(X ≤ 2) = P(0) + P(1) + P(2) = `square`