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

Which of the following is not a statement?

Smoking is injuries to health

2 + 2 = 4

2 is the only even prime number.

Come here

If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______

(–1, 0)

(1, 0)

(1, –1)

(–1, 1)

If y = log `("e"^"x"/"x"^2)`, then `"dy"/"dx" = ?` 

`(2 - "x")/"x"`

`("x" - 2)/"x"`

`("e - x")/"ex"`

`("x - e")/"ex"`

The value of `int ("d"x)/(sqrt(1 - x))` is ______.

`2sqrt(1 - x) + "c"`

`-2sqrt(1 - x) + "c"`

`sqrtx + "c"`

x + c

`int "dx"/(("x" - 8)("x" + 7))`=

`1/15 log |("x" + 2)/("x" - 1)| + "c"`

`1/15 log |("x" + 8)/("x" + 7)| + "c"`

`1/15 log |("x"- 8)/("x" + 7)| + "c"`

(x − 8)(x − 7) + c

`1/15 log |("x" + 2)/("x"+ 1)| + "c"`

(x − 8)(x + 7) + c

The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.

`(d^2y)/dx^2 - y = 0`

`(d^2y)/dx^2 + dy/dx  = 0`

`(d^2y)/dx^2 + ydy/dx  = 0`

`(d^2y)/dx^2 + y  = 0`

`int_a^b f(x) dx = int_a^b f (t) dt`

True

False

For `int ("x - 1")/("x + 1")^3  "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)².

True

False

Order and degree of a differential equation are always positive integers.

True

False

The slope of tangent at any point (a, b) is also called as ______.

If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______

A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

~ p → (p → ~ q)

Find `"dy"/"dx"`, if x = e^3t, y = `"e"^((4"t" + 5))`

If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find A^T + 4B^T

Consider the following statements.

1. If D is dog, then D is very good.
2. If D is very good, then D is dog.
3. If D is not very good, then D is not a dog.
4. If D is not a dog, then D is not very good. 

Identify the pairs of statements having the same meaning. Justify.

Determine the minimum value of the function.

f(x) = 2x³ – 21x² + 36x – 20

Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.

Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`

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

Find the differential equation whose general solution is

x³ + y³ = 35ax.

Find the inverse of `[(3, 1, 5),(2, 7, 8),(1, 2, 5)]` by adjoint method.

The consumption expenditure E_c of a person with the income x. is given by E_c = 0.0006x² + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.

Complete the following activity:

`int_0^2 dx/(4 + x - x^2) `

= `int_0^2 dx/(-x^2 + square + square)`

= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`

= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`

= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`

The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1,00,000, when will the city have population 4,00,000?

Let ‘p’ be the population at time ‘t’ years.

∴ `("dp")/"dt" prop "p"`

∴ Differential equation can be written as  `("dp")/"dt" = "kp"`

where k is constant of proportionality.

∴ `("dp")/"p" = "k.dt"`

On integrating we get

`square` = kt + c   ...(i)

(i) Where t = 0, p = 1,00,000

∴ from (i)

log 1,00,000 = k(0) + c

∴  c = `square`

∴  log `("p"/(1,00,000)) = "kt"`       ...(ii)

(ii) When t = 25, p = 2,00,000

as population doubles in 25 years

∴ from (ii) log2 = 25k

∴  k = `square`

∴  log`("p"/(1,00,000)) = (1/25log2).t`

(iii) ∴ when p = 4,00,000

`log ((4,00,000)/(1,00,000)) = (1/25log2).t`

∴ `log 4 = (1/25 log2).t`

∴ t = `square ` years

The difference between face value and present worth is called ______.

Banker’s discount

True discount

Banker’s gain

Cash value

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

Beginning of each period

End of each period

Mid of each period

Quarterly basis

b_xy and b_yx are _______.

Independent of change of origin and scale

Independent of change of origin but not of scale

Independent of change of scale but not of origin

Affected by change of origin and scale

Dorbish-Bowley’s Price Index Number is given by ______.

`((sum"p"_1"q"_0)/(sum"p"_0"q"_1) + (sum"p"_0"q"_1)/(sum"p"_1"q"_0))/(2) xx 100`

`((sum"p"_1"q"_1)/(sum"p"_0"q"_0) + (sum"p"_0"q"_0)/(sum"p"_1"q"_1))/(2) xx 100`

`((sum"p"_1"q"_0)/(sum"p"_0"q"_0) + (sum"p"_1"q"_1)/(sum"p"_0"q"_1))/(2) xx 100`

`((sum"p"_0"q"_0)/(sum"p"_1"q"_0) + (sum"p"_0"q"_1)/(sum"p"_1"q"_1))/(2) xx 100`

Objective function of LPP is ______.

A constraint

A function to be maximised or minimised

A relation between the decision variables

A feasible region

Equation of straight line

To use the Hungarian method, a profit maximization assignment problem requires ______.

Converting all profits to opportunity losses

A dummy person or job

Matrix expansion

Finding the maximum number of lines to cover all the zeros in the reduced matrix

Broker is an agent who gives a guarantee to seller that the buyer will pay the sale price of goods.

True

False

`sum ("p"_0"q"_0)/("p"_1"q"_1) xx 100` is Value Index Number by Simple Aggregate Method.

True

False

The optimum value of the objective function of LPP occurs at the center of the feasible region.

True

False

The banker’s discount is always _______ than the true discount.

The cost of living index number using Weighted Relative Method is given by ______.

The time interval between starting the first job and completing the last job including the idle time (if any) in a particular order by the given set of machines is called _______.

Deepak’s salary was increased from ₹ 4,000 to ₹ 5,000. The sales being the same, due to reduction in the rate of commission from 3% to 2%, his income remained unchanged. Find his sales.

For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9.

Obtain the trend values for the data in using 4-yearly centered moving averages.

Solve the following problem:

If find x is Walsh’s Price Index Number is 150 for the following data

A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B, C in the order ABC. The time required in hours for each process is given in the following table.

Solve the problem for minimizing the total elapsed time.

A random variable X has the following probability distribution:

Find:

1. k
2. P(X < 3)
3. P(X > 4)

A building is insured for 75% of its value. The annual premium at 0.70 percent amounts to ₹ 2,625. If the building is damaged to the extent of 60% due to fire, how much can be claimed under the policy?

Three new machines M1, M2, M3 are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M2 can not be placed at B. The cost matrix (in hundred rupees) is as follows:

Determine the optimum assignment schedule and find the minimum cost.

The eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg in the lot of 10 eggs.

Following table shows the all India infant mortality rates (per '000) for years 1980 to 2010:

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

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

Minimize: Z = 6x + 2y subject to x + 2y ≥ 3, x + 4y ≥ 4, 3x + y ≥ 3, x ≥ 0, y ≥ 0.

For a bivariate data `barx = 10`, `bary = 12`, V(X) = 9, σ_y = 4 and r = 0.6
Estimate y when x = 5

Solution: Line of regression of Y on X is

`"Y" - bary = square ("X" - barx)`

∴ Y − 12 = `r.(σ_y)/(σ_x)("X" - 10)`

∴ Y − 12 = `0.6 xx 4/square ("X" - 10)`

∴ When x = 5

Y − 12 = `square(5 - 10)`

∴ Y − 12 = −4

∴ Y = `square`

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`