Mathematics and Statistics Official 2022-2023 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

The dual of the statement (p ˅ q) ˄ (r ˅ s) is ______.

(p ˄ q) ˄ (r ˄ s)

(p ˄ q) ˅ (r ˄ s)

(p ˅ q) ˅ (r ˅ s)

(p ˅ q) ˄ (r ˅ s)

If y = x . log x then `dy/dx` = ______.

1

`1/x`

log x

1 + log x

If y = 2x² + 2² + a², then `"dy"/"dx" = ?`

x

4x

2x

-2x

4x + 2a

4x + 4

A function f is said to be increasing at a point c if ______.

f'(c) = 0

f'(c) > 0

f'(c) < 0

f'(c) = 1

`int_0^2 e^x dx` = ______.

e² – 1

1 – e² 

e – 1

1 – e

The integrating factor of `(dy)/(dx) + y` = e^–x is ______.

x

–x

e^x

e^–x

The derivative of f(x) = a^x, where a is constant is x.a^x-1.

True

False

`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`

True

False

The degree of the differential equation `((d^2y)/dx^2)^2 + (dy/dx)^3` = a^x is 3.

True

False

Converse of the statement q `rightarrow` p is ______.

If y = (log x)² the `dy/dx` = ______.

If 0 < η < 1 then the demand is ______.

Construct the truth table for the following statement pattern.

(p ∧ ~ q) ↔ (q → p)

Solve the following:

If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`

Evaluate the following.

`int 1/(7 + 6"x" - "x"^2)` dx

Express the following equations in matrix form and solve them by the method of reduction:

x + 2y + z = 8, 2x + 3y - z = 11, 3x - y - 2z = 5.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the value of x for which Total cost is decreasing.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the values of x for which Revenue is increasing.

Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`

Write the negation of the following statement.

∃ n ∈ N, (n² + 2) is odd number.

Write the negation of the following statement.

Some continuous functions are differentiable.

Write the negation of the following statement:

(p `rightarrow` q) ∨ (p `rightarrow` r)

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.

Evaluate the following : `int x^3.logx.dx`

Find the area of the region bounded by the parabola y² = 4x and the line x = 3.

For the differential equation, find the particular solution (x – y²x) dx – (y + x²y) dy = 0 when x = 2, y = 0

If A = `[(4, 3, 2),(-1, 2, 0)]`, B = `[(1, 2),(-1, 0),(1, -2)]`

Find (AB)^–1 by adjoint method.

Solution:

AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`

AB = [  ]

|AB| =  `square`

M₁₁ = –2  ∴ A₁₁ = (–1)¹⁺¹ . (–2) = –2

M₁₂ = –3     A₁₂ = (–1)¹⁺² . (–3) = 3

M₂₁ = 4       A₂₁ = (–1)²⁺¹ . (4) = –4

M₂₂ = 3       A₂₂ = (–1)²⁺² . (3) = 3

Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`

adj (A) = [  ]

A^–1 = `1/|A| . adj(A)`

A^–1 = `square`

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.

Solution:

Let N be the number of bacteria present at time ‘t’.

Since the rate of increase of N is proportional to N, the differential equation can be written as –

`(dN)/dt αN`

∴ `(dN)/dt` = KN, where K is constant of proportionality

∴ `(dN)/N` = k . dt

∴ `int 1/N dN = K int 1 . dt`

∴ log N = `square` + C   ...(1)

When t = 0, N = N₀ where N₀ is initial number of bacteria.

∴ log N₀ = K × 0 + C

∴ C = log N₀

Also when t = 4, N = 2N₀

∴ log (2 N₀) = K . 4 + `square`   ...[From (1)]

∴ `log((2N_0)/N_0)` = 4K,

∴ log 2 = 4K

∴ K = `square`   ...(2)

Now N = ? when t = 12

From (1) and (2)

log N = `1/4 log 2  . (12) + log N_0`

log N – log N₀ = 3 log 2

∴ `log(N_0/N_0)` = `square`

∴ N = 8 N₀

∴ Bacteria are increased 8 times in 12 hours.

The sum due is also called as ______.

Face value

Present value

Cash value

True discount

______ is a series of constant cash flows over a limited period of time.

Perpetuity

Annuity

Present value

Future value

b_YX is ______.

Regression coefficient of Y on X

Regression coefficient of X on Y

Correlation coefficient between X and Y

Covariance between X and Y

The complicated but efficient method of measuring trend of time series is ______.

graphical method

method of moving average

method of least squares

method of addition

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

`sum("q"_1"w")/("q"_0"w") xx 100`

`sum("q"_0"w")/("q"_1"w") xx 100`

`(sum"q"_1"w")/(sum"q"_0"w") xx 100`

`(sum"q"_0"w")/(sum"q"_1"w") xx 100`

If the corner points of the feasible region are (0, 0), (3, 0), (2, 1) and `(0, 7/3)` the maximum value of z = 4x + 5y is ______.

12

13

`(35)/(2)`

0

The banker’s discount is also called true discount.

True

False

Laspeyre’s Price Index Number uses current year’s quantities as weights.

True

False

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

True

False

The date by which the buyer is legally allowed to pay the amount is known as _______.

Walsh’s Price Index Number is given by _______.

Graphical solution set of the inequations x ≥ 0 and y ≤ 0 lies in ______ quadrant.

An agent places insurance for ₹ 4,00,000 on life of a person. The premium is to be paid annually at the rate of ₹ 35 per thousand per annum. Find the agent’s commission at 15% on the premium.

For a bivariate data:

`sum(x - overlinex)^2` = 1200, `sum(y - overliney)^2` = 300, `sum(x - overlinex)(y - overliney)` = – 250

Find: 

1. b_yx
2. b_xy
3. Correlation coefficient between x and y.

The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Obtain trend values using 5-yearly moving values.

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

Maximize: Z = 4x + 6y

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

A marketing manager has list of salesmen and territories. Considering the travelling cost of the salesmen and the nature of territory, the marketing manager estimates the total of cost per month (in thousand rupees) for each salesman in each territory. Suppose these amounts are as follows:

Find the assignment of salesman to territories that will result in minimum cost.

A random variable X has the following probability distribution:

Find:

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

An agent was paid ₹ 88,000 as a commission on the sales of computers at the rate of 12.5%. If the price of each computer was ₹ 32,000, how many computers did he sell?

The publisher of a magazine wants to determine the rate of increase in the number of subscribers. The following table shows the subscription information for eight consecutive years:

Fit a trend line by graphical method.

Solve the following problem :

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, find variance of X.

For the following data, find the regression line of Y on X

Hence find the most likely value of y when x = 4.

Calculate Marshall – Edgeworth’s price index number for the following data:

Six jobs are performed on Machines M₁ and M₂ respectively. Time in hours taken by each job on each machine is given below:

Determine the optimal sequence of jobs and find total elapsed time. Also find the idle time for machines M₁ and M₂.

Solution:

Given jobs can be arranged in optimal sequence as,

Total Elapsed time = `square` hrs.

Idle time for Machine M₁ = 43 – 42 = 1 hour.

Idle time for Machine M₂ = `square` hrs.

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of getting at least two success.

Solution:

A pair of dice is thrown 3 times.

∴ n = 3

Let x = number of success (doublets)

p = probability of success (doublets)

∴  p = `square`, q = `square`

∴ x ∼ B (n, p)

P(x) = ⁿC_xp^x q^n–x

Probability of getting at least two success means x ≥ 2.

∴ P(x ≥ 2) = P(x = 2) + P(x = 3)

= `square` + `square`

= `2/27`