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

The negation of the proposition “If 2 is prime, then 3 is odd”, is ______.

If 2 is not prime, then 3 is not odd.

2 is prime and 3 is not odd.

2 is not prime and 3 is odd.

If 2 is not prime, then 3 is odd.

If A is a non-singular matrix, then det(A^–1) = ______.

1

0

det(A)

`1/(det(A)`

If y = 2x² + a² + 2² then `dy/dx` = ______.

4x

4x + 2a

4x + 4

2x

−2x

If the elasticity of demand η = 1, then demand is ______.

constant

inelastic

unitary elastic

elastic

`int(1 - x)^(-2) dx` = ______.

(1 − x)⁻¹ + c

(1 + x)⁻¹ + c

(1 − x)⁻¹ − 1 + c

(1 − x)⁻¹ + 1 + c

(1 − x)⁻¹ − x + c

(1 − x)⁻¹ + x + c

`int_(-7)^7 x^3/(x^2 + 7) * dx` = ______.

7

49

0

`(7)/(2)`

p ∧ t ≡ p

True

False

`int(f'(x))/(sqrt(f(x))) = sqrt(f(x)) + c`

True

False

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

True

False

To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.

Using the definite integration area of the circle x² + y² = 25 is ______.

The order and degree of `((dy)/(dx))^3 - (d^3y)/(dx^3) + ye^x` = 0 are ______.

Write the converse of the implication: 

“If Sanjay studies, then he will go to college.”

Write the inverse of the implication: 

“If Sanjay studies, then he will go to college.”

Write the contrapositive of the implication: 

“If Sanjay studies, then he will go to college.”

If A = `[(3, 1),(-1, 2)]`, prove that A² – 5A + 7I = 0, where I is unit matrix of order 2

If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`

Solve the following differential equation.

x²y dx − (x³ + y³) dy = 0

The total cost of producing x units is ₹ (x² + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?

Evaluate:

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

Using the truth table, prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

If x = t . log t, y = t^t, then show that `dy/dx - y = 0`.

Find the area of the region bounded by y = x², the X-axis and x = 1, x = 4.

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

x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4

If the demand function is D = `((p + 6)/(p − 3))`, find the elasticity of demand at p = 4.

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

Solution:

Let I = `int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x)) dx`    ...(i)

By property `int_a^b f(x) dx = int_a^b f(a + b - x)dx`,

I = `int_1^3 square/(root(3)(9 - x) + square)`    ...(ii)

Adding (i) and (ii)

∴ I + 1 = `int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x)) dx + int_1^3 square/(root(3)(9 - x) + square)`

= `int_1^3 (root(3)(x + 5) + root(3)(9 - x))/(root(3)(x + 5) + root(3)(9 - x)) dx`

∴ 2I = `int_1^3 square` dx

= `[square]_1^3`

∴ 2I = `square`

∴ I = 1

Solve the differential equation: `dy/dx - y = 2x`

Solution:

The differential equation `dy/dx - y = 2x` is in the form of `dy/dx + py = Q`, where P = −1 and Q = 2x

∴ I.F. = `e^(intP dx) = square`

∴ The solution of the linear differential equation is

∴ `y square = int 2x square dx + c`

= `2{x int e^(-x) dx - int e^(-x) dx * d/dx (x) dx} + c`

= `2{x int square/square - int square/square dx} + c`

∴ `ye^(−x) = −2xe^(−x) + 2∫e^(−x) dx + c`

∴ `ye^(-x) = -2 xe^(-x) + 2((e^(-x))/-1) + c`

∴ `y + square = ce^x` is the required solution of the given differential equation. 

The payment date after adding 3 days of grace period is known as ______.

The legal due date

The nominal due date

Days of grace

Date of drawing

b_XY . b_YX = ______.

V(X)

σ_x

r²

`(σ_y)^2`

If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0), then the point of minimum z = 3x + 2y is ______.

(2, 2)

(2, 2)

(0, 10)

(0, 10)

(4, 0)

(4, 0)

(3, 4)

(2, 4)

If jobs A to D have processing times as 5, 6, 8, 4 on first machine and 4, 7, 9, 10 on second machine then the optimal sequence is ______.

CDAB

DBCA

BCDA

ABCD

ADBC

If E(X) = m and Var(X) = m then X follows ______.

Binomial distribution

Poisson distribution

Normal distribution

Both Binomial distribution and Poisson distribution

None of the above

If X ∼ B`(20, 1/10)` then Var(X) = ______.

`9/5`

2

`5/9`

`1/2`

The amount of the claim cannot exceed the amount of loss.

True

False

`(sump_0(q_0 + q_1))/(sump_1(q_0 + q_1)) xx 100` is Marshall-Edgeworth’s price index number.

True

False

If r.v. X assumes values 1, 2, 3, ..., n with equal probabilities then E(X) = `(n + 1)/(2)`.

True

False

The difference between the banker’s discount and the true discount is called ______.

A train carries at least twice as many first class passengers (y) as second class passengers (x). The constraint is given by ______.

If F(x) is the distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)` for x = 1, 2, 3 and P(x) = 0 otherwise then F(4) = _______.

A lady plans to save for her daughter’s marriage. She wishes to accumulate a sum of ₹ 4,64,100 at the end of 4 years. What amount should she invest every year if she gets an interest of 10% p.a. compounded annually? [Given (1.1)⁴ = 1.4641]

The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

1. Obtain trend values for the above data using 5-yearly moving averages.
2. Plot the original time series and trend values obtained above on the same graph.

Find x if the price index number by the simple aggregate method is 125.

Minimize z = 7x + y subjected to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0.

A bill of a certain sum drawn on 28^th February 2007 for 8 months was encashed on 26^th March 2007 for ₹ 10,992 at 14% p.a. Find the face value of the bill.

There are five jobs, each of which must go through two machines in the order XY. Processing times (in hours) are given below. Determine the sequence for the jobs that will minimize the total elapsed time. Also find the total elapsed time and idle time for each machine.

The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.

If ∑p₀q₀ = 140, ∑p₀q₁ = 200, ∑p₁q₀ = 350, ∑p₁q₁ = 460, find Laspeyre’s, Paasche’s, Dorbish-Bowley’s and Marshall-Edgeworth’s Price Index Numbers.

Find the expected value and variance of X using the following p.m.f.

The following results were obtained from records of age (X) and systolic blood pressure (Y) of a group of 10 men.

and `sum (x_i - bar x)(y_i - bar y) = 1120`. Find the prediction of blood pressure of a man of age 40 years.

Following table shows the number of traffic fatalities (in a state) resulting from drunken driving for the years 1975 to 1983:

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

A company has a team of four salesmen and there are four districts where the company wants to start its business. After taking into account the capabilities of salesmen and the nature of districts, the company estimates that the profit per day in rupees for each salesman in each district is as below:

Find the assignment of salesmen to various districts which will yield maximum profit.

Solution: It is a maximization problem. Subtract all the elements from `square`.

      `{:(1, 2, 3, 4):}`

`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(4, 3, 1, 1),(1, 1, 5, 2),(3, 2, 2, 1)]`

∴ Subtract the smallest element of each row from the elements of that row:

      `{:(1, 2, 3, 4):}`

`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`

∴ Subtract the smallest element of each column from the elements of that column:

      `{:(1, 2, 3, 4):}`

`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`

∴ Since the number of lines covering zeros is equal to the order of the matrix, the optimal solution has reached:

      `{:(1, 2, 3, 4):}`

`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`

The optimal solution is obtained.

∴ Total profit = ₹ `square`

If X has Poisson distribution with parameter m and P[X = 2] = P[X = 3], then find P[X ≥ 2].

[Given: e⁻³ = 0.0497]

Solution: X ∼ P(m)

∴ P[X = x] = `(e^(-m)m^x)/(x !)`

∴ P[X = 2] = `(e^(-m)m^2)/(2 !)`

∴ P[X = 3] = `square`

Now P[X = 2] = P[X = 3]

∴ `(e^(-m)m^2)/(2 !) = (e^(-m)m^3)/(3 !)`

∴ m = `square`

Now P[X ≥ 2] = 1 − `square`

= 1 − [P(X = 0) + P(X = 1)]

= `1 - [(e^(-3)3^0)/(0!) + (e^(-3)3^1)/(1!)]`

= 1 − e⁻³[1 + 3]

= 1 − 0.0497 × 4

Hence P[X ≥ 2] is `square`.