Mathematics and Statistics Model set 2 by shaalaa.com 2024-2025 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Choose the correct alternative.

If A = `[(α, 4),(4, α)]` and |A³| = 729, then α = ______.

±3

±4

±5

±6

Choose the correct alternative:

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

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

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

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

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

Slope of the tangent to the curve y = 6 – x² at (2, 2) is ______.

4

–4

–2

–1

Choose the correct alternative:

The solution of `dy/dx` = 1 is ______.

x + y = c

xy = c

x² + y² = c

y – x = c

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

constant

inelastic

unitary elastic

elastic

`int_0^1 1/(2x + 5) dx` = ______.

`1/2` log `7/5`

`1/2` log `5/7`

log `7/5`

`1/4` log `7/5`

`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.

True

False

`int(7x - 2)^2dx = (7x -2)^3/21 + c`

True

False

State whether the following is True or False:

The derivative of `log_ax`, where a is constant is `1/(x.loga)`.

True

False

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

Using definite integration, area of the circle x² + y² = 49 is _______.

`int (f^'(x))/(f(x))dx` = ______ + c.

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

y = Ae^3x + Be^–3x

Write the converse, inverse, contrapositive of the following statement.

If a man is bachelor, then he is happy.

Determine the maximum and minimum value of the following function.

f(x) = `x^2 + 16/x`

The sum of the cost of one Economic book, one Co-operation book and one account book is ₹ 420. The total cost of an Economic book, 2 Co-operation books and an Account book is ₹ 480. Also the total cost of an Economic book, 3 Co-operation books and 2 Account books is ₹ 600. Find the cost of each book using matrix method.

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

Solve: `int sqrt(4x^2 + 5)dx`

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

Draw Venn diagram for the following:

No policeman is thief

Draw Venn diagram for the following:

Some doctors are rich

Draw Venn diagram for the following:

Some students are not scholars

Solve the following differential equation.

`dy/dx + y` = 3

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

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

Find `(dy)/(dx)`, if x^y = y^x 

The rate of disintegration of a radioactive element at time t is proportional to its mass at that time. The original mass of 800 gm will disintegrate into its mass of 400 gm after 5 days. Find the mass remaining after 30 days.

Solution: If x is the amount of material present at time t then `dx/dt = square`, where k is constant of proportionality.

`int dx/x = square + c` 

∴ logx = `square`

x = `square` = `square`.e^c

∴ x = `square`.a where a = e^c

At t = 0, x = 800

∴ a = `square`

At t = 5, x = 400

∴ e^–5k = `square`

Now when t = 30 

x = `square` × `square` = 800 × (e^–5k)⁶ = 800 × `square` = `square`.

The mass remaining after 30 days will be `square` mg.

Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0

y = `a + b/x`

`(dy)/(dx) = square`

`(d^2y)/(dx^2) = square`

Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`

= `x square + 2 square`

= `square`

Hence y = `a + b/x` is solution of `square`

If there are n jobs and m machines, then there will be_______ sequences of doing the jobs.

mn

m(n!)

n^m 

(n!)^m

If E(x) > Var(x) then X follows _______.

Binomial distribution

Poisson distribution

Normal distribution

None of the above

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

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

`sum(q_1)/(q_0) xx 100`

`sum(q_0)/(q_1) xx 100`

`(sumq_1)/(sumq_0) xx 100`

`(sumq_0)/(sumq_1) xx 100`

Choose the correct alternative:

There are ______ types of regression equations

4

2

3

1

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

Banker’s discount

True discount

Banker’s gain

Cash value

If X ∼ B(2, 3) then E(X) = 5.

True

False

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

True

False

`(sump_1q_0)/(sump_0q_0) xx 100` is Paasche’s Price Index Number.

True

False

The region represented by the inequality y ≤ 0 lies in _______ quadrants.

Fill in the Blank.

A _______ is an agent who brings together the buyer and the seller.

Conditions under which the object function is to be maximum or minimum are called ______.

In the modification of a plant layout of a factory 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 of locating a machine at a place (in hundred rupees) is as follows.

Find the optimal assignment schedule.

Two cards are randomly drawn, with replacement. from a well shuffled deck of 52 playing cards. Find the probability distribution of the number of aces drawn.

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.

The probability distribution of X is as follows:

Find

1. k
2. P[X < 2]
3. P[X ≥ 3]
4. P[1 ≤ X < 4]
5. P(2)

The Price Index Number for year 2004, with respect to year 2000 as base year. is known to be 130. Find the missing numbers in the following table if ∑p₀ = 320

In a cattle breeding firm, it is prescribed that the food ration for one animal must contain 14, 22, and 1 unit of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit weight of these two contains the following amounts of these three nutrients:

The cost of fodder 1 is ₹ 3 per unit and that of fodder ₹ 2 per unit. Formulate the L.P.P. to minimize the cost.

Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Determine whether each of the following is a probability distribution. Give reasons for your answer.

Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at 5 stations I, II, III, IV, and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Verify whether f(x) is a p.d.f.

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Find P(0 < X ≤ 1).

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Find probability that X is between 1 and 3..

The following results were obtained from records of age (x) and systolic blood pressure (y) of a group of 10 women.

`sum(x_i - barx)(y_i - bary)` = 1170

If X – P(m) with P(X = 1) = P(X = 2), then find the mean and P(X = 2) given that e^–2 = 0.1353

Solution: Since P(X = 1) = P(X = 2)

`(e^-mm^1)/(1!) = square`

∴ m = `square`

∴ mean = `square` = `square`

Then P(X = 2) = `square` = `square`

For an annuity due, C = ₹ 2000, rate = 16% p.a. compounded quarterly for 1 year

∴ Rate of interest per quarter = `square/4` = 4

⇒ r = 4%

⇒ i = `square/100 = 4/100` = 0.04

n = Number of quarters

= 4 × 1

= `square`

⇒ P' = `(C(1 + i))/i [1 - (1 + i)^-n]`

⇒ P' = `(square(1 + square))/0.04 [1 - (square + 0.04)^-square]`

= `(2000(square))/square [1 - (square)^-4]`

= 50,000`(square)`[1 – 0.8548]

= ₹ 7,550.40