Mathematics and Statistics Model set 2 shaalaa.com 2021-2022 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

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

3, 1

1, 3

3, 3

1, 1

The price P for the demand D is given as P = 183 + 120D − 3D², then the value of D for which price is increasing, is ______.

D < 60

D > 60

D < 20

D > 20

Choose the correct alternative :

If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

∼ p ∨ ∼ p

∼ (p ∧ q)

∼ (p ∨ q)

p ∨ ∼ q

Choose the correct alternative:

`int(("e"^(2x) + "e"^(-2x))/"e"^x) "d"x` =

`"e"^x - 1/(3"e"^(3x)) + "c"`

`"e"^x + 1/(3"e"^(3x)) + "c"`

`"e"^(-x) - 1/(3"e"^(3x)) + "c"`

`"e"^(-x) + 1/(3"e"^(3x)) + "c"`

`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))  dx` = ______.

`7/2`

`5/2`

7

2

Choose the correct alternative:

Area of the region bounded by the parabola y² = 25x and the lines x = 5 is ______

`(75sqrt(5))/2` sq.units

`(20sqrt(5))/3` sq.units

`(100sqrt(5))/3` sq.units

`(75sqrt(5))/3` sq.units

State whether the following statement is True or False: 

The degree of a differential equation is the power of highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any

True

False

State whether the following statement is True or False:

The equation of tangent to the curve y = x² + 4x + 1 at (– 1, – 2) is 2x – y = 0 

True

False

State whether the following statement is True or False:

`int sqrt(1 + x^2) *x  "d"x = 1/3(1 + x^2)^(3/2) + "c"`

True

False

If A = [a_ij]_2×2, where a_ij = i – j, then A = ______

A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution

`int (x^2 + x - 6)/((x - 2)(x - 1))  "d"x` = x + ______ + c

Find the values of x such that f(x) = 2x³ – 15x² – 144x – 7 is decreasing function

Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]` 

Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours  `("Given"  sqrt(2) = 1.414)`

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

Evaluate `int_0^"a" x^2 ("a" - x)^(3/2)  "d"x`

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

(p ∧ ~ q) → (~ p ∧ ~ q)

Find the area of the region bounded by the curve y = (x² + 2)², the X-axis and the lines x = 1 and x = 3

Find the differential equation by eliminating arbitrary constants from the relation y = (c₁ + c₂x)e^x 

The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 − 𝑥). Find x for which profit is increasing

If x = `(4"t")/(1 + "t"^2)`, y = `3((1 - "t"^2)/(1 + "t"^2))`, then show that `("d"y)/("d"x) = (-9x)/(4y)` 

Complete the truth table.

The given statement pattern is a `square`

Complete the following activity.

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R₁ ↔ R₂

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R₂ – 4R₁, R₃ – 6R₁

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"_3`

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R₃ – 2R₂

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( )    .....(i)

y + 2z = 24    ......(ii)

–z = ( )      ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

Choose the correct alternative:

The present worth of ₹ 11,660 due 9 months hence is ₹ 11,000. The True discount is ______

₹ 660

₹ 750

₹ 400

₹ 5,940

Multiple choice questions:

If for an immediate annuity r = 10% p.a., P = ₹ 12,679.46 and A = ₹ 18,564, then the amount of each annuity paid is ______

₹ 4,000

₹ 4,500

₹ 3,500

₹ 4,200

Choose the correct alternative:

A distance random variable X is said to have the Poisson distribution with parameter m if its p.m.f. is given by P(x) = `("e"^(-"m")"m"^"x")/("x"!)` the condition for m is ______

m > 0

m ≥ 0

m ≠ 1

m = 0

If there are n jobs and m machines, then there will be _______ sequence of doing jobs.

mn

m(n!)

n^m 

(n!)^m

Choose the correct alternative:

The corner points of feasible region for the inequations, x + y ≤ 5, x + 2y ≤ 6, x ≥ 0, y ≥ 0 are

(0, 3), (5, 0), (0, 5), (6, 0)

(0, 3), (5, 0), (4, 1), (0, 0)

(0, 0), (1, 4), (5, 0), (0, 3)

(3, 0), (0, 5), (0, 0), (4, 1)

Choose the correct alternative:

f(x) is c.d.f. of discete r.v. X whose distribution is

then F(– 3) = ______

0

1

0.2

0.15

State whether the following statement is True or False:

The trade discount is first calculated on the catalogue (list) price

True

False

State whether the following statement is True or False:

If f(x) = `{:("k"x  (1 - x)",", "for"  0 < x < 1),(= 0",", "otherwise"):}`
is the p.d.f. of a r.v. X, then k = 12

True

False

State whether the following statement is True or False:

If the corner points of the feasible region are `(0, 7/3)`, (2, 1), (3, 0) and (0, 0), then the maximum value of Z = 4x + 5y is 12

True

False

If the regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214, the mean value of y is ______

An unbalanced assignment problems can be balanced by adding dummy rows or columns with ______ cost

If X is discrete random variable takes the values x₁, x₂, x₃, … x_n, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______

A merchant takes fire insurance policy to cover 80% of the value of his stock. Stock worth ₹ 80,000 was completely destroyed in a fire. while the rest of stock was reduced to 20% of its value. If the proportional compensation under the policy was ₹ 67,200, find the value of the stock

Find x if the Price Index Number by Simple Aggregate Method is 125

Find the assignments of salesman to various district which will yield maximum profit

Minimize Z = 2x + 3y subject to constraints

x + y ≥ 6, 2x + y ≥ 7, x + 4y ≥ 8, x ≥ 0, y ≥ 0

Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982

Fit a trend line by the method of least squares

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.

Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)

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.

Given the following information about the production and demand of a commodity.
Obtain the two regression lines:

Coefficient of correlation between x and y is 0.8.
What should be the advertising budget if the company wants to attain the sales target of ₹ 150 lakhs?

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has between 5 and 7 rats inclusive. Given e⁻⁵ = 0.0067.

Find the missing wage if the Cost of Living Index for the following data is 150.

Three cars were sold through an agent for ₹ 2,40,000, ₹ 2,22,000 and ₹ 2,25,000 respectively. The rates of commission were 17.5% on the first, 12.5% on the second. If the agent overall received 14% commission on the total sales, find the rate of commission paid on the third car.

Solution: Total selling Price of three cars = 2,40,000 + 2,22,000 + 2,25,000

= `square`

Commision on total sale = 14%

= `14/100 xx square`

Selling price of First car = ₹ 2,40,000

Rate of commission = 17.5% 

= `17.5/100 xx 2,40,000 = square`

∴ Commission on first car = ₹ `square`

Selling price of Second car = ₹ 2,22,000

Rate of commission = 12.5%

= `12.5/100 xx 2,22,000 = square`

∴ Commission on second car = ₹ `square`

Selling price of third car = ₹ 2,25,000

Let the rate of commission be x

Commission on third car = `x/100 xx 2,25,000`

∴ Commission on third car = Total commission − (commission on first car + commission on second car)

∴ `x/100 xx 2,25,000 = square - {square + square}`

∴ x = `square`

Mean of x = `barx = square`

Mean of y = `bary = square`

b_xy = `square/square`

b_yx = `square/square`

Regression equation of x on y is `(x - barx) = "b"_(xy)  (y - bary)`

∴ Regression equation x on y is `square`

Regression equation of y on x is `(y - bary) = "b"_(yx)  (x - barx)`

∴ Regression equation of y on x is `square`