Mathematics and Statistics Model set 3 shaalaa.com 2021-2022 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If A² + 5A + 3I = 0, |A| ≠ 0, then A^–1 = ______

`(-1)/3 ("A" + 5"I")`

`(-1)/5 ("A" + 3"I")`

(A + 15I)

`(-1)/3 ("I" + 5"A")`

Choose the correct alternative:

Negation of p → (p ˅ ~q) is

~p → (~p ˅ q)

p ˄ (~p ˄ q)

~p ˅ (~p ˅ ~q)

~p → (~p → q)

Choose the correct alternative:

Slope of the normal to the curve 2x² + 3y² = 5 at the point (1, 1) on it is 

`-2/3`

`2/3`

`3/2`

`-3/2`

`int_"a"^"b" "f"(x)  "d"x` = ______

`int_"b"^"a" "f"(x)  "d"x`

`- int_"a"^"b" "f"(x)  "d"x`

`- int_"b"^"a" "f"(x)  "d"x`

`int_0^"a" "f"(x)  "d"x`

Choose the correct alternative:

The integrating factor of `("d"^2y)/("d"x^2) - y` = e^x, is e^–x, then its solution is

ye^–x = x + c

ye^x = x + c

ye^x = 2x + c

ye^–x = 2x + c

`int(x + 1/x)^3 dx` = ______.

`1/4(x + 1/x)^4 + c`

`x^4/4 + (3x^2)/2 + 3log x - 1/(2x^2) + c`

`x^4/4 + (3x^2)/2 + 3log x + 1/x^2 + c`

`(x - x^(-1))^3 + c`

The area of the shaded region bounded by two curves y = f(x), and y = g(x) and X-axis is `int_"a"^"b" "f"(x) "d"x + int_"a"^"b" "g"(x)  "d"x`

True

False

State whether the following statement is True or False:

If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1|  + B log|x – 2|, then A + B = 1

True

False

State whether the following statement is True or False:  

The degree of a differential equation `"e"^(-("d"y)/("d"x)) = ("d"y)/("d"x) + "c"` is not defined

True

False

If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______

Order and degree of differential equation`(("d"^3y)/("d"x^3))^(1/6)`= 9 is ______

State whether the following statement is True or False:

If `int x  "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`

True

False

Solve: `("d"y)/("d"x) + 2/xy` = x² 

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

"If it snows, then they do not drive the car"

Find the area of the region bounded by the curve y = `sqrt(9 - x^2)`, X-axis and lines x = 0 and x = 3

The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.

Find MPC, MPS, APC and APS, if the expenditure E_c of a person with income I is given as E_c = (0.0003) I² + (0.075) I ; When I = 1000.

Evaluate `int_2^3 x/((x + 2)(x + 3))  "d"x`

Find `("d"y)/("d"x)`, if y = [log(log(logx))]² 

If A = `[(2, 1),(0, 3),(1, -1)]` and B = `[(0, 3, 5),(1, -7, 2)]`, then verify (BA)^T = A^TB^T

Find the values of x such that f(x) = 2x³ – 15x² + 36x + 1 is increasing function

`int x/((x - 1)^2 (x + 2)) "d"x`

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Find `("d"y)/("d"x)`, if x = e^m, y = `"e"^(sqrt("m"))`

Solution: Given, x = e^m and y = `"e"^(sqrt("m"))`

Now, y = `"e"^(sqrt("m"))`

Diff.w.r.to m,

`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`

∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))`    .....(i)

Now, x = e^m

Diff.w.r.to m,

`("d"x)/"dm" = square`    .....(ii)

Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`

∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`

∴  `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

Find the population of city at any time t given that rate of increase of population is proportional to the population at that instant and that in a period of 40 years the population increased from 30000 to 40000.

Solution: Let p be the population at time t.

Then the rate of increase of p is `"dp"/"dt"` which is proportional to p.

∴ `"dp"/"dt" prop "p"`

∴ `"dp"/"dt"` = kp, where k is a constant.

∴ `"dp"/"p"` = k dt

On integrating, we get

`int "dp"/"p" = "k" int "dt"`

∴ log p = kt + c

Initially, i.e. when t = 0, let p = 30000

∴ log 30000 = k × 0 + c       

∴ c = `square`

∴ log p = kt + log 30000

∴ log p - log 30000 = kt

∴ `log("p"/30000)` = kt          .....(1)     

when t = 40, p = 40000

∴ `log (40000/30000) = 40"k"`

∴ k = `square`

∴ equation (1) becomes, `log ("p"/30000)` = `square`

∴ `log ("p"/30000) = "t"/40 log (4/3)`

∴ p = `square`

Choose the correct alternative:

The feasible region is

common region determined by all the constraints

common region determined by the non-negativity constraints

either common region determined by all the constraints or common region determined by the non-negativity constraints

both common region determined by all the constraints and common region determined by the non-negativity constraints

Choose the correct alternative:

Walsh's Price Index Number is given by

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

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

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

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

Choose the correct alternative:

The date on which the period of the bill expires is called  ______

Legal Due Date

Days of grace

The Nominal Due date

Date of Drawing

Choose the correct alternative:

A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies ______

The trials are independent.

The probability of success remains the same in all trials.

The trials are independent but not the probability of success remains the same in all trials.

both trials are independent but not the probability of success remains the same in all trials.

Multiple choice questions:

A shop valued ₹ 2,00,000 is insured at 80% of it’s value. If the rate of premium is 4%, then the premium is ______

6,400

6,000

6,450

6,500

Choose the correct alternative:

The following trend line equation was developed for annual sales from 1984 to 1990 with 1984 as base or zero year.

Y = 500 + 60X (in 1000 ₹). The estimated sales for 1984 (in 1000 ₹) is

500

560

1,040

1,100

State whether the following statement is True or False:

In sequencing problem the processing times are dependent of order of processing the jobs on machine

True

False

State whether the following statement is True or False: 

If only one discount is given then List price = Invoice price

True

False

State whether the following statement is True or False: 

Moving average method of finding trend is very complicated and involves several calculations

True

False

Heramb requires at most 400 calories from his breakfast. Every morning he likes to take oats and milk. If each bowl of oats and a glass of milk provides him 80 calories and 50 calories respectively, then as a constraint this information can be expressed as ______

The optimum sequence for the above data is ______

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

Find the amount of an ordinary annuity if a payment of ₹ 500 is made at the end of every quarter for 5 years at the rate of 12% per annum compounded quarterly. [Given (1.03)²⁰ = 1.8061]

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.

The Cost of Living Index Numbers for years 2003 and 2008 are 150 and 200 respectively. A person earned ₹ 18,000 per month in year 2003. What should be his earning per month in year 2008, so as to maintain same standard of living as 2003?

If difference between true discount and banker’s discount on a sum due 4 months hence is ₹ 20. Find true discount, banker’s discount and amount of bill, the rate of simple interest charged being 5%p.a.

Use the method of least squares to fit a trend line to the data given below. Also, obtain the trend value for the year 1975.

Two samples from bivariate populations have 15 observations each. The sample means of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148 respectively. The sum of product of deviations from respective means is 122. Obtain the regression equation of x on y

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

Find the sequence that minimizes total elapsed time to complete the following jobs in the order XY. Find the total elasped time and idle times for each machine.

Given P₀₁(M-E) = 120, `sum"p"_1"q"_1` = 300, `sum"p"_0"q"_0` = 120, `sum"p"_0"q"_1` = 320, Find P₀₁(L)

For a certain bivariate data of a group of 10 students, the following information gives the internal marks obtained in English (X) and Hindi (Y):

If r = 0.6, Estimate x when y = 16 and y when x = 10

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 matrix is given below.

Find the optimal assignment schedule

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, using the following activity find the value of m.

Solution: X : Follows Poisson distribution

∴ P(X) = `("e"^-"m" "m"^x)/(x!)`, P(X = 1) = 0.4 and P(X = 2) = 0.2

∴ P(X = 1) = `square` P(X = 2).

`("e"^-"m" "m"^x)/(1!) = square ("e"^-"m" "m"^2)/(2!)`,

`"e"^-"m" = square  "e"^-"m" "m"/2`, m ≠ 0

∴ m = `square`

Solve the following LPP graphically:

Maximize Z = 9x + 13y subject to constraints

2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0

Solution: Convert the constraints into equations and find the intercept made by each one of it.

The feasible region is OAPC, where O(0, 0), A(0, 6),

P( ___, ___ ), C(5, 0)

The optimal solution is in the following table:

∴ Z is maximum at __( ___, ___ ) with the value ___.