Mathematics and Statistics Model set 1 shaalaa.com 2021-2022 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Choose the correct alternative:

General solution of `y - x ("d"y)/("d"x)` = 0 is

`3log x + 7/y` = c

`2log x + 3/y = c`

log x – log y = log c

`3log y + 2/x` = c

Choose the correct alternative:

The function f(x) = x³ – 3x² + 3x – 100, x ∈ R is

increasing for all x ∈ R, x ≠ 1 

decreasing

neither increasing nor decreasing

decreasing for all x ∈ R, x ≠ 1

Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

`1/3`

`1/2`

`1/4`

2

Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

`1/3`

`1/2`

`1/4`

2

The false statement in the following is ______.

p ˄ (∼ p) is contradiction

(p → q) ↔ (∼ q → ∼ p) is a contradiction

∼ (∼ p) ↔ p is a tautology

p ˅ (∼ p) ↔ p is a tautology

Choose the correct alternative:

`int_2^3 x/(x^2 - 1)  "d"x` =

`log (8/3)`

`- log (8/3)`

`1/2 log(8/3)`

`-1/2 log(8/3)`

State whether the following statement is True or False:

Order and degree of differential equation `x ("d"^3y)/("d"x^3) + 6(("d"^2y)/("d"x^2))^2 + y` = 0 is (2, 2)

True

False

State whether the following statement is True or False:

If `[(3, 0),(0, 2)][(x),(y)] = [(3),(2)]`, then x = 1 and y = – 1

True

False

State whether the following statement is True or False:

For `int (x - 1)/(x + 1)^3  "e"^x"d"x` = e^x f(x) + c, f(x) = (x + 1)²

True

False

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

The total cost function for production of articles is given as C = 100 + 600x – 3x², then the values of x for which the total cost is decreasing is  ______

`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x⁵ – ______ x³ + 5x + c

Solve the following differential equation:

`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`

Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°”

Find the area of the region bounded by the curve x = `sqrt(25 - y^2)`, the Y-axis lying in the first quadrant and the lines y = 0 and y = 5

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`

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

If the demand function is D = 50 – 3p – p². Find the elasticity of demand at p = 5 comment on the result

Find k, if A = `[(3, -2),(4, -2)]` and A² = kA – 2I, where I is identity matrix of order 2

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`

Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing

Solution: Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = `square`

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = `square`

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ Profit is increasing for `square`

By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`.

Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`     ......(i)

Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x`, we get

I = `int_2^5 ("(  )")/(sqrt(7 - x) + "(  )")  "d"x`   ......(ii)

Adding equations (i) and (ii), we get

2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x))  "d"x + (   )  "d"x`

2I = `int_2^5 (("(    )" + "(     )")/("(    )" + "(     )"))  "d"x`

2I = `square`

∴ I =  `square`

Choose the correct alternative:

If the lines of regression of Y on X is y = `x/4` and X on Y is x = `y/9 + 1` then the value of r is

`1/6`

0

`- 1/4`

`- 1/6`

Choose the correct alternative:

If there are 3 machines A, B and C, conditions for reducing a 3 machine problem to a 2 machine problem with respect to minimum processing time is ______

Min A_i ≥ Max B_i OR Min C_i ≥ Max B_i, i = 1, 2, 3…n

Min A_i ≤ Max B_i OR Min C ≤ Max B_i, i = 1, 2, 3…n

Max A_i ≥ Min B_i OR Max B ≥ Min A_i, i = 1, 2, 3…n

Max A_i ≤ Min B_i OR Max B ≤ Min A_i, i = 1, 2, 3…n

Choose the correct alternative:

If b_yx < 0 and b_xy < 0, then r is ______

< 0

> 0

c = 0

> 1

Choose the correct alternative:

If A bill of ₹ 6,395 drawn on 15th February 2015 for 10 months was discounted on 28th May 2015 at 8% p.a. interest, then legal due date is ______

15^th December 2015

15^th November 2015

18^th December 2015

18^th November 2015

Multiple choice questions:

The present value of an immediate annuity of ₹ 10,000 paid each quarter for four quarters at 16% p.a. compounded quarterly is ______

40,000

36,300

36,286.75

36289.25

Choose the correct alternative:

The point at which the minimum value of Z = 8x + 12y subject to the constraints 2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0 is obtained at the point

(8, 0)

(9, 1)

(2, 4)

(10, 0)

State whether the following statement is True or False:

The cumulative distribution function (c.d.f.) of a continuous random variable X is denoted by F and defined by

F(x) = `{:(0",",  "for all"  x ≤ "a"),( int_"a"^x  f(x) "d"x",",  "for all"  x ≥ "a"):}`

True

False

State whether the following statement is True or False:

If b_yx = 1.5 and b_xy = `1/3` then r = `1/2`, the given data is consistent

True

False

State whether the following statement is True or False: 

A factor is an agent who is given the possession of goods and enters a contract for sale in his/her own name

True

False

In sequencing problem the time which required to complete all the jobs i.e. entire task is called ______

If n = 5, ∑xy = 76, ∑x² = ∑y² = 90, ∑x = 20 = ∑y, the covariance = ______

A doctor prescribed 2 types of vitamin tablets, T₁ and T₂ for Mr. Dhawan. The tablet T₁ contains 400 units of vitamin and T₂ contains 250 units of vitamin. If his requirement of vitamin is at least 4000 units, then the inequation for his requirement will be ______

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

Solve the following problem :

Five jobs must pass through a lathe and a surface grinder, in that order. The processing times in hours are shown below. Determine the optimal sequence of the jobs. Also find the idle time of each machine.

Obtain the trend values for the data, using 3-yearly moving averages

For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y − 5x + 180 = 0.  The variance of marks in statistics is `(9/16)^"th"` of the variance of marks in accountancy. Find the correlation coefficient between marks in two subjects.

A retailer sold a suit for ₹ 8,832 after allowing 8% discount on marked price and further 4% cash discount. If he made 38% profit, find the cost price and the marked price of the suit.

Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F₁ and F₂. F₁ provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F₂ provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F₁ costs ₹ 3 and F₂ costs ₹ 2 per 10 grams. How many grams of each F₁ and F₂ should buy every day to keep her food bill minimum

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e⁻³ = 0.0497

A 35-year old person takes a policy for ₹ 1,00,000 for a period of 20 years. The rate of premium is ₹ 76 and the average rate of bonus is ₹ 7 per thousand p.a. If he dies after paying 10 annual premiums, what amount will his nominee receive?

Find Price Index Number using Simple Aggregate method by taking 2005 as base year.

The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

Calculate
a) Laspeyre’s
b) Passche’s
c) Dorbish-Bowley’s Price Index Numbers for following data.

For the following assignment problem minimize total man hours:

Subtract the `square` element of each `square` from every element of that `square`

Subtract the smallest element in each column from `square` of that column.

The lines covering all zeros is `square` to the order of matrix `square`

The assignment is made as follows:

Optimum solution is shown as follows:

A → `square, square` → III, C → `square, square` → IV

Minimum hours required is `square` hours

Complete the table using 4 yearly moving average method.