Mathematics and Statistics Set 1 2021-2022 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If A = `[(2, 3),(a, 6)]` is a singular matrix, then a = ______.

6

– 5

3

4

`int 1/sqrt(x^2 - 9) dx` = ______.

`1/3 log |x + sqrt(x^2 - 9)| + c`

`log |x + sqrt(x^2 - 9)| + c`

`3log |x + sqrt(x^2 - 9)| + c`

`log |x - sqrt(x^2 - 9)| + c`

The slope of a tangent to the curve y = 3x² – x + 1 at (1, 3) is ______.

5

– 5

`(-1)/5`

`1/5`

The order and degree of the differential equation `[1 + ((dy)/(dx))^3]^(2/3) = 8((d^3y)/(dx^3))` are respectively ______.

3, 1

1, 3

3, 3

1, 1

The area of the region bounded by the curve y = x², x = 0, x = 3, and the X-axis is ______.

9 sq.units

`26/3` sq.units

`52/3` sq.units

18 sq.units

`int_(-5)^5  x^7/(x^4 + 10)  dx` = ______.

10

5

0

`1/5`

State whether the following statement is true or false.

If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).

True

False

State whether the following statement is true or false.

If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2e^x – 5| + c, where c is the constant of integration, then A = 5.

True

False

State whether the following statement is true or false.

The integrating factor of the differential equation `(dy)/(dx) + y/x` = x³ is – x.

True

False

If p ∨ q is true, then the truth value of ∼ p ∧ ∼ q is ______.

`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`

y² = (x + c)³ is the general solution of the differential equation ______.

Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."

If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`

Find the area between the two curves (parabolas)

y² = 7x and x² = 7y.

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

[(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)

If ax² + 2hxy + by² = 0, then prove that `(d^2y)/(dx^2)` = 0.

Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx

Find x, y, z if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, -2),(1, 3)]}[(2),(1)] = [(x  + 1),(y - 1), (3z)]`

Divide 20 into two ports, so that their product is maximum.

Solve the following differential equation

x²y dx – (x³ + y³)dy = 0

Find the inverse of the matrix A by using adjoint method.

where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`

Evaluate the following definite integral:

`int_1^3 logx.dx`

Complete the following activity to 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)I²

when I = 1000

In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, complete the following activity to find the number of times the bacteria are increased in 12 hours.

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

Banker’s discount

True discount

Banker’s gain

Cash value

b_xy . b_yx = ______.

V(X)

σ_x

r²

`σ_y^2`

Choose the correct alternative :

The assignment problem is said to be balanced if it is a ______.

Square matrix

Rectangular matrix

Unit matrix

Triangular matrix

Choose the correct alternative :

The price Index Number by Weighted Aggregate Method is given by ______.

`sum("p"_1"w")/("p"_0"w") xx 100`

`sum("p"_0"w")/("p"_1"w") xx 100`

`(sum"p"_1"w")/(sum"p"_0"w") xx 100`

`(sum"p"_0"w")/(sum"p"_1"w") xx 100`

The following function represents the p.d.f of a.r.v. X

f(x) = `{{:((kx;, "for"  0 < x < 2, "then the value of K is ")),((0;,  "otherwise")):}` ______ 

`3/2`

`1/2`

1

0

If X ~ B`(20, 1/10)`, then E(X) = ______

2

5

4

3

State whether the following statement is true or false:

lf X ∼ P(m) with P(X = 1) = P(X = 2) then m = 1.

True

False

State whether the following statement is true or false:

Dorbish-Bowley's Price Index Number is the square root of the product of Laspeyre's and Paasche's Index Numbers.

True

False

State whether the following statement is true or false:

To convert a maximization-type assignment problem into a minimization problem, the smallest element in the matrix is deducted from all elements of the matrix.

True

False

A wholesaler allows a 25% trade discount and a 5% cash discount. The net price of an article marked at ₹ 1,600 is ______.

For certain bivariate data on 5 pairs of observations given:

∑x = 20, ∑y = 20, ∑x² = 90, ∑y² = 90, ∑xy = 76 then b_xy = ______.

If P₀₁ (L) = 121, P₀₁ (P) = 100, then P₀₁ (F) = ______.

Find the equation of the line of regression of Y on X for the following data:

n = 8, `sum(x_i - barx).(y_i - bary) = 120, barx = 20, bary = 36, sigma_x = 2, sigma_y = 3`

A job production unit has four jobs P, Q, R, and S which can be manufactured on each of the four machines I, II, III, and IV. The processing cost of each job for each machine is given in the following table:

Find the optimal assignment to minimize the total processing cost.

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.

Calculate the cost of living index number for the following data by aggregative expenditure method:

Five jobs are performed first on machine M₁ and then on machine M₂. Time taken in hours by each job on each machine is given below:

Determine the optimal sequence of jobs and total elapsed time. Also, find the idle time for two machines.

The probability distribution of a discrete r.v. X is as follows:

1. Determine the value of k.
2. Find P(X ≤ 4)
3. P(2 < X < 4)
4. P(X ≥ 3)

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.

Solve the following LP.P.

Maximize z = 13x + 9y,

Subject to 3x + 2y ≤ 12,

x + y ≥ 4,

x ≥ 0,

y ≥ 0.

Obtain the trend values for the following data using 5 yearly moving averages:

A warehouse valued at ₹ 40,000 contains goods worth ₹ 2,40,000. The warehouse is insured against fire for ₹ 16,000 and the goods to the extent of 90% of their value. Goods worth ₹ 80,000 are completely destroyed, while the remaining goods are destroyed to 80% of their value due to a fire. The damage to the warehouse is to the extent of ₹ 8,000. Find the total amount that can be claimed under the policy.

A bill was drawn on 14^th April 2005 for ₹ 3,500 and was discounted on 6^th July 2005 at 5% per annum. The banker paid ₹ 3,465 for the bill. Find the period of the bill. 

Solve the following problem:

An examination consists of 5 multiple choice questions, in each of which the candidate has to decide which one of 4 suggested answers is correct. A completely unprepared student guesses each answer completely randomly. Find the probability that,

1. the student gets 4 or more correct answers.
2. the student gets less than 4 correct answers.

Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:

Complete the following activity to fit a trend line by method of least squares: