Mathematics and Statistics 2013-2014 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Write the following statements in symbolic forms :
(a) Either 49 is a perfect square or 39 is divisible by 11.
(b) It is not true that if 'i is a real number, then '2' is an even prime number.

If A = `[(1,2,3), (2,k,2), (5,7,3)]` is a singular matrix then find the value of 'k'.

If A = `[(7,1), (2,5)]` and B = `[(1,2), (3,-1)]` then verify that |AB| = |A|  |B|.

Discuss the continuity of the function at the point given. If the function is discontinuous, then remove the discontinuity.

f (x) = `(sin^2 5x)/x^2` for x ≠ 0 
= 5   for x = 0, at x = 0

Find the value of x for which the function `f(x) = x^3 - 3x^2 - 9x + 25` is increasing.

Differentiate tan^-1 (cot 2x) w.r.t.x.

Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1

= `-1/3`   for x = 1, at x = 1

Evaluate : `int e^x [(x + 3)/(x + 4)^2] dx`

Without using the truth table, show that p ∧ [(∼p ∨ q) ∨ ∼q] ≡ p

If Y = tan^-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`

Evaluate = `int (tan x)/(sec x + tan x)` . dx

If the function
f(x) = x2 + ax + b,         x < 2

      = 3x + 2,                 2≤ x ≤ 4

      = 2ax + 5b,             4 < x

is continuous at x = 2 and x = 4, then find the values of a and b

The total cost function of a firm is C = x² + 75x + 1600 for output x. Find the  output for which the average cost ls minimum. Is C_A= C_m at this output?  

Find the area of the ellipse `x^2/4 + y^2/25 = 1`

Examine whether the following statement (p ∧ q) ∨ (∼p ∨ ∼q) is a tautology or contradiction or neither of them.

Find `(dy)/(dx)` if x = a cosec θ, y = b cot θ at θ = `π/4`

Evaluate : `int 1/(x^2 + 8x + 20)` dx

Express the following equations in matrix form and solve them by the method of inversion.
x + 2y + 3z = 8,
2x - y + z = 1,
3x + y - 4z = 1

The expenditure Ee of a person with income. x  ls given by Ec = (0.000035).x² + (0.045)x~ Find the marginal propensity to consume and marginal propensity to save when x = 5000. Also find the average propensity to consume and average propensity to save.

Evaluate : `int_(π/6)^(π/3) 1/(1 + sqrt(cotx)` dx

Obtain the Crude Death Rate (CDR) for city A from the data given below.

Compute the product moment coefficient of correlation for the following data: 
n = 100, `bar x` = 62, `bary` = 53, `sigma_x` = 10, `sigma_y` = 12

`Sigma (x_i - bar x) (y_i - bary) = 8000`

Find the age standard death rate (S.D.R.) for the following data:

Using the above information find x, if the CDR = 13.4 per thousand.

The ranks of the 10 students in Mathematics and Accountancy are as follows  : 
Two numbers given in the brackets denote the ranks of the students in Mathematics and Accountancy respectively.
(1. 1). (2. 10). (3. 3). (4. 4). (5. 5). (6. 7). (7. 2). (8. 6). (9. 8). (10. 9) 
Calculate the rank correlation coefficient

If X ∼ N (4,25), then find P(x ≤ 4)

Find common region for the following system of linear inequations :

x ≥ -3, 4x - 5y ≥ -20, 3x + 4y ≤ 12, y ≥ -2

For the following problem, find the sequence that minimize total elapsed time (in hours) required to complete the following jobs on two machines M₁ and M₂ in the order M₁ - M₂: 

A company produces mixers and food processor. Profit on selling one mixer and one food processor is Rs 2,000 and Rs 3,000 respectively. Both the products are processed through three machines A, B, C. The time required in hours for each product and total time available in hours per week on each machine arc as follows:

Formulate the problem as L.P.P. in order to maximize the profit.

The defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have-

(a) No defects
(b) At least one defect 
[Use e^-1 = 0.3678]

A Banker's discount calculated for 1 year is 26 times his gain. Find the rate or interest.

For the two regression equations 4y = 9x + 15 and 25x = 6y + 7 find correlation coefficient r, `barx, bary`

Incomes or Mr. Shah, Mr. Patel and Mr. Mehta are ln the ratio 1 : 2 : 3, while their expenditures are in the ratio 2 : 3 : 4. if Mr. Shah saves 20% or his lncome, find the ratio of their savings.

A sample of 5 items is taken from the production of a firm. Length and weight of the five items arc given below :

Calculate Karl Pearson's coefficient of   correlation between the length and weight and interpret the result.

A departmental head has three jobs and four subordinates. The subordinates differ in their capabilities and the jobs differ in their work
contents. With the help of the performance matrix given below, find out which of the four subordinates should be assigned which jobs ?

Raghu, Madhu and Ramu started a business in partnership by investing Rs 60,000, Rs 40,000 and Rs 75,000 respectively. At the end of the year, they found that they have incurred a loss of Rs 24,500. Find how much loss each one had to bear.

A card is drawn at random and replaced four times from a well shuftled pack of 52 cards. Find the probability that -

(a) Two diamond cards are drawn.
(b) At least one diamond card is drawn.

Find tha agents commission at 15% on the first premium if he places insurance for Rs 2,00,000 on the life of a person, the premium being at the rate of Rs 35 per thousand, per annum, paid annually.

From the following table which relates to the number of animals of a certain
species at age x. complete the life table :

Information on v:ehicles [in thousands) passing through seven different highways during a day (X) and number of accidents reported (Y) is given as follows :   

`Sigmax_i` = 105, `Sigmay_i` = 409, n = 7, `Sigmax_i^2` = 1681, `Sigmay_i^2` = 39350 `Sigmax_iy_i` = 8075

  Obtain the linear regression of Y on X.

In a factory there are six jobs to be performed each of which should go through two machines A and B in the order A - B. The processing timing (in hours) for the jobs arc given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T. What is the value of T? Also find the idle time for machines · A and B.