Date: October 2013
(1) All questions are compulsory.
(2) Answer to every question must be written on new page.
(3) Log table will be provided on demand.
(4) Answer to Section-I and Section-II should be written in two seperate answer books.
(5) Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
(6) Graph paper is necessary for L.P.P.
(7) Figure to the right indicate full marks.
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.
Chapter: [0.14] Random Variable and Probability Distribution
If A = `[(1,2,3), (2,k,2), (5,7,3)]` is a singular matrix then find the value of 'k'.
Chapter: [0.02] Matrices
If A = `[(7,1), (2,5)]` and B = `[(1,2), (3,-1)]` then verify that |AB| = |A| |B|.
Chapter: [0.02] Matrices
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
Chapter: [0.03] Continuity
Find the value of x for which the function `f(x) = x^3 - 3x^2 - 9x + 25` is increasing.
Chapter: [0.05] Applications of Derivative
Differentiate tan^{-1} (cot 2x) w.r.t.x.
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
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
Chapter: [0.03] Continuity
Evaluate : `int e^x [(x + 3)/(x + 4)^2] dx`
Chapter: [0.07] Definite Integrals
Without using the truth table, show that p ∧ [(∼p ∨ q) ∨ ∼q] ≡ p
Chapter: [0.06] Indefinite Integration
If Y = tan^{-1 }`[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`
Chapter: [0.03] Continuity
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
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
Chapter: [0.03] Continuity
The total cost function of a firm is C = x^{2} + 75x + 1600 for output x. Find the output for which the average cost ls minimum. Is C_{A}= C_{m} at this output?
Chapter: [0.04] Differentiation
Find the area of the ellipse `x^2/4 + y^2/25 = 1`
Chapter: [0.06] Indefinite Integration
Examine whether the following statement (p ∧ q) ∨ (∼p ∨ ∼q) is a tautology or contradiction or neither of them.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Find `(dy)/(dx)` if x = a cosec θ, y = b cot θ at θ = `π/4`
Chapter: [0.07] Definite Integrals
Evaluate : `int 1/(x^2 + 8x + 20)` dx
Chapter: [0.06] Indefinite Integration
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
Chapter: [0.02] Matrices
The expenditure Ee of a person with income. x ls given by Ec = (0.000035).x^{2} + (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.
Chapter: [0.1] Insurance and Annuity
Evaluate : `int_(π/6)^(π/3) 1/(1 + sqrt(cotx)` dx
Chapter: [0.06] Indefinite Integration
Obtain the Crude Death Rate (CDR) for city A from the data given below.
Age group (in years) |
City A | |
Population | No. of deaths | |
Below 15 | 800 | 32 |
15 - 25 | 3000 | 12 |
25 - 65 | 4800 | 48 |
65 and above | 1400 | 42 |
Chapter: [0.11] Demography
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`
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
Find the age standard death rate (S.D.R.) for the following data:
Age (in years) |
Population | No. of deaths |
0 - 20 | 40000 | 350 |
20 - 65 | 65000 | 650 |
65 and above | 15000 | x |
Using the above information find x, if the CDR = 13.4 per thousand.
Chapter: [0.15] Management Mathematics
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
Chapter: [0.12] Bivariate Data and Correlation
If X ∼ N (4,25), then find P(x ≤ 4)
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
Find common region for the following system of linear inequations :
x ≥ -3, 4x - 5y ≥ -20, 3x + 4y ≤ 12, y ≥ -2
Chapter: [0.15] Management Mathematics
For the following problem, find the sequence that minimize total elapsed time (in hours) required to complete the following jobs on two machines M_{1} and M_{2} in the order M_{1} - M_{2}:
Jobs | A | B | C | D | E |
Machine M_{1} | 5 | 1 | 9 | 3 | 10 |
Machine M_{2} | 2 | 6 | 7 | 8 | 4 |
Chapter: [0.15] Management Mathematics
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:
Machine | Mixer | Food Processor | Available time |
A | 3 | 3 | 36 |
B | 5 | 2 | 50 |
C | 2 | 6 | 60 |
Formulate the problem as L.P.P. in order to maximize the profit.
Chapter: [0.15] Management Mathematics
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]
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
A Banker's discount calculated for 1 year is 26 times his gain. Find the rate or interest.
Chapter: [0.09] Commission, Brokerage and Discount
For the two regression equations 4y = 9x + 15 and 25x = 6y + 7 find correlation coefficient r, `barx, bary`
Chapter: [0.13] Regression Analysis Introduction
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.
Chapter: [0.08] Ratio, Proportion and Partnership
A sample of 5 items is taken from the production of a firm. Length and weight of the five items arc given below :
Length (cm) | 3 | 4 | 6 | 7 | 10 |
Weight (gm) | 9 | 11 | 14 | 15 | 16 |
Calculate Karl Pearson's coefficient of correlation between the length and weight and interpret the result.
Chapter: [0.12] Bivariate Data and Correlation
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 ?
Subordinates | Jobs | ||
I | II | III | |
A | 7 | 3 | 5 |
B | 2 | 7 | 4 |
C | 6 | 5 | 3 |
D | 3 | 4 | 7 |
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
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.
Chapter: [0.09] Commission, Brokerage and Discount
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.
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
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.
Chapter: [0.09] Commission, Brokerage and Discount
From the following table which relates to the number of animals of a certain
species at age x. complete the life table :
x | 0 | 1 | 2 | 3 | 4 | 5 |
l_{x} | 1000 | 850 | 760 | 360 | 25 | 0 |
Chapter: [0.11] Demography
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.
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
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.
Jobs | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} |
Machine A | 1 | 3 | 8 | 5 | 6 | 3 |
MAchine B | 5 | 6 | 3 | 2 | 2 | 10 |
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
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