Date & Time: 28th February 2015, 11:00 am
Duration: 3h
Express the following statement in symbolic form and write its truth value.
"If 4 is an odd number, then 6 is divisible by 3 "
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Find the values of x and y, if
`2[[1,3],[0,x]]+[[y,0],[1,2]]=[[5,6],[1,8]]`
Chapter: [0.02] Matrices
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , "for " x != 0 `
`=k`, for x=0
is continuos at x=0
Chapter: [0.03] Continuity
Find `dy/dx if y=cos^1(sqrt(x))`
Chapter: [0.04] Differentiation
Price P for demand D is given as P = 183 +120D  3D^{2} Find D for which the price is increasing
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
Evaluate : `int1/(x(3+logx))dx`
Chapter: [0.06] Indefinite Integration
If `"A"=[[2,1],[1,1]] " then show taht " A^23A+I=0`
Chapter: [0.02] Matrices
evaluate : `int xcosxdx`
Chapter: [0.06] Indefinite Integration
Prove that the following statement pattern is equivalent :
(p ∨ q) → r and (p → r) ∧ (q → r)
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Examine the continuity of the following function :
`{:(,,f(x)= x^2 x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Chapter: [0.03] Continuity
find dy/dx if `y=tan^1((6x)/(15x^2))`
Chapter: [0.04] Differentiation
Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,2],[0,2,1],[1,3,0]]`
Chapter: [0.012] Matrices [0.02] Matrices
Find the area of elipse `x^2/a^2+y^2/b^2=1`
Chapter: [0.07] Definite Integrals
The expenditure E_{c} of a person with income I is given by E_{c} = (0.000035) I^{2} + (0.045) I. Find marginal propensity to consume (MPC) and marginal propensity to save (MPS) when I = 5000. Also find A (average) PC and A (average)
PS.
Chapter: [0.05] Applications of Derivative
Express the truth of each of the following statements by Venn diagram:
(a) Some hardworking students are obedient.
(b) No circles are polygons.
(c) All teachers are scholars and scholars are teachers.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
If 'f' is continuous at x = 0, then find f(0).
`f(x)=(15^x3^x5^x+1)/(xtanx) , x!=0`
Chapter: [0.03] Continuity
find dy/dx if x=e^{2t , }y=`e^sqrtt`
Chapter: [0.04] Differentiation
Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`
Chapter: [0.06] Indefinite Integration
Evaluate :`int_0^(pi/2)dx/(1+cotx)`
Chapter: [0.06] Indefinite Integration
A firm wants to maximize its profit. The total cost function is C = 370Q + 550 and revenue is R = 730Q3Q^{2}. Find the output for which profit is maximum and also find the profit amount at this output.
Chapter: [0.05] Applications of Derivative
The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.
Chapter: [0.08] Ratio, Proportion and Partnership
Obtain crude death rates (C. D. R.) for city A and city B from the data given below :
Age groups (in years) 
City A 
City B 

Population 
No. of deaths 
Population 
No. of deaths 

Below 15 
800 
32 
900 
12 
1525 
3000 
12 
1500 
8 
2565 
4800 
48 
4500 
38 
65 and above 
1400 
42 
600 
30 
Chapter: [0.11] Demography
Coefficient of rank correlation between x and y is 0.5 and `sumd_i^2`= 42. Assuming that no ranks are repeated, find the number of pairs of observations.
Chapter: [0.12] Bivariate Data and Correlation
An agent charges 12% commission on the sales. What does he earn if the total sale amounts to Rs. 36,000? What does the seller get?
Chapter: [0.09] Commission, Brokerage and Discount
Find the age standard death rate (S. D. R.) for the following data:
Age groups (in years) 
Population (in '000) 
No. of deaths 
010 
11 
240 
1020 
12 
150 
2060 
9 
125 
60 and above 
2 
90 
Chapter: [0.11] Demography
Following table gives the age of husbands and wives
Age of wives (in years) 
Age of husbands (in years) 

2030 
3040 
4050 
5060 

1525 
5 
9 
3 
— 
2535 
— 
10 
25 
2 
3545 
— 
1 
12 
2 
4555 
— 
— 
4 
16 
5565 
— 
— 
— 
4 
Find:
(a)The marginal frequency distribution of the age of husbands.
(b)The conditional distribution of the age of husbands when the age of wives lies between 25  35
Chapter: [0.11] Demography
The present worth of the sum of Rs. 5,830, due 9 months hence, is Rs. 5,500. Find the rate of interest.
Chapter: [0.09] Commission, Brokerage and Discount
For a binomial distribution mean is 6 and variance is 2. Find n and p.
Chapter: [0.14] Random Variable and Probability Distribution
For the following problem, find the sequence that minimizes total elapsed time (in hours) required to complete jobs on two machines M_{1} and M_{2} in the order M_{1}M_{2} Also find the minimum elapsed time T.
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
Natarajan and Mr. Gopalan are partners in the company having capitals in the ratio 4 : 5 and the profits received by them are in the ratio 5 :4. If Mr. Gopalan invested capital in the company for 16 months, how long was Mr. Natarajan’s investment in the company?
Chapter: [0.09] Commission, Brokerage and Discount
From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain 
(a) exactly 1 defective bulb.
(b) at least 1 defective bulb.
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
Given the following table which relates to the number of parrots at age x, complete the life table for parrots.
X 
0 
1 
2 
3 
4 
5 
l_{x} 
1000 
940 
780 
590 
25 
0 
Chapter: [0.11] Demography
You are given the following information about advertising expenditure and sales:

Advertisement 

Expenditure (Rs. in lakh) (X) 
Sales (Rs. in lakh) (Y) 

Arithmetic mean 
10 
90 
Standard deviation 
3 
12 
Correlation coefficient between X and Y = 0.8.
(a) Obtain the two regression equations.
(b) What is the likely sales when the advertising budget is ? 15 lakh?
(c) What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
Electro Corp. Co. manufactures two electrical products :Air conditioners and Fans. The assembly process for each is similar in which both require a certain amount of wiring and drilling. Each air conditioner takes 4 hours for wiring and 2 hours for drilling. Each fan also takes 2 hours for wiring and 1 hour for drilling. During the next production period, 240 hours of wiring time are available and upto 100 hours of drilling time may be used. Each airconditioner assembled may be sold for Rs. 2,000 profit and each fan assembled may be sold for Rs. 1,000 profit. Formulate this problem as an L.P.P. in order to maximize the profit.
Chapter: [0.15] Management Mathematics
The equations given of the two regression lines are 2x + 3y  6 = 0 and 5x + 7y  12 = 0.
Find:
(a) Correlation coefficient
(b) `sigma_x/sigma_y`
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
Find graphical solution for the following system of linear inequations:
2x + 3y ≥ 2, x + y ≤ 3, x ≤ 4, y ≥ 3
Chapter: [0.15] Management Mathematics
The number of complaints which a bank manager receives per day is a Poisson random variable with parameter m = 4. Find the probability that the manager will receive 
(a) only two complaints on any given day.
(b) at most two complaints on any given day
[Use e^{4} =0.0183]
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
A warehouse valued at Rs. 10,000 contained goods worth Rs. 60,000. The warehouse was insured against fire for Rs. 4,000 and the goods to the extent of 90% of their value. A fire broke out and goods worth Rs. 20,000 were completely destroyed, while the remainder was damaged and reduced to 80% of its value. The damage to the warehouse was to the extent of Rs. 2,000. Find the total amount that can be claimed.
Chapter: [0.1] Insurance and Annuity
In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8
X 
6 
2 
10 
4 
8 
Y 
9 
11 
? 
8 
7 
(a) Estimate the missing observation.
(b) Calculate correlation coefficient.
Chapter: [0.12] Bivariate Data and Correlation
A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job is given in the following table:
Jobs

Machines 

P 
Q 
R 
S 

Processing Cost (Rs.)


A 
31 
25 
33 
29 
B 
25 
24 
23 
21 
C 
19 
21 
23 
24 
D 
38 
36 
34 
40 
How should the jobs be assigned to the four machines so that the total processing cost is minimum?
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
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