Express the following statement in symbolic form and write its truth value.
"If 4 is an odd number, then 6 is divisible by 3 "
Concept: Mathematical Logic  Statement Patterns and Logical Equivalence
Find the values of x and y, if
`2[[1,3],[0,x]]+[[y,0],[1,2]]=[[5,6],[1,8]]`
Concept: Solution of Equations
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , `
`=k`, for x=0
is continuos at x=0
Concept: Continuous Function of Point
Find dy/dx if `y=cos^1(sqrt(x))`
Concept: Derivative  Derivative of Inverse Function
Price P for demand D is given as P = 183 +120D  3D^{2} Find D for which the price is increasing
Concept: Increasing and Decreasing Functions
Evaluate : `int1/(x(3+logx))dx`
Concept: Indefinite Integration  Methods of Integration
If `a=[[2,1],[1,1]] " then show taht " A^23A+I=0`
Concept: Solution of Equations
evaluate : `int xcosxdx`
Concept: Indefinite Integration  Integration by Parts
Prove that the following statement pattern is equivalent :
(p ∨ q) r and (p → r) ∧ (q → r)
Concept: Mathematical Logic  Statement Patterns and Logical Equivalence
Examine the continuity of the following function :
`f(x)=x^2x+9` , for x ≤ 3
`=4x+3` for x > 3`
Concept: Continuous Function of Point
find dy/dx if `y=tan^1((6x)/(15x^2))`
Concept: Derivative  Derivative of Inverse Function
Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,2],[0,2,1],[1,3,0]]`
Concept: Inverse of Matrix
Find the area of elipse `x^2/a^2+y^2/b^2=1`
Concept: Applications of 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.
Concept: Maxima and Minima
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.
Concept: Venn Diagrams
If 'f' is continuous at x = 0, then find f(0).
`f(x)=(15^x3^x5^x+1)/(xtanx) , x!=0`
Concept: Continuous Function of Point
find dy/dx if x=e^{2t , }y=`e^sqrtt`
Concept: Derivatives of Functions in Parametric Forms
Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`
Concept: Indefinite Integration  Methods of Integration
Evaluate :`int_0^(pi/2)dx/(1+cotx)`
Concept: Indefinite Integration  Integration by Parts
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.
Concept: Maxima and Minima
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.
Concept: 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 
Concept: Measurements of Mortality
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.
Concept: Rank 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?
Concept: 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 
Concept: Life Tables
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 
 
— 
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
Concept: Uses of Vital Statistics in Demography
The present worth of the sum of Rs. 5,830, due 9 months hence, is Rs. 5,500. Find the rate of interest.
Concept: Commission, Brokerage and Discount
For a binomial distribution mean is 6 and variance is 2. Find n and p.
Concept: Binomial Theorem and Binomial 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 
Concept: Sequencing in 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?
Concept: 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.
Concept: Random Variables and Its Probability Distributions
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 
Concept: Life Tables
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
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
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.
Concept: Commission, Brokerage and Discount
The equations given of the two regression lines are :
2x + 3y6 = 0 and 5x + 7y12 = 0.
Find :
(a) Correlation coefficient
(b) `sigma_x/sigma_y`
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Find graphical solution for the following system of linear inequations:
2x + 3y ≥ 2, x + y ≤ 3, x ≤ 4, y ≥ 3
Concept: Linear Programming Problem in 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]
Concept: Poisson 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.
Concept: 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.
Concept: Statistics  Karl Pearsonâ€™s Coefficient of 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?
Concept: Assignment Problem