Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2ax)dx`
Concept: Properties of Definite Integrals
A random variable X has the following probability distribution:
then E(X)=....................
Concept: Random Variables and Its Probability Distributions
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) 2
(c) 2
(d) ±2
Concept: Properties of Definite Integrals
Evaluate : `int e^x[(sqrt(1x^2)sin^1x+1)/(sqrt(1x^2))]dx`
Concept: Properties of Definite Integrals
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Concept: Properties of Definite Integrals
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , "for " x != 0 `
`=k`, for x=0
is continuos at x=0
Concept: Continuous Function of Point
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
Examine the continuity of the following function :
`{:(,,f(x)= x^2 x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Concept: Continuous Function of Point
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
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
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
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
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
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?
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
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`
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
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
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
Evaluate : `intlogx/(1+logx)^2dx`
Concept: Properties of Definite Integrals
For the bivariate data r = 0.3, cov(X, Y) = 18, σ_{x} = 3, find σ_{y} .
Concept: Statistics (Entrance Exam) > Bivariate Frequency Distribution