A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B, what is the least number that must have liked both products?

In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

A toy company manufacturers two types of dolls, A and B. Market research and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company make profit of Rs. 12 ad Rs. 16 doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

A toy company manufactures two types of dolls, A and B. Market research and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit ?

(A)John and Jivanti together have 45 marbles, Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (2) A cottage industry produces a cartain number of toys in a day. The cost of production of each toy (in rupee) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs.750. We would like to find out the number of toys produced on that day.

A successful forest conservation strategy should involve... a.protection of animals at the highest trophic level. b.protection of only consumers. c.to stop all the products available from forests. d.comprehensive programme to protect all the physical and biological components.

It is said that elemental composition of living organisms and that of inanimate objects (like earth's crust) are similar in the sense that all the major elements are present in both. Then what would be the difference between these two groups ? Choose a correct answer from among the following. (A) Living organisms have more gold in them than inanimate objects. (B) Living organisms have more water in their body than inanimate objects. (C) Living organisms have more carbon, oxygen and hydrogen per unit mass than inanimate objects. (D) Living organisms have more calcium in them than inanimate objects.

A radioactive sample consists of two distinct species having equal number of atoms initially. The mean life of one species is tau and that of the other is 5 tau . The decay products in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figure best represents the form of this plot? (a), (b), (c), (d)

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

Consider the decay of a free neutron at rest : n to p +e^(-) Show that the two body decay of this type must necessary give an electron of fixed energy and therefore cannot account distribution in the beta - decay of a neutron or a nucleous as shown in figure . [ Note : The simple result of this exercise was one among the several arguments advanced by W . Pauli to perdict the existence of a third particle in the decay products of beta - decay . This particle is known as neutrino spin 1/2 ( like e^(-) p or n ) but is neutral ad either massless or having an extremely small mass (compared to the mass of electron) and which interacts very weakly with matter . The correct decay process of neutron is : n to p+e(-)+v)