Prove that in a sequence of numbers 49,4489,444889,44448889 in which every number is made by inserting 48-48 in the middle of previous as indicated, each number is the square of an integer.

There is a certain sequence of positive real numbers. Beginning from the third term, each term of the sequence is the sum of all the previous terms. The seventh term is equal to 1000 and the first term is equal to 1 . The second term of this sequence is equal to

Prove that the number of the sequence 121,12321,1234321,...... are each a perfect square of odd integer.

Prove that the number square root of a positive integer which is not a square number, cannot be rational.

A G.P. has first term 3 and last term 48. If each term is twice the preceding, find the number of terms and the sum of the G.P.

Give one example each to the following statements. i. A number which is rational but not an integer

There are p copies each of n different books. Find the number of ways in which a nonempty selection can be made from them.

Four different movies are running in a town. Ten students go to watch these four movies. The number of ways in which every movie is watched by atleast one student, is (Assume each way differs only by number of students watching a movie)

The initial number of radioactive atoms in a radioactive sample is N_0 . If after time t the number of becomes N, then N=N_0e^(-lambdat) , where lambda is known as the decay constant of the element. The time in which the number of radioactive atoms becomes half of its initial number is called the half-life (T) of the element. The time in which the number of atoms falls to 1/e times of its initial number is the mean life (tau) of the element. The product lambdaN is the activity (A) of the radioactive sample when the number of atoms is N. The SI unit of activity is bequerel (Bq)' where 1 Bq= 1 decay. s^(-1) . After how many days the activity of Iodine-131 will be 1/16 th of its initial value. [The half-life of Iodine-131 is 8 d.] (A) 24 d (B) 32 d (C) 40 d (D) 48 d

Which one of the following is false : (i) Every rational number is a real number (ii) Every integer is a real number (iii) sqrt2 is a rational number

State with reasons wheater the statemnts given below is negation of to each other or not : (a) There exists a natural number which is not an integer . (b) Some natural numbers are not integers .