Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
(i) For what value of n, 4ⁿ ends in 0?

(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, aⁿ is a rational number?

(iii) If x and yare two odd positive integers, then which of the following is true?

(iv) The statement 'One of every three consecutive positive integers is divisible by 3' is

(v) If n is any odd integer, then n2 - 1 is divisible by

Real numbers are extremely useful in everyday life. That is probably one of the main reasons we all learn how to count and add and subtract from a very young age. Real numbers help us to count and to measure out quantities of different items in various fields like retail, buying, catering, publishing etc. Every normal person uses real numbers in his daily life. After knowing the importance of real numbers, try and improve your knowledge about them by answering the following questions on real life based situations.
(i) Three people go for a morning walk together from the same place. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance travelled when they meet at first time after starting the walk assuming that their walking speed is same?

(ii) In a school Independence Day parade, a group of 594 students need to march behind a band of 189 members. The two groups have to march in the same number of columns. What is the maximum number of columns in which they can march?

(iii) Two tankers contain 768litres and 420 litres of fuel respectively. Find the maximum capacity of the container which can measure the fuel of either tanker exactly.

(iv) The dimensions of a room are 8 m 25 cm, 6 m 75 crn and 4 m 50 cm. Find the length of the largest measuring rod which can measure the dimensions of room exactly.

(v) Pens are sold in pack of 8 and notepads are sold in pack of 12. Find the least number of pack of each type that one should buy so that there are equal number of pens and notepads

In a classroom activity on real numbers, the students have to pick a number card from a pile and frame question on it if it is not a rational number for the rest of the class. The number cards picked up by first 5 students and their questions on the numbers for the rest of the class are as shown below. Answer them.
(i) Suraj picked up \(\sqrt{8}\) and his question was - Which of the following is true about \(\sqrt{8}\)?

(ii) Shreya picked up 'BONUS' and her question was - Which of the following is not irrational?

(iii) Ananya picked up \(\sqrt{5}\)  -.\(\sqrt{10}\) and her question was - \(\sqrt{5}\)  -.\(\sqrt{10}\) _________is number.

(iv) Suman picked up \(\frac{1}{\sqrt{5}}\) and her question was - \(\frac{1}{\sqrt{5}}\) is __________ number.

(v) Preethi picked up \(\sqrt{6}\) and her question was - Which of the following is not irrational?

Decimal form of rational numbers can be classified into two types.
(i) Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form \(\frac{p}{\sqrt{q}}\) where p and q are co-prime and the prime faetorisation of q is of the form 2ⁿ·5^m, where n, mare non-negative integers and vice-versa.
(ii) Let x = \(\frac{p}{\sqrt{q}}\) be a rational number, such that the prime faetorisation of q is not of the form 2ⁿ 5^m, where n and m are non-negative integers. Then x has a non-terminating repeating decimal expansion.
(i) Which of the following rational numbers have a terminating decimal expansion?

(ii) 23/(2³ x 5²) =

(iii) 441/(2² x 5⁷ x 7²) is a_________decimal.

(iv) For which of the following value(s) of p, 251/(2³ x p²) is a non-terminating recurring decimal?

(v) 241/(2⁵ x 5³) is a _________decimal.

HCF and LCM are widely used in number system especially in real numbers in finding relationship between different numbers and their general forms. Also, product of two positive integers is equal to the product of their HCF and LCM. Based on the above information answer the following questions.
(i) If two positive integers x and yare expressible in terms of primes as x = p2q3 and y = p3 q, then which of the following is true?

(ii) A boy with collection of marbles realizes that if he makes a group of 5 or 6 marbles, there are always two marbles left, then which of the following is correct if the number of marbles is p?

(iii) Find the largest possible positive integer that will divide 398, 436 and 542 leaving remainder 7, 11, 15 respectively.

(iv) Find the least positive integer which on adding 1 is exactly divisible by 126 and 600.

(v) If A, Band C are three rational numbers such that 85C - 340A :::109, 425A + 85B = 146, then the sum of A, B and C is divisible by

The department of Computer Science and Technology is conducting an International Seminar. In the seminar, the number of participants in Mathematics, Science and Computer Science are 60, 84 and 108 respectively. The coordinator has made the arrangement such that in each room, the same number of participants are to be seated and all of them being in the same subject. Also, they allotted the separate room for all the official other than participants.

(i) Find the total number of participants.

(ii) Find the LCM of 60, 84 and 108.

(iii) Find the HCF of 60, 84 and 108.

(iv) Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject.

(v) Based on the above (iv) conditions, find the minimum number of rooms required for all the participants and officials.

Aditya works as a librarian in Bright Children International School in Indore. He ordered for books on English, Hindi and Mathematics. He received 96 English books, 240 Hindi Books and 336 Maths books. He wishes to arrange these books in stacks such that each stack consists of the books on only one subject and the number of books in each stack is the same. He also wishes to keep the number of stacks minimum.

(a) Find the number of books in each stack.

(b) Find the total number of stacks formed.

(c) How many stacks of Mathematics books will be formed?

(d) If the thickness of each English book is 3 cm, then the height of each stack of English books is

(e) If each Hindi book weighs 1.5 kg, then find the weight of books in a stack of Hindi books.