Q.1. Check whether 13 and 31 are coprimes .
Ans- yes.( क्योंकि 13 और 31 का HCF 1 है)
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Q.2. Check whether 150 and 295 are coprimes.
Ans- No. ( दोनों का H.CF. 5 इसलिए यह coprime नही है)
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Q.3. What is the difference between prime and coposite numbers?
Ans- prime numbers has 2 factors, composit numbers has more than 2 factors
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Q.4. Is 1 coprime to all numbers?
Ans-yes. ( Ex. 1,3. 5,1. etc)
-----------------------------------------------------------------------------------------------------------------------------------------Q.5. What is theSum of first n natural numbers:
- Ans- n(n+1)/2
For example:- 1+2+3+………..+99 = 99(99+1)/2 = (99×100)/2=4950
---------------------------------------------------------------------------------------------------------------------------------------Q.6.What is Sum of first n even numbers:-
- Ans : n(n+1)
For example:- 2+4+6+………….+56 (28th even number) = 28(28+1) = 28×29 = 812
-----------------------------------------------------------------------------------------------------------------------------------------Q.7. What is the Sum of first n odd numbers:-
Ans :n2
For example: 1+3+5+………+13 = 72 = 49. ( क्योंकि यहां 1 से लेकर 13 तक कुल 7 नंबर odd हैं)
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Q.8. What is the Sum of squares of first n natural numbers:-
- Ans: n(n+1)(2n+1)/6
For example:- 12 + 22 +…….+82 = 8×9×17/6 = 204
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Q.9 What is the Sum of cubes of first n natural numbers:-
- Ans: [n(n+1)/2]2
For example:- 13 + 23 +…….+73 = [7×8/2]2 = 789
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Relation Between Divisor, Dividend, Quotient and Reminder in a Division Sum:
Dividend = Divisor*Quotient + Reminder
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Q.10. When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?
a) 4
b) 3
c) 2
d) 1
Solution:- Let integer m = 10, because it satisfies the first statement of the question i.e 70/6 we get remainder 4
⇒ 7m = 7*10 = 70
now divided by 3:- 70/3 = Remainder:- 1
option d) is correct.
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Q.11..What are rational numbers? Give Examples.
A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to 0. Some of the examples of rational numbers include 1/3, 2/4, 1/5, 9/3, and so on.
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Q.12. Is 0 is rational numbers?
Yes, 0 is a rational number because it is an integer that can be written in any form such as 0/1, 0/2, where b is a non-zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.
-------------------------------------------------------------------------------------------------------------------------------------------------------------Q.13. 7 a rational number?
Q.16.What is the denominator of the rational number?
The denominator of the rational number can be any real number except 0.
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Q.17. Is Pi(π) a rational number?
No, Pi (π) is not a rational number. It is an irrational number and its value equals 3.142857…
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Q.18. The total number of factors of a prime even number is_____________.
Ans . Two
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Q.19. The addition of rational and irrational number is equal to _____________.
Ans. Irrational.
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Q.20. whai is the first prime number of two digit ?
Ans. 11
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Q.21. 2.35 is
(a) An integer
(b) A rational number
(c) An irrational number
(d) None of these
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Q.22. What is the smallest composite no.?
Ans. 4
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Q.23. The addition of a rational numbers and an irrational numbers ?
Ans . Irrational numbers
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Q.24. Product of a rational and irrational no. is always..?
Ans .irrational no.
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Q.25. .What is the smallest prime no?
Ans. 2
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Q.26 what is is the smallest composite no.?
Ans. 4
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Q.27. what is the smallest composite no of 2-digit.?
Ans. 10
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Q.28 Give an example of consecutive odd no?
Ans. 1,3
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Q.29 Which number is only even and also prime.
Ans. 2
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Q.30.How many prime no.are there from 1 to 100?
Ans. 25
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Q.31.What is the smallest natural no.?
Ans. 1
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Q.32. Which no. is smallest odd and also prime.
Ans. 3
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Q.33. Is π rational or irrational?
Ans .Irrational
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Q.34. which is the smallest four digit number?
Ans. 1000
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Q.35. What is the first even number of two digit?
Ans. 10
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Q.36.How many prime numbers are there between 100 and 120?
( A).6
(B).7
(C).5
(D).More than one of the above.
(we find that the prime numbers between 100 and 120 are: 101, 103, 107, 109, and 113 as these are not divisible by any number except 1).
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Q.37.Rationalise the denominator of 1/√3.
Ans: : To rationalise the denominator of 1/√3, we need to multiply the numerator and denominator by √3
1/√3 x (√3/√3) = √3/3.
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Q.38. Why do we use numbers?
Ans: The numbers are used to count the surrounding thing. Numbers are used for expressing money, time, date, and so on. Without numbers, we could not be able to understand the value of many things.
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Q.39. What are the four different types of number systems?
The four major types of number system are:
Binary number system (base 2 number system)
Octal number system (base 8 number system)
Decimal number system (base 10 number system)
Hexadecimal number system (base 16 number system)
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Q.40. How many rational number and Irrational number can be interested between 2 and 3 ?
Ans- There are infinite number of rational and Irrational between 2 and 3.
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Q.41. Insert a rational and an irrational number between 2 and 3.
Ans:
rational number between 2 and 3 = (2+3)/2 =5/2
irrational number between 2 and 3 = √5
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Q. 42. Convert 0.235555..., in the form of p/q.
Ans . 0.2355555 =( 235-23)/900 = 212/900
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Q.43. Which of the following numbers will completely divide 7¹⁰ + 7¹¹ + 7¹² + 7¹³?
A. 11. B.13. C.14. D.15.
Ans- C.
=7¹⁰ + 7¹¹ + 7¹² + 7¹³
⇒ (1 + 7 + 72 + 73) 710
⇒ (1 + 7 + 49 + 343) 710
⇒ 400 × 710
⇒ 24 × 52 × 710
So, the factors are 2, 4, 5, 7, 8, 10 etc.
So, out of given options, required factor = 2 × 7 = 14
∴ 14 will completely divide 710 + 711 + 712 + 713.
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Q.44..Find the total number of factors of 10500.
Ans. 10500= 2×2×3×5×5×5×7 =2²×3¹×5³×7¹
(Power में एक-एक जोड़कर गुणा करते हैं)
Total number of factors =(2+1)×(1+1)×(3+1)×(1+1)
= 3×2×4×2 =48
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Q.45 . Three numbers are in the ratio 3 ∶ 5 ∶ 7. If LCM of the numbers is 25200, then what is the sum of smallest and largest number?
A.1200. B2400. C.3600.. D.2880.
Ans :Let the number be 3x, 5x and 7x.
LCM of (3x, 5x and 7x)
⇒ 3 × 5 × 7 × (x) = 105x
According to the question,
105x = 25200
⇒ x = 25200/105 = 240
Now, the sum of the smallest and the largest number.
⇒ 7x + 3x = 10x
⇒ 240 × 10 = 2400
∴ The answer is 2400.
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Q.46. Find the sum of the factors of 3240.
A. 10890. B.11000. C. 10800. D. 10190.
Ans. 3240 = 2×2×2×3×3×3×3×5 =23 × 34 × 51
Sum of factors = (2⁰ + 2¹ + 2² + 2³) (30 + 31 + 32 + 33 + 34) (50 + 51)
⇒ (1 + 2 + 4 + 8) (1 + 3 + 9 + 27 + 81) (1 + 5)
⇒ 15 × 121 × 6
⇒ 10890
∴ required sum is 10890
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Q.47.Find which of the following are twin Primes.
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Q.48. Four bells ring simultaneously at starting and an interval of 6 sec, 12 sec, 15 sec and 20 sec respectively. How many times they ring together in 2 hours?
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Q.49..Find the total number of prime factors of 10500.
Ans. 10500= 2×2×3×5×5×5×7 =2²×3¹×5³×7¹
(Power को जोड़कर लिखते हैं )
Total number of prime factors =2+1+3+1 =7
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Q.50.
If a number is in the form of 810 × 97 × 78, find the total number of prime factors of the given number.
-----------do it self.--------------------------------------------------------------------------------------------------------------------
51. Express 0.777777… in p/q form.
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52. Express 0.757575… in p/q form.
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53. Express 0.752752752… in p/q form.
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54. Express 0.7533333… in p/q form.
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55. Express 0.12388888… in p/q form.
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56.. Express 0.75333… in p/q fo
57. Express 2.12222… in p/q form.
58. Express 2.33333...+ 3.8888… in p/q form.
59. Express 0.12388888… + 2.541444..in p/q form.
60. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a Non-Terminating decimal expansion
1. 31/ 2²×5³
2. 121/2³×3²×7⁵
3. 13/3150
4. 17/8
5. 64/455
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