Worksheet of A.P

 

1. The common difference of the AP 1/p, (1 -p) /p ,(1 - 2p)/p is..

 (a) p.        (b) -p.       (c) -1        (d) 1

2. If the nth term of an AP is (2n + 1) then the sum of its first three terms 

     is.. 

 (a) 6n + 3. (b) 15. (c) 12. (d) 21

3. The sum of first n terms of an AP is (3n² + 6n) The common difference of the AP is.. 

 (a) 6. (b) 9. (c) 15. (d)-3

4. The sum of first n terms of an AP is (5n-n²). The nth term of the AP is..

 (a) (5 - 2n) (b) (6 - 2n) (c) (2n - 5) (d) (2n - 6)

5. The sum of first n terms of an AP is (4n² + 2n) . The nth term of this AP.is..               

 (a) (6n - 2) (b) (7n - 3) (c) (8n - 2) (d) (8n + 2)

6. The 7th term of an AP is -1 and its 16th term is 17. The nth term of the  AP is.. 

(a) (3n + 8) (b) (4n - 7) (c) (15 - 2n) (d) (2n - 15)

7. The 5th term of an AP is -3 and its common difference is. - 4. The sum of its first 10 terms is… 

 (a) 50 (b) -50. (c) 30. (d) -30

8. The 5th term of an AP is 20 and the sum of its 7th and 11th terms is  64 The common difference of the AP is.. 

(a) 4. (b) 5. (c) 3 (d) 2

9. The 13th term of an AP' is 4 times its 3rd term. It its 5th term is 16 then the sum of its first ten terms is .

(a) 150 (b) 175 (c) 160 (d) 135

10. An AP 5, 12, 19….., has 50 terms. Its last term is..(

a) 343. (b) 353. (c) 348 (d) 362

11. The sum of first 20 odd natural numbers is…..

  (a)100. (b) 210. (c) 400. (d) 420

12. The sum of first 40 positive integers divisible by 6 is…

 (a) 2460. (b) 3640. (c) 4920. (d) 4860

13. How many two-digit numbers are divisible by 3? 

(a) 25. (b) 30. (c) 32. (d) 36

14. How many three-digit numbers are divisible by 97

(a) 86. (b) 90. (c) 9% (d) 100

15. If a, denotes the nth term of the AP 3,8, 13, 18, then what is the value

(b) 36. (c) 50. (d) 56

16. Which term of the AP 72, 63,54, is 0?

          (b) 9th. (c) 10th. (d) 11th

17: Which term of the AP 25, 20, 15, is the first negative term?

      (a) 10th. (b) 9th. (c) 8th. (d) 7th

18. Which term of the AP 21, 42, 63, 84, is 210?

      (a) 9th. (b) 10th. (c) 11th. (d) 12th

19. What is 20th term from the end of the AP 3, 8, 13………..,2537

    (a) 163. (b) 158. (c) 153. (d) 148

20. The sum of first 16 terms of the AP 10, 6, 2... is

(a) 320. (b)-320. (C)-352. (d)-400

21. How many terms of the AP 3,7, 11, 15, will make the sum 4067

    (a) 10. (b) 12. (c) 14. (d) 20

22. The 2nd term of an AP is 13 and its 5th term is 25. What is its 17th term?

       (a) 69. (b) 73. (c) 77. (d) 81

23. The 17th term of an Al exceeds its 10th term by 21. The common

       difference of the AP is

    (a) 3. (b) 2. (c)-3. (d)-2

24. Which term of the AP :4,9,14,19, ....... is 109?

       (A) 14th. (B) 18th. (C) 22nd. (D) 16th

25. The sum of the arithmetic series 51 + 52 + 53 +.......+100 is:

      (A) 3775. (B) 4025. (C) 4275. (D) 5050

WORKSHEET-1 OF REAL NUMBER CLASS-10

 Q.1. Check whether 13 and 31 are coprimes .

Ans-   yes.( क्योंकि 13 और 31 का HCF 1 है)

-----------------------------------------------------------------------------------------------------------------------------------------

Q.2. Check whether 150 and 295 are coprimes.

Ans- No.          ( दोनों का H.CF. 5 इसलिए यह coprime नही  है)

-----------------------------------------------------------------------------------------------------------------------------------------

Q.3. What is the difference between prime and coposite numbers?

Ans- prime numbers has 2 factors, composit numbers has more than 2 factors 

-----------------------------------------------------------------------------------------------------------------------------------------

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 (28^th 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)

-----------------------------------------------------------------------------------------------------------------------------------------

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

-----------------------------------------------------------------------------------------------------------------------------------------

 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

-----------------------------------------------------------------------------------------------------------------------------------------

Relation Between Divisor, Dividend, Quotient and Reminder in a Division Sum:

  
Dividend = Divisor*Quotient + Reminder

 

-----------------------------------------------------------------------------------------------------------------------------------------

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. 

---------------------------------------------------------------------------------------------9-------------------------------------------

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.

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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?

Yes, 7 is a rational number because it can be written in the form of a ratio such as 7/1.

----------------------------------------------------------------------------------------------------------------------------------------------------------Is Q.14. 3.14 a rational number?

Yes, 3.14 is a rational number because it is terminating. But is not a rational number because the exact value of is 3.141592653589793238…which is non terminating non recurring.

---------------------------------------------------------------------------------------------------------------------------------------

Q.15. Find a rational number between 3 and 4.

Rational number between 3 and 4 = (3+4)/2 = 7/2

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.16.What is the denominator of the rational number?

The denominator of the rational number can be any real number except 0.

---------------------------------------------------------------------------------------------------------------------------------------------------------

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…

---------------------------------------------------------------------------------------------------------------------------

Q.18. The total number of factors of a prime even number is_____________.        

Ans . Two              

---------------------------------------------------------------------------------------------------------------------------------------------                

Q.19. The addition of rational and irrational number is equal to _____________.     

Ans. Irrational.

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.20. whai is the first prime number of two digit ?

Ans. 11

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.21. 2.35 is

 (a) An integer

 (b) A rational number

 (c) An irrational number

 (d) None of these

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.22. What is the smallest composite no.?

Ans. 4

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.23. The addition of a rational numbers and an irrational numbers ?

Ans . Irrational numbers

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.24. Product of a rational and irrational no. is always..?

Ans .irrational no.

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.25. .What is the smallest prime no?

Ans. 2

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.26 what is is the smallest composite no.?

Ans. 4

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.27. what is the smallest composite no of 2-digit.?

Ans. 10

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.28 Give an example of consecutive odd no?

Ans. 1,3

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.29 Which number is only even and also prime.

Ans. 2

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.30.How many prime no.are there from 1 to 100?

Ans. 25

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.31.What is the smallest natural no.?

Ans. 1

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.32. Which no. is smallest odd and also prime.

Ans. 3

-------------------------------------------------------------------------------------------------------------------------------------------------------------

Q.33. Is π rational or irrational?

Ans .Irrational

-------------------------------------------------------------------------------------------------------------------------------------------------------------

 Q.34. which is the smallest four digit number?

Ans. 1000

  ----------------------------------------------------------------------------------------------------------------------------------------------------------                                                                                      

  Q.35.  What is the first even number of two digit?

Ans. 10

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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).

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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.

-----------------------------------------------------------------------------------------------------------------------------

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.

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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)

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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.

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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

-----------------------------------------------------------------------------------------------------------------------------------------------------------

Q. 42. Convert  0.235555..., in the form of p/q.

Ans .          0.2355555 =( 235-23)/900 = 212/900

-------------------------------------------------------------------------------------------------------------------------------------------------------------

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.

-----------------------------------------------------------------------------------------------------------------------------

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

-----------------------------------------------------------------------------------------------------------------------------

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.

-----------------------------------------------------------------------------------------------------------------------------

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

-----------------------------------------------------------------------------------------------------------------------------

Q.47.Find which of the following are twin Primes.

1. (37, 41)
2. (3 , 7)
3. (43 , 47)
4. (71, 73).               ( क्योंकि इन दोनों का HCF  1 है तथा दोनों में 2 का अंतर है ) 

-----------------------------------------------------------------------------------------------------------------------------

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?

1. 120
2. 60
3. 121
4. 112. .                  Ans.       LCM of (6, 12, 15, 20) = 60

   All 4 bells ring together again after every 60 seconds

   Now,

   In 2 Hours, they ring together = [(2 × 60 × 60)/60] times + 1 (at the starting) = 121 times

   ∴ In 2 hours they ring together for 121

--------------------------------------------------------------------------------------------------------

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

                                               

-----------------------------------------------------------------------------------------------------------------------------

Q.50.

If a number is in the form of 8¹⁰ × 9⁷ × 7⁸, find the total number of prime factors of the given number.

1. 52.  
2. 560
3. 3360
4. 25.           

-----------do it self.--------------------------------------------------------------------------------------------------------------------

51. Express 0.777777… in p/q form.

-----------------------------------------------------------------------------------------------------------------------------

52. Express  0.757575… in p/q form.

-----------------------------------------------------------------------------------------------------------------------------

53. Express   0.752752752… in p/q form.

-----------------------------------------------------------------------------------------------------------------------------

54.   Express 0.7533333… in p/q form.

-----------------------------------------------------------------------------------------------------------------------------

55. Express  0.12388888… in p/q form.

-----------------------------------------------------------------------------------------------------------------------------

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

61.  Prove that  2 + √5 is an irrational number.

62. Find the simplest form of 1095 / 1168.

63. π is rational or irrational number and its decimal expression is terminating or non terminating. ?.

64. If HCF and LCM of two numbers are 4 and 9696 then find the product of two numbers.

65. What will be the HCF  and  L.C.M. of two prime numbers?

66. What will be the HCF.and LCM of two odd number?.

67 what will be the HCF and LCM of two even numbers ?.

68. What will be the combination of a rational and irrational number?. 

69. Find the largest number which divides 546 and 764 leaving reminders 6 and 8 respectively.  (Ans.  108 ) .

70.  Find the HCF and LCM of 7 / 9 ,  10 /27 and 16 / 81.

71. Find the HCF of 1008 and 1080 by prime factorization method. 

72 . Give an example of two irrational whose sum is rational. 

73. Examine 17 / 30 is a terminating decimal.

74 . The HCF of two numbers is 27 and their LCM is 162 if one of the number is 81 find the other number. 

75 . Show that (15) power to n can never end with the digit 0.

76. find the value of 0.68686868...  +0.737373...  .

78. What is the least number that is divisible by all the natural numbers from 1 to 10.

79 . The decimal expression of the rational number 37 / 2⁵ × 5 will it terminate after ? .

80. A number when divided 143 leaves 31 as remainder what will be the remainder when the same number is divided by 13.    (Ans. 5 ) .

81. Write a rational number between √3 and 2 .

82. Give an example of two irrational whose product is rational. 

83. Give an example of two irrational whose sum is rational. 

84. Is it possible to have two numbers whose HCF is 25 and LCM is 520 ?.

86. 6 bells commence tolling together and toll at interval 2 ,4 ,6 ,8 ,10 , and 12 minutes respectively. In 30 hours how many times do they toll together ?  ( Ans 30/2  +1= 16) .

87. An electric device makes a beep after every 60 seconds another device makes a beef after every 62 seconds they beeped together at 10:00 a.m. at what time will they together at the earliest ?.     (Ans.  1860/60 =31 min ,    10:31am)

88. Find the largest number of four digits which is exactly divisible by 15 , 24 and 36 . ( Ans (9720 is exactly divisible by 360 which is LCM)

89. Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520. ( Ans.  4680-17= 4663 ).

90. Find the least number which when divides 35 ,56 and 91 leaves the same remainder 7 in each case. ( 3640+7=3647 )

91. The H.C.F.  of  85 and 238 is expressible in the form 85m - 238 . find the value of m.

92. In the following question a statement of Assassin (A)  is followed by a statement of Reason (R) .

Choose the correct option.

(A)  Both assertion (A) and reason (R) are true and reason(R)  is the correct explanation of assertion (A).

(B)  Both assertion (A) and reason (R) are true and reason(R)  is  not the correct explanation of assertion (A).

(C) Assertion (A) is true but reason (R) is false.

(D) Assertion (A) is false but reason (R) is true.

®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️