We provide real numbers practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on real numbers skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of real numbers Questions

Question No | Questions | Class |
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1 | Which one of the following statements is correct? ( mathbf{A} cdot ) If ( g(x)(neq 0) ) and ( f(x) ) are two polynomials ( epsilon F(x), ) then there exists unique polynomials ( q(x) ) and ( r(x) epsilon F(x) ) such that ( f(x)=g(x) q(x)+r(x), ) where deg. ( r(x)< ) ( operatorname{deg} . g(x) ) B. If ( g(x)(neq 0) ) and ( f(x) ) are two polynomials ( epsilon F(x) ), then there exists unique polynomials ( q(x) ) and ( r(x) epsilon F(x) ) such that ( f(x)=g(x) q(x)+r(x), ) where deg. ( r(x) leq ) ( operatorname{deg} . g(x) ) C . If ( g(x)(neq 0) ) and ( f(x) ) are two polynomials ( epsilon F(x) ), then there exists unique polynomials ( q(x) ) and ( r(x) epsilon F(x) ) such that ( f(x)=g(x) q(x)+r(x), ) where either ( r(x)=0 ) or deg. ( r(x)operatorname{deg} . g(x) ) | 10 |

2 | 53. The sum of a pair of positive in- tegers is 336 and their H.C.F. is 21. The number of such possi- ble pairs is (1) 2 (2) 3 (3) 4 (4) 5 | 10 |

3 | 72. L.C.M. of x-1, -1 and x8 – 1 will be (1) (x – 1)(x4 – 1)(x8 – 1) (2) (x – 1)(x4 + 1)(x – 1) (3) x16 – 1 (4) x8 – 1 | 10 |

4 | Prove that one of every three consecutive positive integer is divisible by 3 | 10 |

5 | The LCM of 5490 and a third number is 1890 and their HCF is 18 The third number is A . 36 B. 180 ( c cdot 126 ) D. 108 | 10 |

6 | use Euclid’s division algorithm to find the H.C.F. of 135 and 714 | 10 |

7 | 70. A number when divided by 899 gives a remainder 63. If the same number is divided by 29, the re- mainder will be: (1) 10 (2) 5 (3) 4 (4)2 | 10 |

8 | Find the multiplicative inverse of: ( frac{2-sqrt{3}}{2+sqrt{3}} ) | 10 |

9 | Prove that ( 5-sqrt{3} ) is an irrational number. | 10 |

10 | State the following statement is True or False ( frac{7}{9} ) has a value of non terminating decimal number A. True B. False | 10 |

11 | The positive integer whose product with -1 is A. positive B. negative c. 0 ( D cdot ) none | 10 |

12 | We need blocks to build a building. In the same way are basic blocks to form all natural numbers. A. prime numbers B. real numbers c. unique numbers D. negative numbers | 10 |

13 | What are terminating decimals? | 10 |

14 | The length and breadth of a rectangular field is ( 110 mathrm{m} ) and ( 30 mathrm{m} ) respectively. Calculate the length of the longest rod which can measure the length and breath of the field exactly. | 10 |

15 | Without doing any actual division, find which of the following rational numbers have terminating decimal representation: (i) ( frac{7}{16} ) (ii) ( frac{23}{125} ) (iii) ( frac{mathbf{9}}{mathbf{1 4}} ) (iv) ( frac{32}{45} ) ( (v) frac{43}{50} ) (vi) ( frac{17}{40} ) (vii) ( frac{61}{75}(text { viii }) frac{123}{250} ) A ( . ) (i), (iii), (v), (vi) and (vii) B. (i), (ii), (v), (vi) and (viii) C . (i), (iii), (v), (vi) and (viii) D. (i), (ii), (v), (vi) and (vii) | 10 |

16 | The HCF of 256,442 and 940 is ( A cdot 2 ) B. 14 ( c cdot 142 ) D. none | 10 |

17 | Using Euclid’s division algorithm, find the HCF of 56,96 and 404 | 10 |

18 | A number when divided by 156 gives 29 as remainder. If the same number is divided by ( 13, ) what will be the remainder? A .4 B. 3 c. 5 D. 6 | 10 |

19 | Find the ( H C F ) of 65 and 117 and express it in the form ( 65 m+117 n ) | 10 |

20 | In a division sum a student took 63 as divisor instead of ( 36 . ) His answer was 24. What is the correct answer? A .42 B . 24 ( c .36 ) D. 63 | 10 |

21 | Identify a non-terminating repeating decimal. A ( cdot frac{24}{1600} ) в. ( frac{171}{800} ) c. ( frac{123}{2^{2} times 5^{3}} ) D. ( frac{145}{2^{3} times 5^{2} times 7^{2}} ) | 10 |

22 | Find the ( H C F ) of 81 and 237 and express it as a linear combination of 81 and 237 | 10 |

23 | Express 32844 as a product of prime factors | 10 |

24 | states the possibility of the prime factorization of any natural number is unique. The numbers can be multiplied in any order A. Pythagora’s theorem B. Remainder theorem c. Fundamental theorem of arithmetic D. none of the above | 10 |

25 | Use Euclid’s division lemma to show that the square of any positive integer is either of the form ( 3 m ) or ( 3 m+1 ) for some integer ( m ), but not of the form ( 3 m+2 ) | 10 |

26 | The square of any positive odd integer for some integer ( m ) is of the form ( A cdot 7 m+1 ) B. ( 8 m+1 ) ( c cdot 8 m+3 ) ( D cdot 7 m+2 ) | 10 |

27 | Product of two integers with unlike signs is A. Negative B. c. Positive D. None of these | 10 |

28 | Use Euclid’s division lemma to find the HCF of the following 27727 and 53124 A .233 B . 200 c. 154 D. 212 | 10 |

29 | Use Euclids division algorithm to find the HCF of: 867 and 225 | 10 |

30 | State whether the following statement is true/false. ( frac{2375}{375} ) is not a terminating decimal A. True B. False | 10 |

31 | Divide as directed. ( mathbf{5} 2 p q r(p+q)(q+r)(r+p) div ) ( mathbf{1 0 4 p q}(boldsymbol{q}+boldsymbol{r})(boldsymbol{r}+boldsymbol{p}) ) | 10 |

32 | 53. Given : 34. 13. 625 and 12289, the greatest and least of them are respectively (1) 14289 and 14 (2) 13 and 34 (3) 925 and 13 (4) 74 and $25 | 10 |

33 | Use Euclid’s division algorithm to find the HCF of : ( mathbf{8 6 7} ) and ( mathbf{2 5 5} ) A . 50 B. 51 c. 41 D. 52 | 10 |

34 | Find the HCF of 92690,7378 and 7161 by Euclid’s division algorithm. A . 29 B. 30 c. 31 D. 32 | 10 |

35 | Find whether it is a terminating or a non-terminating decimal ( 2.4 div 0.072 ) A. Terminating B. Non-terminating c. Ambiguous D. Data insufficient | 10 |

36 | When a natural number ( x ) is divided by ( 5, ) the remainder is ( 2 . ) When a natural number y is divided by 5, the remainder is ( 4 . ) The remainder is ( z ) when ( x+y ) is divided by 5. The value of ( frac{2 z-5}{3} ) is A . – в. ( c cdot-2 ) D. | 10 |

37 | Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of English books is ( 96, ) the number of Hindi books is 240 and the number of Mathematics books is ( 336 . ) Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books. | 10 |

38 | Convert the following fraction into simple decimal recurring form. ( frac{5}{6}=? ) A . ( 0.4 overline{3} ) B. ( 0.1 overline{3} ) c. ( 0 . overline{4} ) D. 0.83 | 10 |

39 | Fundamental theorem of arithmetic is also called as Factorization | 10 |

40 | Use Euclids division lemma to show that the square of any positive integer is either of the form ( 3 m ) or ( 3 m+1 ) for some integer ( boldsymbol{m} ) | 10 |

41 | Number ( frac{3}{625} ) is a terminating decimal or a non-terminating repeating decimal? Write it in decimal form | 10 |

42 | Prove that ( 7 sqrt{5} ) is irrationals number. | 10 |

43 | Write three numbers whose decimal expansions are non terminating and non recurring | 10 |

44 | Solve:- ( sqrt[3]{boldsymbol{y}}(2 sqrt[3]{boldsymbol{y}}-sqrt[3]{boldsymbol{y}^{8}}) ) | 10 |

45 | Express the following in a recurring decimal form. ( 2 frac{1}{6} ) A ( .2 .1 overline{6} ) B. ( 2.1 overline{8} ) c. ( 2.1 overline{4} ) D. 2.19 | 10 |

46 | What is the 25 th digit to the right of the decimal point in the decimal form of ( frac{mathbf{6}}{mathbf{1 1}} ) ( ? ) A . 3 B. 4 c. 5 D. 6 E. 7 | 10 |

47 | The HCF of 455 and 42 using Euclid algorithm is A. 7 B. 6 c. 5 ( D ) | 10 |

48 | Evaluate ( sqrt{11 sqrt{11 sqrt{11 sqrt{11 dots dots infty}}}} ) | 10 |

49 | State whether the given statement is true/false: ( sqrt{boldsymbol{p}}+sqrt{boldsymbol{q}}, ) is irrational, where ( p, q ) are primes. A . True B. False | 10 |

50 | H.C.F. of 26 and 91 is: A . 13 в. 2366 ( c .91 ) D. 182 | 10 |

51 | Prove that the following are irrational ( sqrt{3}+sqrt{5} ) | 10 |

52 | ( 3.24636363 ldots ) is A. An integer B. An irrational number C. A rational number D. Not a real number | 10 |

53 | Which option will have a terminating decimal expansion? A ( cdot frac{77}{210} ) в. ( frac{23}{30} ) c. ( frac{125}{441} ) D. ( frac{23}{8} ) | 10 |

54 | ( sqrt{7} ) is a A. an integer B. an irrational number c. a rational number D. none of these | 10 |

55 | The ………… when multiplied always give a new unique natural number. A. decimal numbers B. fractions c. irrational numbers D. prime numbers | 10 |

56 | Is ( pi ) a rational number? Justify your answer | 10 |

57 | Classify the following numbers as rational or irrational. ( mathbf{1 1} . mathbf{2 1 3 2} mathbf{4 3 5 4 6 5} ) | 10 |

58 | In a division operation the divisor is 5 times the quotient and twice the remainder. If the remainder is ( 15, ) then what is the dividend? A . 175 в. 185 c. 195 D. 205 | 10 |

59 | Show that the cube of any positive integer is of the form ( 9 m, 9 m+1 ) or ( 9 m+8 . ) Using Euclid’s division lemma. | 10 |

60 | Evaluate ( sqrt[3]{left(frac{1}{64}right)^{-2}} ) ( A ) B . 16 ( c .32 ) D. 64 | 10 |

61 | Prove that, if ( x ) and ( y ) are odd positive integers, then ( x^{2}+y^{2} ) is even but not divisible by 4. | 10 |

62 | Consider the following statements: ( frac{1}{22} ) can not be written as terminating decimal 2. ( frac{2}{15} ) can be written as a terminating decimal 3. ( frac{1}{16} ) can be written as a terminating decimal Which of the statements given above is/are correct? | 10 |

63 | 53. The sum of a pair of positive in- tegers is 336 and their H.C.F. is 21. The number of such possi- ble pairs is (2) 3 (3) 4 (4) 5 (1) 2 | 10 |

64 | Which of the following will have a terminating decimal expansion? A ( cdot frac{77}{210} ) в. ( frac{23}{30} ) c. ( frac{125}{441} ) D. ( frac{23}{8} ) | 10 |

65 | Which of the following numbers has the terminal decimal representation? A ( cdot frac{1}{7} ) B. ( frac{1}{3} ) ( c cdot frac{3}{5} ) D. ( frac{17}{3} ) | 10 |

66 | The H.C.F. of two expressions is ( x ) and their L.C.M is ( x^{3}-9 x ) IF one of the expression is ( x^{2}+3 x ) then,the other expression is A ( cdot x^{2}-3 x ) B. ( x^{3}-3 x ) c. ( x^{2}+9 x ) D. ( x^{2}-9 x ) | 10 |

67 | Prove that square of any positive integer is always of the form ( 4 m, 4 m+ ) ( 1,4 m+4 ) or ( 4 m+9 ) | 10 |

68 | Show that ( 1.272727 ldots=1 . overline{27} ) can be expressed in the form ( underline{boldsymbol{p}} ) ( overline{boldsymbol{q}} ) | 10 |

69 | Is ( pi ) an irrational number? Why? | 10 |

70 | The additive identity for integers is ( A cdot 0 ) B. ( c .0 . ) D. 2 | 10 |

71 | If ( a=107, b=13 ) using Euclid’s division algorithm find the values of ( q ) and ( r ) such that ( a=b q+r ) A ( cdot q=8, r=8 ) В . ( q=8, r=3 ) c. ( q=0, r=3 ) D. None of these | 10 |

72 | Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively. | 10 |

73 | Prove ( 6+3 sqrt{5} ) is irrational | 10 |

74 | The non terminating non-recurring decimal cannot be represented as A. irrational numbers B. rational numbers c. real numbers D. none of these | 10 |

75 | 20 is written as the product of primes as: ( mathbf{A} cdot 2 times 5 ) в. ( 2 times 2 times 3 times 5 ) c. ( 2 times 2 times 5 ) D. ( 2 times 2 times 3 ) | 10 |

76 | The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form ( p ), you say about the prime factors of ( q ? ) (i) 43.123456789 (ii) ( 0.120120012000120000 ldots ) ( 43 . overline{123456789} ) | 10 |

77 | Without actual division, check whether ( frac{47}{14} ) is terminating or not. | 10 |

78 | ( left(y^{2}+7 y+10right) div(y+5) ) | 10 |

79 | Use Euclid’s division algorithm to find the HCF of 135 and 225 | 10 |

80 | The divisor when the quotient, dividend and the remainder are respectively 547,171282 and 71 is equal to A . 333 в. 323 c. 313 D. 303 | 10 |

81 | Use Euclid’s division algorithm to find the ( H C F ) of 196 and 38220 | 10 |

82 | In a division sum, the divisor is 10 times the quotient and five times the remainder. What is the dividend, if the remainder is ( 46 ? ) A .5636 в. 5536 ( c .5336 ) D. 5436 | 10 |

83 | Prove the following are irrational numbers ( mathbf{3}+sqrt{mathbf{5}} ) | 10 |

84 | According to Euclid’s division algorithm, HCF of any two positive integers a and b with ( a>b ) is obtained by applying Euclid’s division lemma to a and b to find ( q ) and ( r ) such that ( a=b q+ ) ( r, ) where ( r ) must satisfy A. ( 1<r<b ) В. ( 0<r<b ) c. ( 0 leq r<b ) D. ( 0<r leq b ) | 10 |

85 | Find the HCF of 1656 and 4025 by Euclids division theorem | 10 |

86 | 53. The last digit, that is, the digit in the unit’s place of the number [(57)25 – 1) is (1) 6 (3) 0 (4) 5 (2) 8 | 10 |

87 | Use Euclid’s division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and ( 255 . ) Enter the highest HCF amongst the following. | 10 |

88 | Use Euclid’s algorithm to find the HCF of 4052 and 12576 | 10 |

89 | Use Euclid’s division algorithm to find the HCF of : ( mathbf{1 9 6} ) and ( mathbf{3 8 2 2 0} ) A. 169 в. 196 c. 206 D. 192 | 10 |

90 | Show that ( 3 sqrt{2} ) is irrational. | 10 |

91 | One and only one out of ( boldsymbol{n}, boldsymbol{n}+mathbf{4}, boldsymbol{n}+ ) ( 8, n+12 ) and ( n+16 ) is ( dots dots ) (where n is any positive integer) A. Divisible by 5 B. Divisible by 4 c. Divisible by 10 D. Divisible by 12 | 10 |

92 | 57. Product of two co-prime numbers is 117. Then their L.C.M. is (1) 117 (2) 9 (3) 13 (4) 39 | 10 |

93 | Say true or false: A positive integer is of the form ( 3 q+1 ) ( boldsymbol{q} ) being a natural number, then you write its square in any form other than ( 3 m+1, ) i.e.,3m or ( 3 m+2 ) for some integer ( boldsymbol{m} ) A. True B. False | 10 |

94 | Calculate the HCF of ( 3^{3} times 5 ) an ( 3^{2} times 5^{2} ) | 10 |

95 | 35 has a non-terminating decimal ( overline{mathbf{5 0}} ) expansion A. True B. False | 10 |

96 | Find HCF of 45 and 72 | 10 |

97 | Prove that ( sqrt{3}+sqrt{8} ) is an irrational number. | 10 |

98 | Show that one and only one out of ( boldsymbol{n}, boldsymbol{n}+2 ) or, ( boldsymbol{n}+boldsymbol{4} ) is divisible by ( boldsymbol{3} ) where ( n ) is any positive integer. | 10 |

99 | If ( d ) is the ( H C F ) of 56 and ( 72, ) find ( x, y ) satisfying ( boldsymbol{d}=mathbf{5 6 x}+mathbf{7 2 y} . ) Also, show that ( x ) and ( y ) are not unique. | 10 |

100 | If ( d ) is the ( H C F ) of 45 and ( 27, ) then ( x, y ) satisfying ( boldsymbol{d}=mathbf{2 7} boldsymbol{x}+mathbf{4 5} boldsymbol{y} ) are : A ( . x=2, y=1 ) в. ( x=2, y=-1 ) c. ( x=-1, y=2 ) D. ( x=-1, y=-2 ) | 10 |

101 | ( boldsymbol{n} ) is a whole number which when divided by 4 gives 3 as remainder. What will be the remainder when ( 2 n ) is divided by ( 4 ? ) ( A cdot 7 ) B. 5 ( c cdot 4 ) D. 2 | 10 |

102 | Write the following in decimal form and say what kind of decimal expansions has: ( frac{2}{11} ) | 10 |

103 | Use Euclid’s division lemma to show that cube of any positive integer is either of the form ( 9 mathrm{m}, 9 mathrm{m}+1 ) or ( 9 mathrm{m}+8 ) for some integer ‘m’ | 10 |

104 | Write the following in decimal form and say what kind of decimal expansions has: ( frac{329}{400} ) | 10 |

105 | Using Euclid’s division algorithm find the HCF of the following numbers. ( mathbf{2 0 2 4} ) and ( mathbf{1 8 7 2} ) | 10 |

106 | Use Euclid’s division algorithm to find the ( H C F ) of 210 and 55 | 10 |

107 | Express 120 as a product of prime factors | 10 |

108 | 51. The greatest four digit number which is exactly divisible by each one of the numbers 12, 18, 21 and 28 is (1) 9828 (2) 9288 (3) 9882 (4) 9928 | 10 |

109 | Show that ( sqrt{2}+sqrt{3} ) is an irrational number. | 10 |

110 | We know that any odd positive integer is of the form ( 4 q+1 ) or ( 4 q+3 ) for some integer ( boldsymbol{q} ) Thus, we have the following two cases. A ( cdot n^{2}-1 ) is divisible by 8 B . ( n^{2}+1 ) is divisible by 8 c. ( n-1 ) is divisible by 8 D. ( n+1 ) is divisible by 8 | 10 |

111 | In Euclid’s Division Lemma, when ( a=b q+r ) where ( a, b ) are positive integers then what values r can take? | 10 |

112 | Divide as directed. ( 26 x y(x+5)(y-4) div 13 x(y-4) ) | 10 |

113 | For finding the greatest common divisor of two given integers. A method based on the division algorithm is used called A. Euclid’s division algorithm B. Euclid’s addition algorithm C. Euclid’s subtraction algorithm D. Euclid’s multiplication algorithm | 10 |

114 | If the denominator of a fraction has factors other then 2 and 5 , the decimal expression A. repeats B. is that of a whole number c. has equal numerator and denominator D. terminates | 10 |

115 | Euclids division lemma can be used to find the ( ldots ldots ldots . . ) of any two positive integers and to show the common properties of numbers. A. None of the common factors B. Lowest common factor c. Highest common factor D. common factor | 10 |

116 | Prove that ( log _{2} 3 ) is an irrational number. | 10 |

117 | As the decimal of ( frac{1}{3} ) repeats, ( frac{1}{3} ) is a decimal. A. exact B. negative c. terminating D. non-terminating | 10 |

118 | Prove that ( 3 sqrt{2} ) is irrational | 10 |

119 | Use Euclid’s division lemma to find the HCF of 13281 and 15844 | 10 |

120 | Prove that one and only one out of ( boldsymbol{n}, boldsymbol{n}+2 ) or ( boldsymbol{n}+boldsymbol{4} ) is divisible by ( boldsymbol{3} ; ) where ( n ) is any positive integer. | 10 |

121 | Use Euclid’s division lemma to find the HCF of the following 65 and 495 A . 5 B. 10 c. 15 D. 0 | 10 |

122 | If the H.C.F. of ( A ) and ( B ) is 24 and that of ( C ) and ( D ) is ( 56, ) then the H.C.F. of ( A, B, C ) and ( D ) is A .4 B. 12 ( c cdot 8 ) D. 3 | 10 |

123 | Show that every positive even integer is of the form ( 2 q ) and every positive odd integer is of the form ( 2 q+1 ), where q is a whole number. | 10 |

124 | Find the H.C.F. of the following numbers using prime factorization method. 81,117 | 10 |

125 | Use Euclid’s division algorithm to find the HCF of the following number: 55 and 210 | 10 |

126 | Prove that ( 5 sqrt{3} ) is an irrational. | 10 |

127 | Prove that ( 4-3 sqrt{2} ) is an irrational | 10 |

128 | Prove that ( sqrt{2} ) is an irrational number. | 10 |

129 | Use Euclid’s division algorithm to find the ( H C F ) of 4052 and 12576 | 10 |

130 | What is the square of ( (2+sqrt{2}) ? ) A. A rational number B. An irrational number c. A natural number D. A whole number | 10 |

131 | Use Euclid’s Division Lemma to show that the cube of any positive integer is of the form ( 9 m, 9 m+1 ) or ( 9 m+8, ) for some integer ( boldsymbol{m} ) | 10 |

132 | 49. The least number, which is to be added to the greatest number of 4 digits so that the sum may be divisible by 345, is (1) 50 (2) 6 (3) 60 (4) 5 | 10 |

133 | Use Euclids division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 225 | 10 |

134 | In a division sum the divisor is 12 times the quotient and 5 times the remainder. If the remainder is 48 then what is the dividend? A . 2404 в. 4808 c. 3648 D. 4848 | 10 |

135 | If these numbers form positive odd integer ( 6 q+1, ) or ( 6 q+3 ) or ( 6 q+5 ) for some ( q ) then ( q ) belongs to: This question has multiple correct options A. integers B. rational c. real D. none of these | 10 |

136 | Square root of ( sqrt{13-4 sqrt{3}} ) | 10 |

137 | State true or false of the following. If a and b are natural numbers and ( a< ) ( b, ) than there is a natural number c such that ( a<c<b ) A. True B. False | 10 |

138 | Find the highest common factor of the monomial. ( 6 a^{2} b^{2} c ) and ( 27 a b c^{3} ) | 10 |

139 | Use Euclid’s division algorithm to find the H.C.F. of 196 and 38318 | 10 |

140 | Represent ( frac{-3}{4} ) on number line | 10 |

141 | Show that the square of any positive integer is of the form ( 4 mathrm{m} ) or ( 4 mathrm{m}+1 ) for any integer ( m ) | 10 |

142 | Express 3825 as a product of prime factors | 10 |

143 | Show that ( n^{2}-1 ) is divisible by 8 , if ( n ) is an odd positive integer. | 10 |

144 | State whether the given statement is True or False: ( 3+sqrt{2} ) is an irrational number. A . True B. False | 10 |

145 | Given that ( sqrt{2} ) is irrational, prove that ( (5+3 sqrt{2}) ) is an irrational number. | 10 |

146 | In a question on division if four times the divisor is added to the dividend then how will the new remainder change in comparison with the original remainder? A. Remainder remains unchanged. B. Remainder increases by 4 c. Remainder decreases by 4. D. Remainder becomes 4 times the older remainder | 10 |

147 | Use Euclid’s division lemma to find the HCF of the following 16 and 176 A . 16 B. 14 c. 10 D. 12 | 10 |

148 | The numbers ( 7.478478 ldots . . ) and ( 1.101001000100001 ldots ) are A. Rational and irrational respectively B. Both rationals c. Both irrationals D. None of these | 10 |

149 | Prove that ( sqrt{3} ) is a irrational number. | 10 |

150 | If the denominator of a fraction has only factors of 2 and factors of 5 , the decimal expression A. has equal numerator and denominator B. becomes a whole number c. does not terminate D. terminates | 10 |

151 | Given that ( sqrt{3} ) and ( sqrt{5} ) are irrational numbers, prove that ( sqrt{3}+sqrt{5} ) is an irrational number. | 10 |

152 | Use Euclid’s division lemma to show that the cube of any positive integer is of the form ( 9 m, 9 m+1 ) or ( 9 m+8 ) | 10 |

153 | State true or false. ( sqrt{3}+sqrt{4} ) is an rational number. A. True B. False | 10 |

154 | According to Euclid’s division algorithm, using Euclid’s division lemma for any two positive integers ( a ) and ( b ) with ( a>b ) enables us to find the A. нс в. Lсм c. Decimal expansion D. Probability | 10 |

155 | In a question on division the divisor is 7 times the quotient and 3 times the remainder. If the remainder is 28 then what is the dividend? A . 1008 B. 1516 c. 1036 D. 2135 | 10 |

156 | If ( 10^{2 y}=25, ) then ( 10^{-y} ) equals to A. ( -frac{1}{5} ) в. ( frac{1}{50} ) c. ( frac{1}{625} ) D. ( frac{1}{5} ) | 10 |

157 | Using Euclid’s division of lemma, find H.C.F. of 15 and 575 | 10 |

158 | A book seller has 28 Kannada and 72 English books. The books are of the same size. These books are to be packed in separate bundles and each bundle must contain the same number of books. Find the least number of bundles which can be made and also the number of books in each bundle | 10 |

159 | Write the decimal forms of ( frac{1}{3} ) and ( frac{1}{9} ) | 10 |

160 | Show that cube of any position integer will be in the form of ( 8 m ) or ( 8 m+1 ) or ( 8 m+3 ) or ( 8 m+5 ) or ( 8 m+7, ) where ( m ) is a whole number. | 10 |

161 | f ( 25025=p_{1}^{x_{1}} cdot p_{2}^{x_{2}} cdot p_{3}^{x_{3}} cdot p_{4}^{x_{4}} ) find the value of ( p_{1}, p_{2}, p_{3}, p_{4} ) and ( x_{1}, x_{2}, x_{3}, x_{4} ) | 10 |

162 | 61. The greatest number by which 2300 and 3500 are divided leav- ing the remainders of 32 and 56 respectively, is (1) 136 (2) 168 (3) 42 (4) 84 | 10 |

163 | Factorise the expression and divide them as directed. ( left(m^{2}-14 m-32right) div(m+2) ) | 10 |

164 | Show that the number ( 3-sqrt{5} ) is irrational. | 10 |

165 | Sum of digits of the smallest number by which 1440 should be multiplied so that it becomes a perfect cube is A . 4 B. 6 ( c cdot 7 ) D. | 10 |

166 | Use Euclid’s Division Lemma, prove that for any positive integer ( n, n^{3}-n ) is divisible by 6 | 10 |

167 | Use Euclid’s division lemma to show that the square of any positive integer is either of the form ( 3 m ) or ( 3 m+1 ) for some integer ( boldsymbol{m} ) | 10 |

168 | Prime factors of 140 are : A. ( 2 times 2 times 7 ) В. ( 2 times 2 times 5 ) c. ( 2 times 2 times 5 times 7 ) D. ( 2 times 2 times 5 times 7 times 3 ) | 10 |

169 | Find H.C.F of 81 and 237 Also express it as a linear combination of 81 and 237 i.e, H.C.F of ( 81,237= ) ( 81 x+237 y ) for some ( x, y ) [Note: Values of ( x ) and ( y ) are not unique] | 10 |

170 | Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) ( frac{mathbf{1 3}}{mathbf{3 1 2 5}} ) (ii) ( frac{17}{8} ) (iii) ( frac{mathbf{6 4}}{mathbf{4 5 5}} ) (iv) ( frac{mathbf{1 5}}{mathbf{1 6 0 0}}(mathbf{v}) ) ( frac{mathbf{2 9}}{mathbf{3 4 3}} ) (vi) ( frac{mathbf{2 3}}{mathbf{2}^{3} mathbf{5}^{2}} ) (vii) ( frac{mathbf{1 2 9}}{mathbf{2 ^ { 2 }} mathbf{5}^{7} mathbf{7}^{mathbf{5}}} ) (viii) ( frac{mathbf{6}}{mathbf{1 5}}(text { ix }) frac{mathbf{3 5}}{mathbf{5 0}} ) (x) ( frac{77}{210} ) | 10 |

171 | Find the product: ( left(frac{1}{2} p^{3} q^{6}right)left(-frac{2}{3} p^{4} qright)left(p q^{2}right) ) | 10 |

172 | The number ( frac{13}{15} ) is correctly represented on number line ( A ) ( begin{array}{cccccc}< & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ frac{1}{15} & & & frac{13}{15} & frac{15}{15} & frac{17}{15}end{array} ) B. ( begin{array}{ccccccc}langle 1 & & 1 & 1 & 1 & 1 & 1 \ 10 & & frac{13}{15} & frac{14}{15} & frac{15}{15} & frac{16}{15} & frac{17}{15}end{array} ) ( c ) ( begin{array}{lllllllll}nwarrow_{1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ frac{2}{15} & frac{3}{15} & frac{4}{15} & & frac{13}{15} & frac{14}{15} & frac{15}{15} & frac{16}{15} & cdots & .end{array} ) D. begin{tabular}{ccccccccc} hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ ( frac{14}{15} ) & ( frac{15}{15} ) & ( frac{16}{15} ) & ( frac{17}{15} ) & ( frac{18}{15} ) & ( frac{6}{15} ) & ( frac{13}{15} ) & ( cdot ) & ( cdot ) end{tabular} | 10 |

173 | State whether the following statement is true or not: ( 7-sqrt{2} ) is irrational. A. True B. False | 10 |

174 | Euclids division lemma, the general equation can be represented as A. ( a=b times q+r ) B. ( a=b div q+r ) c. ( a=b-q-r ) D. ( a=b+q-r ) | 10 |

175 | If any positive’ even integer is of the form ( 4 q ) or ( 4 q+2, ) then ( q ) belongs to: A. whole number B. rational number c. real number D. none of these | 10 |

176 | The equation ( sqrt{x+4}-sqrt{x-3}+1=0 ) has: A. no root B. one real root C. one real root and one imaginary root D. two imaginary roots E. two real roots | 10 |

177 | The number ( 0.211211121111211111 ldots ) is a. A. Terminating decimal B. Non-terminating repeating decimal c. Non-terminating and non-repeating decimal D. None of these | 10 |

178 | State true or false: ( sqrt{2} ) is not a rational number. A. True B. False | 10 |

179 | =p+g+r, then x= If – +- pq qr +- pr (a) par (c) e | 10 |

180 | Euclid’s division lemma states that for two positive integers a and b, there exist unique integers ( q ) and ( r ) such that ( boldsymbol{a}=boldsymbol{b} boldsymbol{q}+boldsymbol{r}, ) where ( r ) must satisfy ( A cdot 1<r<b ) B. ( 0<r leq b ) c. ( 0 leq r<b ) ( D cdot 0<r<b ) | 10 |

181 | A number ( x ) when divided by 7 leaves a remainder 1 and another number ( y ) when divided by 7 leaves the remainder 2. What will be the remainder if ( x+y ) is divided by ( 7 ? ) A . B. 2 ( c .3 ) D. 4 | 10 |

182 | Prove that, one and only one out of ( boldsymbol{n}, boldsymbol{n}+mathbf{2}, ) or ( boldsymbol{n}+mathbf{4} ) is divisible by ( boldsymbol{3} ) where ( n ) is any positive integer. | 10 |

183 | Show that any positive odd integer is of the form ( 4 q+1 ) or ( 4 q+3 ) where ( q ) is some integer. | 10 |

184 | Prove that ( n^{2}-n ) is divisible by 2 for every positive integer ( n ) | 10 |

185 | For any integers ( a ) and 3 , there exists unique integers ( q ) and ( r ) such that ( a= ) ( 3 q+r . ) Find the possible value of ( r ) | 10 |

186 | Write the following in decimal form and say what kind of decimal expansion each has: (i) ( frac{mathbf{3 6}}{mathbf{1 0 0}} ) (ii) ( frac{mathbf{1}}{mathbf{1 1}} ) (iii) ( 4 frac{1}{8} ) (iv) ( frac{mathbf{3}}{mathbf{1 3}} ) ( mathbf{2} ) (vi) ( frac{329}{400} ) ( overline{mathbf{1 1}} ) | 10 |

187 | State whether the given statement is True or False: ( 2 sqrt{3}-1 ) is an irrational number A . True B. False | 10 |

188 | The decimal expansion of the number ( sqrt{2} ) is? A . a finite decimal B. 1.41421 c. non-terminating recurring D. non-terminating non-recurring | 10 |

189 | ( frac{1}{2}=0.5 ) It is a terminating decimal because the denominator has a factor as ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 6 | 10 |

190 | The product of ( 4 sqrt{6} ) and ( 3 sqrt{24} ) is? A .124 B. 134 ( c cdot 144 ) D. 154 | 10 |

191 | 52. L.C.M. of two numbers is 120 and their H.C.F. is 10. Which of the following can be the sum of those two numbers ? (1) 140 (2) 80 (3) 60 (4) 70 | 10 |

192 | Prove that the sum of two successive odd numbers is divisible by 4 | 10 |

193 | Find the multiplicative inverse of: (0.11)( (sqrt{0.11}) ) | 10 |

194 | The correct number in the ( 5^{t h} ) decimal place of the number ( frac{2}{7}=0 . overline{285714} ) | 10 |

195 | If the quotient is terminating decimal, the division is complete only when A. we get the remainder 1 B. we get the remainder zero c. we get the remainder as the repeated numbers D. All of the above | 10 |

196 | Apply Euclid”s theorem for 17,5 A. ( 17=5 times 3+2 ) В. ( 17=5 times 2+7 ) ( mathbf{c} cdot 17=5 times 4-3 ) D. None of the above | 10 |

197 | The H. C. F. of 252,324 and 594 is A . 36 B. 18 c. 12 D. 6 | 10 |

198 | Use Euclid’s division algorithm to find the H.C.F. of 6265 and 76254 | 10 |

199 | ( mathbf{2} times mathbf{2} times mathbf{2} times mathbf{3} times mathbf{3} times mathbf{1 3}=mathbf{2}^{mathbf{3}} times mathbf{3}^{mathbf{2}} times mathbf{1 3} ) is equal to A. 1004 в. 828 c. 724 D. 936 | 10 |

200 | Give two rational numbers lying between ( 0.232332333233332 ldots . ) and 0.2121121112111122 Enter 1 if the answer is 0.221,0.222 otherwise enter 0 | 10 |

201 | Find the largest number that will divide ( mathbf{3 9 8}, mathbf{4 3 6} ) and ( mathbf{5 4 2} ) leaving remainders 7,11 and 15 respectively. | 10 |

202 | Use Euclid division Lemma to show that the cube of any positive integer Is either of the form ( 9 m, 9 m+1 ) or, ( 9 m+8 ) for some integer ( boldsymbol{m} ) | 10 |

203 | While representing ( frac{2}{3} ) on a number line, between which 2 integers does the point lie? A. 1 and 2 B. 0 and 1 ( c cdot 2 ) and 3 D. 1 and 3 | 10 |

204 | 1.2348 is: A. An integer B. A rational number C. An irrational number D. A natural number | 10 |

205 | Represent ( frac{7}{4} ) on the number line. | 10 |

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