Real Numbers Questions

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.

Real Numbers Questions

List of real numbers Questions

Question NoQuestionsClass
1Which 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
253.
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
372. 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
4Prove that one of every three
consecutive positive integer is divisible by 3
10
5The 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
6use Euclid’s division algorithm to find
the H.C.F. of 135 and 714
10
770. 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
8Find the multiplicative inverse of: ( frac{2-sqrt{3}}{2+sqrt{3}} )10
9Prove that ( 5-sqrt{3} ) is an irrational
number.
10
10State the following statement is True or False ( frac{7}{9} ) has a value of non terminating decimal number
A. True
B. False
10
11The positive integer whose product with
-1 is
A. positive
B. negative
c. 0
( D cdot ) none
10
12We 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
13What are terminating decimals?10
14The 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
15Without 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
16The HCF of 256,442 and 940 is
( A cdot 2 )
B. 14
( c cdot 142 )
D. none
10
17Using Euclid’s division algorithm, find
the HCF of 56,96 and 404
10
18A 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
19Find the ( H C F ) of 65 and 117 and
express it in the form ( 65 m+117 n )
10
20In 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
21Identify 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
22Find the ( H C F ) of 81 and 237 and
express it as a linear combination of 81
and 237
10
23Express 32844 as a product of prime factors10
24states 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
25Use 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
26The 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
27Product of two integers with
unlike signs is
A. Negative
B.
c. Positive
D. None of these
10
28Use Euclid’s division lemma to find the
HCF of the following
27727 and 53124
A .233
B . 200
c. 154
D. 212
10
29Use Euclids division algorithm to find
the HCF of: 867 and 225
10
30State whether the following statement
is true/false.
( frac{2375}{375} ) is not a terminating decimal
A. True
B. False
10
31Divide 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
3253. 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
33Use 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
34Find the HCF of 92690,7378 and 7161
by Euclid’s division algorithm.
A . 29
B. 30
c. 31
D. 32
10
35Find 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
36When 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
37Three 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
38Convert 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
39Fundamental theorem of arithmetic is
also called as

Factorization
Theorem.
B. Ambiguous
c. Unique
D. None of these

10
40Use 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
41Number ( frac{3}{625} ) is a terminating decimal or a non-terminating repeating decimal? Write it in decimal form10
42Prove that ( 7 sqrt{5} ) is irrationals number.10
43Write three numbers whose decimal
expansions are non terminating and non recurring
10
44Solve:-
( sqrt[3]{boldsymbol{y}}(2 sqrt[3]{boldsymbol{y}}-sqrt[3]{boldsymbol{y}^{8}}) )
10
45Express 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
46What 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
47The HCF of 455 and 42 using Euclid algorithm is
A. 7
B. 6
c. 5
( D )
10
48Evaluate ( sqrt{11 sqrt{11 sqrt{11 sqrt{11 dots dots infty}}}} )10
49State 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
50H.C.F. of 26 and 91 is:
A . 13
в. 2366
( c .91 )
D. 182
10
51Prove 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
53Which 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
55The ………… when multiplied always give a new unique natural number.
A. decimal numbers
B. fractions
c. irrational numbers
D. prime numbers
10
56Is ( pi ) a rational number? Justify your
answer
10
57Classify the following numbers as rational or irrational.
( mathbf{1 1} . mathbf{2 1 3 2} mathbf{4 3 5 4 6 5} )
10
58In 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
59Show 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
60Evaluate ( sqrt[3]{left(frac{1}{64}right)^{-2}} )
( A )
B . 16
( c .32 )
D. 64
10
61Prove that, if ( x ) and ( y ) are odd positive integers, then ( x^{2}+y^{2} ) is even but not
divisible by 4.
10
62Consider 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?
A. 1 only
B. 2 only
c. 1 and 3
D. 2 and 3

10
6353. 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
64Which 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
65Which 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
66The 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
67Prove that square of any positive integer is always of the form ( 4 m, 4 m+ ) ( 1,4 m+4 ) or ( 4 m+9 )10
68Show that ( 1.272727 ldots=1 . overline{27} ) can be
expressed in the form ( underline{boldsymbol{p}} ) ( overline{boldsymbol{q}} )
10
69Is ( pi ) an irrational number? Why?10
70The additive identity for integers is
( A cdot 0 )
B.
( c .0 . )
D. 2
10
71If ( 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
72Find the largest number that divides
2053 and 967 and leaves a remainder of
5 and 7 respectively.
10
73Prove ( 6+3 sqrt{5} ) is irrational10
74The non terminating non-recurring decimal cannot be represented as
A. irrational numbers
B. rational numbers
c. real numbers
D. none of these
10
7520 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
76The 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
77Without actual division, check whether ( frac{47}{14} ) is terminating or not.10
78( left(y^{2}+7 y+10right) div(y+5) )10
79Use Euclid’s division algorithm to find
the HCF of 135 and 225
10
80The 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
81Use Euclid’s division algorithm to find
the ( H C F ) of 196 and 38220
10
82In 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
83Prove the following are irrational
numbers
( mathbf{3}+sqrt{mathbf{5}} )
10
84According 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
85Find the HCF of 1656 and 4025 by
Euclids division theorem
10
8653. 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
87Use 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
88Use Euclid’s algorithm to find the HCF of
4052 and 12576
10
89Use 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
90Show that ( 3 sqrt{2} ) is irrational.10
91One 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
9257. Product of two co-prime numbers
is 117. Then their L.C.M. is
(1) 117
(2) 9
(3) 13
(4) 39
10
93Say 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
94Calculate the HCF of ( 3^{3} times 5 ) an ( 3^{2} times 5^{2} )10
9535
has a non-terminating decimal
( overline{mathbf{5 0}} )
expansion
A. True
B. False
10
96Find HCF of 45 and 7210
97Prove that ( sqrt{3}+sqrt{8} ) is an irrational
number.
10
98Show 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
99If ( 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
100If ( 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
102Write the following in decimal form and say what kind of decimal expansions
has:
( frac{2}{11} )
10
103Use 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
104Write the following in decimal form and say what kind of decimal expansions has:
( frac{329}{400} )
10
105Using Euclid’s division algorithm find
the HCF of the following numbers.
( mathbf{2 0 2 4} ) and ( mathbf{1 8 7 2} )
10
106Use Euclid’s division algorithm to find
the ( H C F ) of 210 and 55
10
107Express 120 as a product of prime factors10
10851. 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
109Show that ( sqrt{2}+sqrt{3} ) is an irrational
number.
10
110We 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
111In Euclid’s Division Lemma, when
( a=b q+r ) where ( a, b ) are positive integers then what values r can take?
10
112Divide as directed.
( 26 x y(x+5)(y-4) div 13 x(y-4) )
10
113For 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
114If 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
115Euclids 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
116Prove that ( log _{2} 3 ) is an irrational
number.
10
117As the decimal of ( frac{1}{3} ) repeats, ( frac{1}{3} ) is a
decimal.
A. exact
B. negative
c. terminating
D. non-terminating
10
118Prove that ( 3 sqrt{2} ) is irrational10
119Use Euclid’s division lemma to find the
HCF of 13281 and 15844
10
120Prove 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
121Use Euclid’s division lemma to find the
HCF of the following 65 and 495
A . 5
B. 10
c. 15
D. 0
10
122If 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
123Show 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
124Find the H.C.F. of the following numbers using prime factorization method.
81,117
10
125Use Euclid’s division algorithm to find
the HCF of the following number: 55 and
210
10
126Prove that ( 5 sqrt{3} ) is an irrational.10
127Prove that ( 4-3 sqrt{2} ) is an irrational10
128Prove that ( sqrt{2} ) is an irrational number.10
129Use Euclid’s division algorithm to find
the ( H C F ) of 4052 and 12576
10
130What is the square of ( (2+sqrt{2}) ? )
A. A rational number
B. An irrational number
c. A natural number
D. A whole number
10
131Use 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
13249. 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
133Use Euclids division algorithm to find
the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 225
10
134In 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
135If 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
136Square root of ( sqrt{13-4 sqrt{3}} )10
137State 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
138Find the highest common factor of the
monomial.
( 6 a^{2} b^{2} c ) and ( 27 a b c^{3} )
10
139Use Euclid’s division algorithm to find the H.C.F. of 196 and 3831810
140Represent ( frac{-3}{4} ) on number line10
141Show that the square of any positive integer is of the form ( 4 mathrm{m} ) or ( 4 mathrm{m}+1 ) for any integer ( m )10
142Express 3825 as a product of prime factors10
143Show that ( n^{2}-1 ) is divisible by 8 , if ( n ) is
an odd positive integer.
10
144State whether the given statement is True or False:
( 3+sqrt{2} ) is an irrational number.
A . True
B. False
10
145Given that ( sqrt{2} ) is irrational, prove that ( (5+3 sqrt{2}) ) is an irrational number.10
146In 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
147Use Euclid’s division lemma to find the
HCF of the following
16 and 176
A . 16
B. 14
c. 10
D. 12
10
148The 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
149Prove that ( sqrt{3} ) is a irrational number.10
150If 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
151Given that ( sqrt{3} ) and ( sqrt{5} ) are irrational numbers, prove that ( sqrt{3}+sqrt{5} ) is an
irrational number.
10
152Use 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
153State true or false.
( sqrt{3}+sqrt{4} ) is an rational number.
A. True
B. False
10
154According 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
155In 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
156If ( 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
157Using Euclid’s division of lemma, find
H.C.F. of 15 and 575
10
158A 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
159Write the decimal forms of ( frac{1}{3} ) and ( frac{1}{9} )10
160Show 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
161f ( 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
16261.
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
163Factorise the expression and divide them as directed.
( left(m^{2}-14 m-32right) div(m+2) )
10
164Show that the number ( 3-sqrt{5} )
is irrational.
10
165Sum 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
166Use Euclid’s Division Lemma, prove that
for any positive integer ( n, n^{3}-n ) is
divisible by 6
10
167Use 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
168Prime 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
169Find 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
170Without 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
171Find the product:
( left(frac{1}{2} p^{3} q^{6}right)left(-frac{2}{3} p^{4} qright)left(p q^{2}right) )
10
172The 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
173State whether the following statement
is true or not:
( 7-sqrt{2} ) is irrational.
A. True
B. False
10
174Euclids 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
175If 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
176The 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
177The 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
178State 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
180Euclid’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
181A 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
182Prove 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
183Show that any positive odd integer is of
the form ( 4 q+1 ) or ( 4 q+3 ) where ( q ) is
some integer.
10
184Prove that ( n^{2}-n ) is divisible by 2 for
every positive integer ( n )
10
185For 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
186Write 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
187State whether the given statement is True or False:
( 2 sqrt{3}-1 ) is an irrational number
A . True
B. False
10
188The 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
190The product of ( 4 sqrt{6} ) and ( 3 sqrt{24} ) is?
A .124
B. 134
( c cdot 144 )
D. 154
10
19152. 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
192Prove that the sum of two successive
odd numbers is divisible by 4
10
193Find the multiplicative inverse of:
(0.11)( (sqrt{0.11}) )
10
194The correct number in the ( 5^{t h} ) decimal
place of the number ( frac{2}{7}=0 . overline{285714} )
10
195If 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
196Apply 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
197The H. C. F. of 252,324 and 594 is
A . 36
B. 18
c. 12
D. 6
10
198Use 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
200Give 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
201Find 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
202Use 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
203While 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
2041.2348 is:
A. An integer
B. A rational number
C. An irrational number
D. A natural number
10
205Represent ( frac{7}{4} ) on the number line.10

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