# Number Systems Questions

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

Question NoQuestionsClass
1Simplify and express with positive zponents ( left[left(frac{mathbf{9}}{mathbf{1 1}}right)^{-mathbf{3}} timesleft(frac{mathbf{9}}{mathbf{1 1}}right)^{-mathbf{7}}right] div )
( left(frac{9}{11}right)^{-3} )
9
2Which of the following is a rational number?
( mathbf{A} cdot 3^{29 / 27} .3^{27 / 29} )
B ( cdot sqrt{left(3^{50 / 51} .3^{52 / 51}right)^{3}} )
( mathbf{c} cdot 3^{15 / 6}-3^{7 / 8} )
D. ( sqrt{left(3^{3}right)^{1 / 5}} )
9
3Represent on number line ( frac{1}{6} )9
4Classify the result as rational or irrationals. ( (3+sqrt{23})-sqrt{23} )
A. Rational number
B. Irrational number
c. Data Insufficient
D. None of the above
9
5Draw the number line and represent the following rational number on it:
( frac{-5}{8} )
9
6Which of the following statements is
true?
A ( cdot frac{-5}{8} ) lies to the left of 0 on the number line
B. ( frac{3}{7} ) lies to the right of 0 on the number line.
C the rational numbers ( frac{1}{3} ) and ( frac{-7}{3} ) are on opposite sides of 0 on the number line
D. All the above
9
7Is zero a rational number? Can you write it in the form ( frac{p}{q}, ) where ( p ) and ( q ) are integers and ( boldsymbol{q} neq mathbf{0} ? )9
8Find:
(i) ( 9^{frac{3}{2}} )
(ii) ( 32^{frac{2}{5}} )
(iii) ( 16^{frac{3}{4}} )
(iv) ( 125^{frac{-1}{3}} )
9
9A rational number ( frac{-2}{3} )
A. Lies to the left side of 0 on the number line.
B. Lies to the right side of 0 on the number line.
C. Is not possible to represent on the number line.
D. None of these.
9
10State True or False.
A rational number can always be written in what as a fraction ( frac{a}{b}, ) where a
and b are not integers ( (boldsymbol{b} neq mathbf{0}) )
A. True
B. False
9
11Is zero a rational number? Justify9
12Express ( 0 . overline{38} ) as a rational number in
simplest form.
9
13n exponential form ( 729=3^{a}, ) what is
the value of ( a ? )
9
14Mark the following rational numbers on number line.
( frac{-8}{3} )
9
15Rewrite the following rational numbers in the simplest form:
( frac{25}{45} )
9
16By what number should ( left(frac{-2}{3}right)^{-3} ) be divided so that the quotient may be ( left(frac{4}{27}right)^{-2} )9
1752. Arranging the following in de-
scending order, we get
44, 12, 93, 45
(1) 14 > 45>2 > 3
(2) 15 > 14 > 3 > V2
(3) V2 > 3 > 4 > 45
(4) 03 > 45 > 34 > 2
9
18Find the value of ( (27)^{-frac{2}{3}}+ )
( left(left(2^{-frac{2}{3}}right)^{-frac{5}{3}}right)^{-frac{9}{10}} )
A ( cdot frac{1}{9} )
B. ( frac{2}{9} )
c. ( frac{11}{18} )
( D )
9
19The number of rational numbers
between two given rational numbers is
A . Infinite
B. Finite
c. Two
D. one
9
20( left[5left{left(frac{1}{8}right)^{frac{-1}{3}}+left(frac{1}{27}right)^{frac{-1}{3}}right}right]^{frac{1}{2}} )
( mathbf{A} cdot 5 )
B. ( sqrt{13} )
( mathbf{c} .25 )
D. 13
9
21If ( 10^{4 x}=625 ) then find the value of ( 10^{-x} )9
22Evaluate:
( frac{1}{(216)^{frac{-2}{3}}} div frac{1}{(27)^{frac{-4}{3}}} ) is equal to ( frac{4}{m} )
value of ( m ) is
9
23Solve and find the value of ( x )
( mathbf{2}^{boldsymbol{x}-mathbf{5}}=mathbf{2 5 6} )
9
24Is this a negative rational number? ( frac{-2}{-9} )9
2552. Arranging the following in de-
scending order, we get
34, 12, 93, 45
(1) 4 > 45 > 2> 93
(2) 45 > 4> 93>
(3) V2> 93 > 34 > 45
(4) 93 > 45 > 34 >
9
26If ( boldsymbol{x}=mathbf{3} ) and ( boldsymbol{y}=-mathbf{2}, ) the value of ( boldsymbol{x}^{boldsymbol{x}}+ )
( boldsymbol{y}^{boldsymbol{y}} ? )
A . 27
B. 9
c. 8
D. none of the above
9
27( left(frac{1}{64}right)^{0}+(64)^{frac{-1}{2}}+(32)^{frac{4}{5}}-(32)^{frac{-4}{5}} ) is
equal to
A ( cdot_{16} frac{1}{8} )
в. ( 17 frac{1}{8} )
c. ( _{17} frac{1}{16} )
D. ( -17 frac{1}{16} )
9
28The number 0 is not the reciprocal of
any number.
A. True
B. False
c. Ambiguous.
D. None
9
29The values of ( A ) and ( B ) represented on
the number line are
begin{tabular}{cccccccccc}
hline & & & & & & & & & \
( frac{-6}{1} ) & ( frac{-5}{1} ) & ( frac{-4}{1} ) & ( frac{-3}{mathrm{A}} ) & ( frac{-2}{1} ) & ( frac{-1}{1} ) & ( frac{0}{1} ) & ( frac{1}{mathrm{B}} ) & ( frac{2}{1} ) & ( frac{3}{1} )
end{tabular}
A . 1,1
B. -1,1
( c cdot 1,-1 )
D. -1,-1
9
30Fill in the blank:
The quotient when a rational number is
divided by its additive inverse is
9
31Solve :
( left(14 x^{3} times 2 x^{4} times 8 x^{8}right) div 7 x^{3} )
9
32Find the value of ( left(x^{3} times x^{7}right) div x^{12} ) for ( x= )
(-2)
9
33Given: ( left(frac{1}{2^{3}}right)^{2}=frac{1}{2^{m}} . ) The value of ( m ) is9
34Are the following statements true or false? Given reasons for your answer. Every rational number is a whole
number.
A. True
B. False
9
35Solve ( left[left(frac{-2}{3}right)^{4} timesleft(frac{-2}{3}right)^{2} divleft(frac{4}{9}right)^{3}right] )9
36Express each of the following exponential expressions as a rational number. ( left(frac{2}{3}right)^{(-1)}+left(frac{3}{2}right)^{(-2)} )9
37There exists …….. number of rational numbers between ( frac{2}{5} ) and ( frac{4}{5} )
A. 0
B. 1
c. 5
D. infinite
9
38Rationalising the denominator of ( frac{5}{sqrt{3}-sqrt{5}} ) is –
A ( cdotleft(frac{5}{2}(sqrt{3}+sqrt{5})right. )
B ( cdotleft(-frac{5}{2}(sqrt{3}+sqrt{5})right. )
c. ( left(frac{5}{2}(sqrt{3}-sqrt{5})right. )
D ( cdotleft(-frac{5}{2}(sqrt{3}-sqrt{5})right. )
9
39Simplify:
( 2^{2} times frac{3^{2}}{2^{-2}} times 3^{-1} )
9
40State, whether the following number is rational,

If rational then enter 1 and if false then
enter 0 ( (5+sqrt{5})(5-sqrt{5}) )

9
41There are ………… rational numbers
between two rational numbers.
A . infinite
B. two
c. one
D. none of these
9
42Is ( frac{mathbf{6 3}}{mathbf{9 0}} ) a terminating rational number?
Justify.
9
43Simplify the following using laws of
exponents.
( left(3^{2}right) timesleft(3^{2}right)^{4} )
9
44Mark the following rational numbers on
the number line.
( frac{3}{2} )
9
45Find any two rational numbers between ( frac{-1}{2} ) and ( frac{-2}{3} )9
46( frac{5}{7}+frac{4}{7}-frac{3}{7} ) is equal to
A ( cdot frac{3}{7} )
B. ( frac{5}{7} )
( c cdot frac{6}{7} )
D. ( frac{12}{7} )
9
47Find 9 rational numbers between 2 and
3
( mathbf{A} cdot 2<2.1<2.2<3.3<2.4<ldots<2.9<3 )
B . ( 2<4.1<2.2<2.3<2.4<ldots<2.9<3 )
c. ( 2<2.1<2.2<2.3<2.4<ldots<2.9<3 )
D. ( 2<2.1<2.2<2.3<2.4<ldots<0.9<3 )
9
48( sqrt{9} ) is a rational number. It is equal to
A . 4.5
B. 3
( c cdot 27 )
D. 18
9
49Find the value of ( x ) if ( 2^{4} times 2^{5}=left(2^{3}right)^{x} )9
50Find the value of
i) ( 2^{6} )
ii) ( 9^{3} )
iii) ( 11^{2} )
9
51Find the value of ( m ) for which ( 5^{m} div )
( 5^{-3}=5^{5} )
9
52Find the five rational numbers between
( frac{1}{6} ) and ( frac{1}{3} )
9
53Find whether the following statement
are true or false.
In a rational number of the form ( frac{boldsymbol{p}}{boldsymbol{q}}, boldsymbol{q} )
must be a non zero integer.
A. True
B. False
9
54Find two rational and two irrational
number between 2 and 5
9
55Convert in ( frac{boldsymbol{p}}{boldsymbol{q}} ) form:
(i) ( 0 . overline{47} )
(ii) ( 0 . overline{001} )
(iii) ( 0 . overline{9} )
(iv) ( 2.3 overline{5} )
9
56Represent ( -frac{7}{5} ) on no line9
57( 4^{3.5}: 2^{5} ) is the same as9
58The rational number lying between the numbers ( frac{1}{3} ) and ( frac{3}{4} ) are
A. ( frac{97}{300}, frac{299}{500} )
в. ( frac{99}{300}, frac{301}{400} )
c. ( frac{95}{300}, frac{301}{400} )
D. ( frac{117}{300}, frac{287}{400} )
9
59Find the value of following ( (625)^{frac{-3}{4}} )9
60If ( 2^{x}-2^{x-1}=4, ) then what is the value
of ( 2^{x}+2^{x-1} ? )
A . 8
B. 12
c. 10
D. 16
9
61Assertion
2 is a rational number.
Reason
The square roots of all positive integers
are irrationals.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
9
62Find the 12 rational number between
-1 and ( 2 ? )
9
63Which of the following number lines represents only natural numbers?
( A )
( begin{array}{lllllllllll}nwarrow & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10end{array} )
B.
( c )
D.
( frac{1}{10} frac{2}{10} frac{3}{10} frac{4}{10} frac{5}{10} frac{6}{10} frac{7}{10} frac{8}{10} frac{9}{10} frac{10}{10} )
9
64Write 5 rational number between ( frac{2}{5} ) and ( frac{3}{5}, ) having the same denominators9
65Find the value of ( x ) for which ( left(frac{4}{9}right)^{x} times )
( left(frac{4}{9}right)^{-7}=left(frac{4}{9}right)^{2 x+1} )
9
66The value of ( (-3)^{0}-(-3)^{3}- )
( (-3)^{-1}+(-3)^{4}-(-3)^{-2} ) is
A ( cdot_{109} frac{2}{9} )
в. ( _{109} frac{9}{2} )
( c .109 )
D. None of these
9
67Find two rational numbers between ( frac{-3}{4} ) and ( frac{1}{2} )9
68Which of the following are not rational?
This question has multiple correct options
A ( cdot frac{2}{sqrt{3}} )
B. ( sqrt{16} )
( c . pi )
D. ( 2+sqrt{25} )
9
69If ( 25^{x+1}=frac{125}{5^{x}}: ) then the value of ( x ) is
1
( bar{m} )
Value of ( m ) is
A .2
B. 4
( c cdot 3 )
D. none of the above
9
70Classify the following number as rational or irrational –
(i) ( 2-sqrt{5} )
(ii) ( (3+sqrt{23})-sqrt{23} )
(iii) ( frac{2 sqrt{7}}{7 sqrt{7}} )
(iv) ( frac{1}{sqrt{2}} )
(v) ( 2 pi )
9
71Find 4 rational numbers between
( frac{1}{6} ) and ( frac{3}{8} )
9
72Represent on number line ( frac{3}{7} )9
73State True or False.
The five rational numbers between ( frac{3}{5} ) and ( frac{4}{5} ) are ( frac{19}{30}, frac{20}{30}, frac{21}{30}, frac{22}{30}, frac{23}{30} )
A. True
B. False
9
74The value of ( left(8^{-25}-8^{-26}right) ) is
A ( cdot 7 times 8^{-25} )
В. ( 7 times 8^{-26} )
c. ( 8 times 8^{-26} )
D. None of these
9
75Represent on number line ( frac{mathbf{3}}{mathbf{4}} )9
76Insert three rational numbers between
( frac{2}{3} ) and ( frac{3}{5} )
9
77Simplify ( frac{1}{177} cdot 11 frac{1}{17} )9
78ff ( 4^{x+3}=112+8 times 4^{x} ); find the value
of ( (18 x)^{3 x} )
9
79If ( sqrt{sqrt{mathbf{2 5 0 0}}+sqrt{mathbf{9 6 1}}}=(x)^{2}, ) then ( boldsymbol{x} )
equals
( mathbf{A} cdot 81 )
в.
c. 6561
( D )
9
80A rational number can be expressed
as a terminating decimal if the denominator has factors:
( A cdot 2 ) or 5
в. 2,3 от 5
c. 3 or 5
D. None of these
9
81Mark the following rational numbers on
the number line.
( frac{1}{2} )
9
82State true or false:
( left(frac{2}{3}right)^{4} divleft(frac{2}{3}right)^{6}=left(frac{2}{3}right)^{2} )
A. True
B. False
c. Ambiuous
D. Data insufficient
9
83Simpify and express following as a
rational number:
( left(frac{3}{4}right)^{2} timesleft(frac{-1}{2}right)^{5} times 2^{3} )
9
84Draw the number line and represent the following rational numbers on it:
( frac{-5}{8} )
9
85Following are the five rational numbers
which are smaller than ( 2 Rightarrow ) ( 1, frac{1}{2}, 0,-1, frac{-1}{2} )
If true then enter 1 and if false then
enter ( mathbf{0} )
9
86The product of ( x^{1 / 2} cdot x^{1 / 4} cdot x^{1 / 8} dots infty )
equals
( mathbf{A} cdot mathbf{0} )
B.
c. ( x )
( D cdot infty )
9
87Write a rational number between 7 and
87
9
88Which of the following rational number lies between ( frac{4}{9} ) and ( frac{4}{5} ? )
A. -1
в. ( frac{28}{45} )
c. 0
( D )
9
89How many rational numbers are there
between ( -frac{3}{2} ) and 0 with denominator as
1?
9
90Draw the number line and represent the following rational numbers on it:
( frac{3}{4} )
9
91( f(-4)^{-2} ) is ( frac{1}{m}, ) then the value of ( m ) is9
92Which of the following numbers lies between ( 2 frac{1}{7} ) and ( 3 frac{1}{7} ? )
A ( frac{37}{7} )
B. ( frac{14}{7} )
( c cdot frac{37}{14} )
D. 2
9
93( frac{3^{x} times 3}{3^{2 x-2}}=9^{-2}, ) find ( x )9
94Draw number lines and locate the
points on them:
( frac{2}{5}, frac{3}{5}, frac{8}{5}, frac{4}{5} )
9
95If we divide a positive integer by another positive integer, what is the resulting
number?
A. Always a natural number
B. Always an integer
c. A rational number
D. An irrational number
9
96( f frac{4 a b^{2}left(-5 a b^{3}right)}{10 a^{2} b^{2}}=-K b^{3} ) then value of
( boldsymbol{K} ) is
9
97( frac{2^{n+4}-2 times 2^{n}}{2 times 2^{n+3}}+2^{-3}=? )
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot frac{1}{2} )
D.
9
98Simplify.
( frac{25 times t^{-4}}{5^{-3} times 10 times t^{-8}}(t neq 0) ) is ( frac{m t^{4}}{2} )
The value of ( m ) is
9
99Represent on number line ( frac{2}{3} )9
100Simplify and give reasons:
( (-2)^{7} )
( mathbf{A} cdot-128 )
в. 128
( c cdot-28 )
D. None of these
9
10154. Among the following num-
bers 512, 34, 45, 73 the
least one is :
(1) C12 (2) 44
(3) 45 (4) 13
9
102Find whether the following statements
are true or false.
( pi ) is an irrational number.
A. True
B. False
9
103Write ( frac{120 m^{2} n^{-3}}{60 m^{5} n^{-2}} ) in simplest form
using only positive exponents. Assume that ( m neq 0 ) and ( n neq 0 )
A ( cdot frac{2 n}{m^{3}} )
в. ( frac{2}{m^{3} n} )
c. ( frac{2 m^{3}}{n} )
D. ( frac{1}{2 m^{3} n} )
9
104Find the value of ( x ) if ( x^{3}=left(frac{6}{5}right)^{-3} x ) ( left(frac{6}{5}right)^{6} )9
105Which are two rational number between ( frac{6}{5} ) and ( frac{7}{5} )9
106Write three rational numbers that lie
between the two given numbers. ( frac{7}{9},-frac{5}{9} )
9
107Zero is a rational number.
if true then enter 1 and if false then
enter 0
9
108Which of the following rational numbers is in the standard form?
A ( -frac{8}{-36} )
в. ( frac{-7}{56} )
c. ( frac{3}{-4} )
D. None
9
109Find five rational numbers between ( frac{2}{3} ) and ( frac{mathbf{3}}{mathbf{5}} )9
110If ( sqrt{9^{x}}=sqrt{9^{2}}, ) then ( x= )
A ( cdot frac{2}{3} )
B. ( frac{4}{3} )
( c cdot frac{1}{3} )
D.
9
111Mark the following rational numbers on the number line.
( frac{10}{3} )
9
112The 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 from ( frac{p}{q}, ) what can you say about the prime factors of ( q ? )
¡) 43.123456789
9
113Which of the following statement is true about a rational number ( frac{-2}{3} ? )
A . It lies to the left side of ( ^{prime} 0^{prime} ) on the number line
B. It lies to the right side of ‘ ( 0^{prime} ) on the number line
c. It is not possible to represent on the number line.
D. It cannot be determined on which side the number lies
9
114Find the value of ( m ) for which
(i) ( 5^{m} div 5^{-3}=5^{5} )
(ii) ( 4^{m}=64 )
(iii) ( 8^{m-3}=1 )
(iv) ( left(a^{3}right)^{m}=a^{9} )
(v) ( left(5^{m}right)^{2} times(25)^{3} times 125^{2}=1 )
( (mathrm{vi}) 2^{m}=(8)^{frac{1}{3}} divleft(2^{3}right)^{2 / 3} )
9
115Zero can be written in the form of ( frac{boldsymbol{p}}{boldsymbol{q}} ) where p and q are integers and ( mathrm{q} neq 0 ? ) If above statement is true then enter 1
and if false then enter 0
9
116Find the value of:
(i) ( left(3^{0}+4^{-1}right) times 2^{2} )
(ii) ( left(2^{-1} times 4^{-1}right) div 2^{-2} )
(iii) ( left(frac{1}{2}right)^{-2}+left(frac{1}{3}right)^{-2}+left(frac{1}{4}right)^{-2} )
(iv) ( left(3^{-1}+4^{-1}+5^{-1}right)^{0} )
( (v)left[left(frac{-2}{3}right)^{-2}right]^{2} )
( (v i) 7^{-20}-7^{-21} )
9
117The exponential form of 512 is ( 2^{k} ) then
value of ( k ) is
9
118Out of the following numbers, which cannot be represented on a number line? ( mathbf{0}, frac{mathbf{5}}{6}, 1, frac{2}{4} )
( mathbf{A} cdot mathbf{0} )
в. ( frac{5}{6} )
c. 1
D. None of these
9
119By expressing the following in the form
( frac{p}{q}, ) where ( p ) and ( q ) are integers and ( q neq )
( mathbf{0} )
0.2 is equal to ( frac{1}{m}, ) Find the value of ( m )
9
120Identify the rational numbers represented by ( mathrm{B}, mathrm{C} ) and ( mathrm{D} ) if ( mathrm{AB}=mathrm{CD}=mathrm{DE} )
as drawn on the number line.
[
begin{array}{|c|c|c|}
hline E D & C & A \
hline-2 & 0 & 2 \
hline
end{array}
]
A. 0,-1,-2
в.
[
frac{-1}{2},-1, frac{-3}{2}
]
c.
[
frac{1}{2}, 1, frac{3}{2}
]
D.
[
frac{-1}{4},-1, frac{-5}{2}
]
9
121List five rational numbers between:
( frac{-4}{5} ) and ( frac{-2}{3} )
9
122( frac{1}{text { The vaue of } 13} frac{1}{5} .17^{frac{1}{5}}-(221)^{frac{1}{5}} ) is9
123Give four rational numbers equivalent
to:
( frac{mathbf{5}}{-3} )
9
124State whether the following statement is true or false.
The rational numbers are whole
numbers, fractions, mixed numbers, and decimals, together with their
negative images.
A. True
B. False
9
125Find the value of ( x ) in the number line.
[
begin{array}{ccc}
1 & 1 & 1 & 1 \
0 & sqrt{x} & 6
end{array}
]
9
126If ( a b c=1, ) then ( frac{1}{1+a+b^{-1}}+ )
( frac{1}{1+b+c^{-1}}+frac{1}{1+c+a^{-1}} ) is equal to
( A )
B. 0
( c )
D. – 5
9
127Mark the following rational numbers on
the number line.
( frac{3}{4} )
9
128Does the following pairs represent the same rational number:
( frac{-7}{21} ) and ( frac{3}{9} )
9
129Explain the density of rational numbers with examples.9
130Simplify and give reasons:
( (-3)^{-4} )
( ^{A} cdot frac{1}{81} )
в. ( frac{-1}{81} )
c. ( frac{3}{81} )
D. None of these
9
131The rational number lying exactly in between the numbers ( frac{1}{5} ) and ( frac{1}{3} ) is
A ( cdot frac{1}{2} )
в.
c. ( frac{2}{15} )
D. ( frac{4}{15} )
E. ( frac{8}{15} )
9
132If ( frac{mathbf{9}^{n} times mathbf{3}^{5} times(mathbf{2 7})^{mathbf{3}}}{mathbf{3} times(mathbf{8 1})^{4}}=mathbf{2 7}, ) then the value
of ( n ) is
( mathbf{A} cdot mathbf{0} )
B . 2
( c cdot 3 )
D.
9
133Solve ( 17^{2} cdot 17^{-5} )9
134Find any two rational numbers ( frac{1}{4} ) and ( frac{3}{4} )9
135Rational numbers on the number line
represented by ( boldsymbol{A}, boldsymbol{E} ) and ( boldsymbol{H} )
( operatorname{are} boldsymbol{A}=frac{mathbf{1}}{mathbf{1 0}}, boldsymbol{E}=frac{mathbf{5}}{mathbf{1 0}}, boldsymbol{H}=frac{mathbf{8}}{mathbf{1 0}} . ) Are the
values correct?
A. Yes
B. No
c. cannot be determined
D. None
9
136Rewrite the following rational numbers in the simplest form:
( frac{-44}{72} )
9
13713.
2 4 .5 6
1546-56-28 + 2
=?
1 11 (2) – 5
(3) 3 + 2 (4),3-5
9
138Is this a negative rational number? ( frac{6}{11} )9
139Evaluate: ( [sqrt{sqrt{5^{9}}}]^{4}[sqrt{sqrt{5^{9}}}]^{4} )
( A cdot 5^{2} )
в. ( 5^{4} )
( c cdot 5^{8} )
D. ( 5^{12} )
9
140( (-3)^{-2}=frac{1}{k} ) then value of ( k ) is9
141Find the nine rational numbers
between 0 and 1
A. ( 0.1,0.2,0.3, ldots, 0.9 )
в. ( 1.1,0.2,10.3, ldots, 0.9 )
c. ( 0.1,0.2,0.3, ldots, 20.9 )
D. ( 0.1,0.2,10.3, ldots, 0.9 )
9
142State whether the statement is
true/false.
8 can be written as a rational number
with any integer as denominator.
A. True
B. False
9
143State whether the following number is rational,

If rational then enter 1 and if false then
enter 0
( left(frac{sqrt{7}}{6 sqrt{2}}right)^{2} )

9
144Is this a negative rational number? ( frac{-2}{3} )9
145Which of the following numbers are rational?
( mathbf{A} cdot mathbf{1} )
B. – 6
( c cdot_{3} frac{1}{2} )
D. All above are rationa
9
146State true or false:
Rational number from following real
numbers ( -8.0, sqrt{5}, frac{5}{7},-sqrt{18}, sqrt{32}, 4.28, pi, 3, frac{8}{15} )
( operatorname{are}-8.0 . frac{5}{7}, 4.28,3, frac{8}{15}, 0.075 )
A. True
B. False
9
147The following real numbers have decimal expansions as given below. In each case, decide whether they are
rational or not. If they are rational or not. If they are rational, and of them ( frac{p}{q}, ) what can you say about the prime factors of ( q )
( ? )
¡) 43.123456789
¡i) 0.120120012000120000….
iii) ( 43 . overline{123456789} )
9
148Evaluate ( (sqrt{3}+sqrt{2})^{6}-(sqrt{3}+sqrt{2})^{6} ? )9
149State whether the following number is
rational, If rational then enter 1 and if
false then enter 0
( (2+sqrt{2})^{2} )
9
150Fill in the blanks so as to make the
statement true:

Two rational numbers with different
numerators are equal, if their numerators are in the same
as their denominators.

9
151All rational numbers are real numbers.
A . True
B. False
9
152List five rational numbers between:
¡) -1 and 0
ii) -2 and -1
9
153Express in the simplest form. ( sqrt{frac{mathbf{1 7 5}}{mathbf{2 7}}} )
A. ( -frac{5}{3} sqrt{frac{7}{3}} )
в. ( frac{5}{3} sqrt{frac{7}{3}} )
c. ( frac{5}{3} sqrt{frac{1}{3}} )
D. Not possible
9
154Between two rational numbers, there
exists-
A. No rational number
B. Only one rational number
c. Infinite numbers of rational numbers
D. No irrational number
9
155Solved:
( 65536=4^{n-1} )
9
156Find four rational numbers between ( frac{3}{5} ) and ( frac{4}{5} )9
157Simplify:
( frac{8^{-1} times 5^{3}}{2^{-4}} )
9
158Evaluate : ( left(frac{5}{3}right)^{x} cdotleft(frac{9}{25}right)^{x^{2}+2 x-11}= )
( left(frac{5}{3}right)^{9} )
9
159( sqrt{frac{54}{250}} ) equals:
A ( cdot frac{9}{25} )
B. ( frac{3}{5} )
c. ( frac{27}{125} )
D. ( frac{sqrt{2}}{5} )
9
160Find the value of the following:
( left(frac{3}{8}right)^{5} timesleft(frac{3}{8}right)^{4} divleft(frac{3}{8}right)^{9} )
9
161Prove that ( frac{left(boldsymbol{a}^{boldsymbol{p}+boldsymbol{q}}right)^{2}left(boldsymbol{a}^{boldsymbol{q}+boldsymbol{r}}right)^{2}left(boldsymbol{a}^{boldsymbol{r}+boldsymbol{p}}right)^{mathbf{2}}}{left(boldsymbol{a}^{boldsymbol{p}} cdot boldsymbol{a}^{boldsymbol{q}} boldsymbol{a}^{boldsymbol{r}}right)^{boldsymbol{4}}}=mathbf{1} )9
162Write five rational numbers greater
than -2
9
163Given ( 2^{-3} times(-7)^{-3}=-frac{1}{m} ) Value of ( m )
is
9
164On the real number line below, numbers increase in value from left to right. If ( B>0, ) then the value of ( A ) must be:
A. Negative
B. Positive
c. Less than ( B )
D. Greater than ( B )
E. Between 0 and ( B )
9
165Solve:
( x^{11} div x^{15} )
9
166Write the rational numbers which are
smaller than 2
9
1675 is a rational number. It can be written
as
A ( cdot frac{5}{1} )
B. ( frac{1}{5} )
( c cdot frac{5}{5} )
D. ( frac{5}{25} )
9
168The rational number, which equals the number 2.357 with
recurring decimal is
(1983 – 1 Mark)
(a) 2355 (b) 2379 (c) 2355
1001 (0) 907 (C) 000 (d) none of these
9
169Simplification of ( left(frac{3}{5}right)^{3} timesleft(frac{15}{2}right)^{3} ) is ( frac{729}{8} )
A. True
B. False
9
170State where ( (sqrt{6}+sqrt{9}) ) is rational or
not
9
171The points ( P, Q, R, S, T, U, A ) and ( B ) on
the number line are such that ( boldsymbol{T} boldsymbol{R}= )
( boldsymbol{R S}=boldsymbol{S U} ) and ( boldsymbol{A P}=boldsymbol{P Q}=boldsymbol{Q B} . ) Name
the rational numbers represented by
( boldsymbol{P}, boldsymbol{Q}, boldsymbol{R}, boldsymbol{S} )
9
172State true or false:
If ( a^{x}=b, b^{y}=c ) and ( c^{z}=a, ) then ( x y z= )
1
A. True
B. False
9
173Simplify: ( 3^{0}+2^{-2} )9
174If ( 3^{n-2}=frac{1}{81}, n= )
A .
B.
( c cdot 2 )
D. –
E. -4
9
175Find ( boldsymbol{a}^{3} ) if ( left(frac{2}{mathbf{7}}right)^{boldsymbol{a}}=left(frac{mathbf{1 6}}{mathbf{2 1}}right)^{-mathbf{5}} timesleft(frac{mathbf{3}}{mathbf{8}}right)^{-5} )9
176State the following statement is True or
False

The reciprocal of 0 lie on the real line.
A. True
B. False

9
177Write the following rational numbers in
ascending order:
( frac{-3}{7}, frac{-3}{2}, frac{-3}{4} )
9
178Which number is represented by ( A ), in the following number line?
A . 13
в. ( frac{6}{13} )
c. 6
D. ( frac{0}{13} )
9
17968. If p-S
and q = 3.
then p + q- pq will be equal to
43
9
180What can you say about the prime factorisations of the denominators of
the following rationals:
(i) 43.123456789
(ii) ( 43 . overline{123456789} ) (iii) ( mathbf{2 7} mathbf{.} overline{mathbf{1 4 2} mathbf{8 5 7}} ) (iv) 0.120120012000120000
9
181Represented ( frac{-3}{4} & frac{1}{8} ) on the number line9
182Find the rational numbers ( a ) and ( b )
( frac{sqrt{5}+sqrt{3}}{sqrt{5}-sqrt{3}}=a-sqrt{15} b )
9
183Express in terms of bases to the power of exponenets
( 8.9^{2} )
9
184Simplify:( 2 times(9)^{frac{3}{2}} times(9)^{frac{-1}{2}} )9
185State true or false:
If ( 4^{2 x}=frac{1}{32}, ) then the value of ( x ) is ( -frac{5}{4} )
A. True
B. False
9
186State whether true or false. ( frac{5}{11} ) is an irrational number.
A. True
B. False
9
18701 Lese.
12
The expression 3+ 55+25 is equal to (1980)
(2) 1- √5 + √2 + √10 (6) 1+ √5 + √2-10
o 1+ √5 – √2 + √10 (d) 1- √5-√2+ To
at the correct alternative in each ofthef.11
9
188Represent the following rational numbers on the number line:
( frac{-8}{5} ; frac{3}{8} ; frac{2}{7} ; frac{12}{5} ; frac{45}{13} )
9
189Find the product. ( left(a^{2}right)left(2 a^{22}right)left(4 a^{26}right) )
( mathbf{A} cdot 8 a^{40} )
B. ( 8 a^{50} )
( c cdot 8 a^{30} )
D. ( 8^{20} )
9
190f ( 2^{x+1}=3^{1-x} ) then find the value of ( x )9
191Simplify : ( frac{left(2^{4}right)^{2} times 7^{3}}{8^{2} times 7} )9
192If ( x ) is so small that ( x^{3} ) and higher
powers of ( x ) may be neglected, then ( frac{(1+x)^{3 / 2}-left(1+frac{1}{2} xright)^{3}}{(1-x)^{1 / 2}} ) may be approximated
as
A ( cdot frac{x}{2}-frac{3}{8} x^{2} )
B. ( -frac{3}{8} x^{2} )
c. ( _{3 x+frac{3}{8} x^{2}} )
D. ( 1-frac{3}{8} x^{2} )
9
193Simplify: ( left(-frac{4}{5}right)^{4} )9
194Find the factor which will rationalise:
( 2+sqrt{7} )
9
195Solve: ( 2^{2 x+1}=8 )9
196Simplify: ( frac{left(3^{3}right)^{-2} timesleft(2^{2}right)^{-3}}{left(2^{4}right)^{-2} times 3^{-4} times 4^{-2}} )9
197What fraction lies exactly halfway between ( frac{2}{3} ) and ( frac{3}{4} ? )
A ( cdot frac{3}{5} )
в. ( frac{5}{6} )
c. ( frac{7}{12} )
D. ( frac{9}{16} )
E ( frac{17}{24} )
9
198List five rational numbers between:
( frac{1}{2} ) and ( frac{2}{3} )
9
199Find six rational number between ( frac{1}{2} ) and ( frac{2}{3} )9
200Which of the following is a rational number ( (s) ? )
A ( cdot frac{-2}{9} )
B. ( frac{4}{-7} )
( c cdot frac{-3}{17} )
D. All the three given numbers
9
201Simplify the following using laws of
exponents. ( left(frac{3}{5}right)^{4} timesleft(frac{3}{5}right)^{3} timesleft(frac{3}{5}right)^{8} )
9
202Choose the rational number which does
not lie between rational numbers ( -frac{2}{5} )
and ( -frac{1}{5} )
( A cdot-frac{1}{4} )
B. ( -frac{3}{10} )
c. ( frac{3}{10} )
D. ( -frac{7}{20} )
9
203Find the value of ( x ) for which
( left(frac{3}{4}right)^{6} timesleft(frac{16}{9}right)^{5}=left(frac{4}{3}right)^{x+2} )
9
204Which of the following numbers lies between ( frac{-5}{2} ) and ( frac{3}{4} ? )
A .
B. 0
c. -3
D. 3
9
205Insert three rational of number between
( frac{3}{7} ) and ( frac{4}{7} )
9
206Evaluate:-
( left(frac{2}{7}right)^{2} timesleft(frac{7}{2}right)^{-3} divleft{left(frac{7}{5}right)^{-2}right}^{-4} )
9
207Find one rational number between the
following pairs of rational numbers
(i) ( frac{4}{3} ) and ( frac{2}{5} )
(ii) ( frac{-2}{7} ) and ( frac{5}{6} )
(iii) ( frac{5}{11} ) and ( frac{7}{8} )
(iv) ( frac{7}{4} ) and ( frac{8}{3} )
9
208State true or false:
Every rational number is a whole
number.
A. True
B. False
9
209Is zero a rational number? Can you write it in the form ( frac{p}{q}, ) where ( p ) and ( q ) are integers and ( boldsymbol{q} neq mathbf{0} ? )
A. True
B. False
9
210Can we represent every decimal number
in the form of ( frac{p}{q} ) or not? Explain it.
9
211Write any two rational numbers which
are between 2.3 and 2.4
9
212Write a rational number between ( sqrt{2} )
and ( sqrt{mathbf{3}} )
A ( cdot frac{3}{2} )
B. ( frac{4}{2} )
( c cdot frac{5}{2} )
D. 5
9
213There are – m. rational numbers between
two given rational numbers.
A . 2
B. 5
c. none
D. infinite
9
214are rational numbers between ( -frac{3}{4} ) and ( frac{1}{2} )
A ( cdot frac{-7}{16}, frac{-1}{8}, frac{9}{16} )
в. ( frac{-15}{16}, frac{-1}{8}, frac{3}{16} )
c. ( frac{-7}{16}, frac{-1}{8}, frac{3}{16} )
D. none of the above
9
215A rational number ( -2 / 3 )
A. Lies to the left side of 0 on the number line.
B. Lies to the right side of 0 on the number line.
C. It is not possible to represent on the number line.
D. Cannot be determined on which side the number lies.
9
216Find the following product.
( operatorname{str} times 7 r^{2} s^{2} )
9
217Find any ten rational numbers between
( frac{-5}{6} ) and ( frac{5}{8} )
9
218( left(frac{-1}{2}right)^{2}=frac{1}{2^{m}} )9
21957. A rational number between
is
(1) 2
colo vs OTCO
(2)
9
220Express each of the following exponential expressions as a rational number. ( left(frac{2}{5}right)^{(-3)} )9
221Show that ( 7-sqrt{5} ) is irrational given that ( sqrt{5} ) is irrational.9
222Simplify the following:
( left(frac{1}{4}right)^{-2}-3(8)^{2 / 3}(4)^{0}+left(frac{9}{16}right)^{frac{-1}{2}} )
9
223Are ( frac{-1}{2} ) and ( frac{-3}{6} ) represented by the same point on the number line?9
224If ( left(a b^{-1}right)^{2 x-1}=left(b a^{-1}right)^{x-2}, ) then what is
the value of ( x ? )
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D.
9
22555. The difference of 5.76 and 2.3
(1) 2.54
(3) 3.46
(2) 3.73
(4) 3.43
9
226( left[(13)^{2}+sqrt{1728}-sqrt{441}right] div 4^{2} times 15^{3} )
equals
A ( cdot frac{2}{675} )
B. 33750
( c cdot frac{2}{45} )
D. 485
9
227The rational number lies between ( frac{3}{7} ) and ( frac{2}{3} ) is
A ( cdot frac{2}{5} )
B. ( frac{4}{7} )
( c cdot frac{3}{7} )
D.
9
228Express ( 1 . overline{32}+0 . overline{35} ) in the form of ( frac{p}{q} ) where ( p ) and ( q ) are integers and ( q neq 0 )
Find ( p+q )
9
229State whether true or false.
( frac{2}{7} ) is an irrational number.
A. True
B. False
9
230( (64)^{frac{-2}{3}} timesleft(frac{1}{4}right)^{-3} ) equals to
( A cdot 4 )
B.
( c )
D. 16
9
231Given: ( boldsymbol{x}, boldsymbol{y} ) and ( boldsymbol{z} ) are integers and
( mathbf{3}^{x+5}=mathbf{2 7}^{y+1}, ) then ( boldsymbol{x} ) is even
A. True
B. False
9
232The rational number of the form ( frac{p}{q}, q neq ) ( 0, p ) and ( q ) are positive integers, which represents ( 0.1 overline{34} ) i.e., ( (0.1343434 ldots . .) ) is
A ( cdot frac{134}{999} )
в. ( frac{134}{990} )
c. ( frac{133}{999} )
D. ( frac{133}{990} )
9
233The value of ( left[2-3(2-3)^{-1}right]^{-1} ) is
A ( cdot frac{-1}{5} )
B.
c. -5
D. 5
9
234Find whether the following statement is true or false:
Equivalent rational numbers of a positive rational numbers are all positive.
A. True
B. False
9
235If we divide a positive integer by another positive integer, what is the resulting
number?
A. It is always a natural number
B. It is always an integer
c. It is a rational number
D. It is an irrational number
9
236Find three rational numbers between 5
and -2
9
237Apply laws of exponents and simplify. ( left(3^{0} times 2^{5}right)+5^{0} )9
238Ten rational numbers between ( frac{-2}{5} ) and ( frac{1}{2} )
( operatorname{are} frac{-7}{20}, frac{-6}{20}, frac{-5}{20}, frac{-4}{20}, frac{-3}{20}, frac{-2}{20}, frac{-1}{20}, 0 ldots, frac{1}{20}, frac{2}{20} )
If true then enter 1 and if false then
enter ( mathbf{0} )
9
239Write 0,7,10,-4 in ( frac{p}{q} ) form9
240The sum of two rational numbers is ( frac{-2}{3} ) If one of them is ( frac{-8}{15}, ) find the other.9
241If ( 2=10^{m} ) and ( 3=10^{n} ) then find the
value of 0.15
A ( cdot 10^{n-m-1} )
B . ( 10^{n-m+1} )
c. ( 10^{n-m-2} )
D. ( 10^{n-m}-m )
9
242Represent these number on number line: ( frac{7}{8}, frac{-5}{3} )9
243Evaluate ( 3 times(9)^{5 / 2} times(9)^{-1 / 2} )9
244State whether the statement is True or
False.
( (boldsymbol{m}+boldsymbol{n})^{-1}left(boldsymbol{m}^{-1}+boldsymbol{n}^{-1}right)=(boldsymbol{m} boldsymbol{n})^{-1} )
A. True
B. False
9
245Find the value of ‘ ( n ) ‘ in the following:
( left(frac{2}{3}right)^{3} timesleft(frac{2}{3}right)^{5}=left(frac{2}{3}right)^{n-2} )
9
246Solve:3 ( x+8>2 ) when ( x ) is an real
number.
9
247If ( boldsymbol{x}=mathbf{2} ) and ( boldsymbol{y}=mathbf{3}, ) then find the value of ( left[frac{1}{x^{x}}+frac{1}{y^{y}}right] )
A ( cdot frac{-31}{108} )
в. ( frac{31}{108} )
c. ( frac{125}{171} )
D. ( frac{153}{222} )
9
248Does the following pairs represent the same rational number:
( frac{-16}{20} ) and ( frac{20}{-25} )
9
24953. The value of
32 43
13+V6 T6+T2 T3+ Te is
(1) 4 (2) 0
(3) 2 0 (4) 3 76
9
25067. If x= + va and y = = =
1
then value of x +1
be
1
y +1 Will
9

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