R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 1 – Number Systems
Chapter 1 – Number Systems Exercise Ex. 1A
Is zero a rational number? Justify.
A
number which can be expressed as , where ‘a’ and ‘b’ both are integers and b ≠ 0, is called
a rational number.
Since,
0 can be expressed as , it is a rational number.
Represent each of the following rational numbers on the number line:
(i)
(i)
Represent each of the following rational numbers on the number line:
(ii)
(ii)
Represent each of the following rational numbers on the
number line:
Represent each of the following rational numbers on the number line:
(iv) 1.3
(iv) 1.3
Represent each of the following rational numbers on the number line:
(v) -2.4
(v) -2.4
Find a rational number lying between
Find a rational number lying between
1.3 and 1.4
Find a rational number lying between
-1 and
Find a rational number lying between
Find a rational number between
Find three rational numbers lying between
How many rational numbers can be determined between these
two numbers?
Infinite rational
numbers can be determined between given two rational numbers.
Find four rational numbers between
We have
We know that 9 < 10 < 11 < 12 < 13 <
14 < 15
Find six rational numbers between 2 and 3.
2 and 3 can be
represented asrespectively.
Now six rational numbers
between 2 and 3 are
.
Find five rational numbers between
Insert 16 rational numbers between 2.1 and 2.2.
Let x = 2.1 and y = 2.2
Then, x < y because 2.1 < 2.2
Or we can say that,
Or,
That is, we have,
We know that,
Therefore, we can have,
Therefore, 16 rational numbers between, 2.1 and 2.2 are:
So, 16 rational numbers between 2.1 and 2.2 are:
2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175, 2.18
State whether the given statement is true or false. Give reasons. for your answer.
Every natural number is a whole number.
True. Since the collection of natural number is a sub collection of whole numbers, and every element of natural numbers is an element of whole numbers
Write, whether the given statement is true or false. Give reasons.
Every whole number is a natural number.
False. Since 0 is whole number but it is not a natural number.
State whether the following statements are true or false.
Give reasons for your answer.
Every integer is a whole number.
False,
integers include negative of natural numbers as well, which are clearly not
whole numbers. For example -1 is an integer but not a whole number.
Write, whether the given statement is true or false. Give reasons.
Ever integer is a rational number.
True. Every integer can be represented in a fraction form with denominator 1.
State whether the following statements are true or false.
Give reasons for your answer.
Every rational number is an integer.
False,
integers are counting numbers on both sides of the number line i.e. they are
both positive and negative while rational numbers are of the form . Hence, Every rational number is not an integer but every
integer is a rational number.
Write, whether the given statement is true or false. Give reasons.
Every rational number is a whole number.
False. Since division of whole numbers is not closed under division, the value of , may not be a whole number.
Chapter 1 – Number Systems Exercise Ex. 1B
Without actual division, find which of the following rationals are terminating decimals.
_{}
If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.
_{Since, 80 has prime factors 2 and 5, }_{ }is a terminating decimal.
Without actual division, find which of the following rationals are terminating decimals.
_{}
_{}
If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.
Since, 24 has prime factors 2 and 3 and 3 is different from 2 and 5,
_{ }_{}is not a terminating decimal.
Without actual division, find which of the following rationals are terminating decimals.
_{}
If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.
Since 12 has prime factors 2 and 3 and 3 is different from 2 and 5,
_{ }is not a terminating decimal.
Without actual division, find which of the following
rational numbers are terminating decimals.
Since the denominator of a given rational number is
not of the form 2^{m} × 2^{n}, where m and n are whole
numbers, it has non-terminating decimal.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
terminating decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion each
has.
Hence, it has
terminating decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has non-terminating
recurring decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
non-terminating recurring decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
non-terminating recurring decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
terminating decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
terminating decimal expansion.
Write
each of the following in decimal form and say what kind of decimal expansion
each has.
Hence, it has
non-terminating recurring decimal expansion.
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.2222…. ….(i)
⇒ 10x =
2.2222…. ….(ii)
On subtracting (i) from (ii), we get
9x = 2
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.5353…. ….(i)
⇒ 100x =
53.535353…. ….(ii)
On subtracting (i) from (ii), we get
99x = 53
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 2.9393…. ….(i)
⇒ 100x =
293.939……. ….(ii)
On subtracting (i) from (ii), we get
99x = 291
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 18.4848…. ….(i)
⇒ 100x =
1848.4848……. ….(ii)
On subtracting (i) from (ii), we get
99x = 1830
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 32.123535..…
⇒ 100x =
3212.3535…… ….(i)
⇒ 10000x =
321235.3535……. ….(ii)
On subtracting (i) from (ii), we get
9900x = 318023
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.235235..… ….(i)
⇒ 1000x =
235.235235……. ….(ii)
On subtracting (i) from (ii), we get
999x = 235
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.003232..…
⇒ 100x =
0.323232……. ….(i)
⇒ 10000x =
32.3232…. ….(ii)
On subtracting (i) from (ii), we get
9900x = 32
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 1.3232323..… ….(i)
⇒ 100x =
132.323232……. ….(ii)
On subtracting (i) from (ii), we get
99x = 131
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.3178178..…
⇒ 10x =
3.178178…… ….(i)
⇒ 10000x =
3178.178……. ….(ii)
On subtracting (i) from (ii), we get
9990x = 3175
Express each of the following decimals in the form , where p, q are integers and q ≠
0.
Let x =
i.e. x = 0.40777..…
⇒ 100x =
40.777…… ….(i)
⇒ 1000x =
407.777……. ….(ii)
On subtracting (i) from (ii), we get
900x = 367
Express as a fraction in simplest form.
Let x =
i.e. x = 2.3636…. ….(i)
⇒ 100x =
236.3636……. ….(ii)
On subtracting (i) from (ii), we get
99x = 234
Let y =
i.e. y = 0.2323…. ….(iii)
⇒ 100y =
23.2323…. ….(iv)
On subtracting (iii) from (iv), we get
99y = 23
Express in the form of
Let x =
i.e. x = 0.3838…. ….(i)
⇒ 100x = 38.3838…. ….(ii)
On subtracting (i) from (ii), we get
99x = 38
Let y =
i.e. y = 1.2727…. ….(iii)
⇒ 100y = 127.2727……. ….(iv)
On subtracting (iii) from (iv), we get
99y = 126
Without actual division, find which of the following rationals are terminating decimals.
_{}
If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.
Since 125 has prime factor 5 only
is a terminating decimal.
Chapter 1 – Number Systems Exercise Ex. 1C
What are irrational numbers? How do they differ from rational numbers? Give examples.
Irrational number: A number which cannot be expressed either as a terminating decimal or a repeating decimal is known as irrational number. Rather irrational numbers cannot be expressed in the fraction form,
For example, 0.101001000100001 is neither a terminating nor a repeating decimal and so is an irrational number.
Also, etc. are examples of irrational numbers.
Find two rational and two irrational numbers between 0.5
and 0.55.
Since 0.5 < 0.55
Let x = 0.5, y = 0.55
and y = 2
Two irrational numbers
between 0.5 and 0.55 are 0.5151151115……. and 0.5353553555….
Find three different irrational numbers between the
rational numbers .
Thus, three different irrational
numbers between the rational numbers are as follows:
0.727227222…..,
0.757557555….. and 0.808008000…..
Find two rational numbers of the form between the numbers 0.2121121112… and 0.2020020002……
Let a and
b be two rational numbers between the numbers 0.2121121112… and 0.2020020002……
Now, 0.2020020002…… <0.2121121112…
Then, 0.2020020002…… < a
< b < 0.2121121112…
Find two irrational numbers between 0.16 and 0.17.
Two irrational numbers between
0.16 and 0.17 are as follows:
0.1611161111611111611111…… and
0.169669666…….
State in each case, whether the given statement is true or false.
The sum of two rational numbers is rational.
True
State in each case, whether the given statement is true or false.
The sum of two irrational numbers is irrational.
False
State in each case, whether the given statement is true or false.
The product of two rational numbers is rational.
True
State in each case, whether the given statement is true or false.
The product of two irrational numbers is irrational.
False
State in each case, whether the given statement is true or false.
_{}is irrational and _{}is rational.
True
State in each case, whether the given statement is true or false.
The sum of a rational number and an irrational number is irrational.
True
State in each case, whether the given statement is true or false.
The product of a nonzero rational number and an irrational number is a rational number.
False
State in each case, whether the given statement is true or false.
Every real number is rational.
False
State in each case, whether the given statement is true or false.
Every real number is either rational or irrational.
True
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
Since quotient of a rational and an irrational is
irrational, the given number is irrational.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
Classify the following numbers as rational or irrational. Give reasons to support you answer.
_{}
_{}
We know that, if n is a not a perfect square, then is an irrational number.
Here, is a not a perfect square number.
So, _{}is irrational.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
3.040040004…..
The given number 3.040040004….. has non-terminating and non-recurring decimal expansion.
Hence, it is an irrational number.
Classify the following numbers as rational or irrational. Give reasons to support you answer.
is the product of a rational number and an irrational number .
Theorem: The product of a non-zero rational number and an irrational number _{is an irrational number}.
Thus, by the above theorem, is an irrational number.
So, is an irrational number.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
4.1276
The given number 4.1276 has terminating
decimal expansion.
Hence, it is a rational number.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
Since the given number has non-terminating recurring
decimal expansion, it is a rational number.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
1.232332333….
The given number 1.232332333…. has non-terminating and non-recurring decimal expansion.
Hence, it is an irrational number.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
2.356565656…..
The given number 2.356565656….. has non-terminating recurring decimal expansion.
Hence, it is a rational number.
Classify the following numbers as rational or irrational.
Give reasons to support your answer.
6.834834….
The given number 6.834834…. has non-terminating recurring decimal expansion.
Hence, it is a rational number.
Let x be a rational number and y be an irrational number.
Is x + y necessarily an irrational number? Give an example in support of your
answer.
We
know that the sum of a rational and an irrational is irrational.
Hence,
if x is rational and y is irrational, then x + y is necessarily an irrational
number.
For
example,
Let a be a rational number and b
be an irrational number. Is ab necessarily an
irrational number? Justify your answer with an example.
We
know that the product of a rational and an irrational is irrational.
Hence,
if a is rational and b is irrational, then ab is necessarily an irrational number.
For
example,
Is the product of two irrationals always irrational?
Justify your answer.
No, the product of two irrationals
need not be an irrational.
For example,
Give an example of two irrational numbers whose
(i) difference is an irrational number.
(ii) difference is a rational number.
(iii) sum is an irrational number.
(iv) sum is an rational number.
(v) product is an irrational number.
(vi) product is a rational number.
(vii) quotient is an irrational number.
(viii) quotient is a rational number.
(i) Difference is an irrational number:
(ii) Difference is a rational number:
(iii) Sum is an irrational number:
(iv) Sum is an rational number:
(v) Product is an irrational number:
(vi) Product is a rational number:
(vii) Quotient is an irrational number:
(viii) Quotient is a rational number:
Examine whether the following numbers are rational or
irrational.
Insert a rational and an irrational number between 2 and
2.5
Rational
number between 2 and 2.5 =
Irrational
number between 2 and 2.5 =
How many irrational numbers lie between? Find any three irrational numbers lying between .
There are infinite irrational
numbers between.
We have
Hence, three irrational numbers lying between are as follows:
1.5010010001……., 1.6010010001…… and 1.7010010001…….
Chapter 1 – Number Systems Exercise Ex. 1D
Add:
_{}
_{}
We have:
_{}
Add:
_{}
_{}
We have:
_{}
Add:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
Divide:
_{}
_{}
_{}
Divide:
_{}
_{}
Divide:
_{}
_{}
_{}
Simplify
= 9 – 11
= -2
Simplify
= 9 – 5
= 4
Simplify:
_{}
_{}
Simplify:
_{}
_{}
_{}
Simplify
Simplify:
_{}
_{}
_{}
Simplify
Examine whether the following numbers are rational or
irrational:
Thus, the given number is rational.
Examine whether the following numbers are rational or
irrational:
Clearly, the given number is irrational.
Examine whether the following numbers are rational or
irrational:
Thus, the given number is rational.
Examine whether the following numbers are rational or
irrational:
Thus, the given number is irrational.
On her birthday Reema distributed chocolates in an orphanage. The total
number of chocolates she distributed is given by .
(i) Find the number of chocolates distributed by her.
(ii) Write the moral values depicted here by Reema.
(i) Number of chocolates distributed
by Reema
(ii) Loving, helping and caring
attitude towards poor and needy children.
Simplify
Simplify
Simplify
Chapter 1 – Number Systems Exercise Ex. 1E
Represent on the number line.
Draw a number line as
shown.
On the number line, take
point O corresponding to zero.
Now take point A on number
line such that OA = 2 units.
Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB^{2} = OA^{2}
+ AB^{2} = 2^{2} + 1^{2 }= 4 + 1 = 5
⇒ OB =
Taking O as centre and OB = as radius draw an
arc cutting real line at C.
Clearly, OC = OB =
Hence, C represents on the number line.
Visualize the representation of on the number line
up to 4 decimal places.
Locate on the number line.
Draw a number line as
shown.
On the number line, take
point O corresponding to zero.
Now take point A on number
line such that OA = 1 unit.
Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB^{2} = OA^{2}
+ AB^{2} = 1^{2} + 1^{2 }= 1 + 1 = 2
⇒ OB =
Taking O as centre and OB = as radius draw an
arc cutting real line at C.
Clearly, OC = OB =
Thus, C represents on the number line.
Now, draw perpendicular
CY at C on the number line and cut-off arc CE = 1 unit.
By Pythagoras Theorem,
OE^{2} = OC^{2}
+ CE^{2} = ^{2} + 1^{2 }= 2 + 1 = 3
⇒ OE =
Taking O as centre and OE = as radius draw an
arc cutting real line at D.
Clearly, OD = OE =
Hence, D represents on the number line.
Locate on the number line.
Draw a number line as
shown.
On the number line, take
point O corresponding to zero.
Now take point A on number
line such that OA = 3 units.
Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB^{2} = OA^{2}
+ AB^{2} = 3^{2} + 1^{2 }= 9 + 1 = 10
⇒ OB =
Taking O as centre and OB = as radius draw an
arc cutting real line at C.
Clearly, OC = OB =
Hence,
C represents on the number line.
Locate on the number line.
Draw a number line as
shown.
On the number line, take
point O corresponding to zero.
Now take point A on number
line such that OA = 2 units.
Draw perpendicular AZ at
A on the number line and cut-off arc AB = 2 units.
By Pythagoras Theorem,
OB^{2} = OA^{2}
+ AB^{2} = 2^{2} + 2^{2 }= 4 + 4 = 8
⇒ OB =
Taking O as centre and OB = as radius draw an
arc cutting real line at C.
Clearly, OC = OB =
Hence,
C represents on the number line.
Represent geometrically on the
number line.
Draw
a line segment AB = 4.7 units and extend it to C such that BC = 1 unit.
Find
the midpoint O of AC.
With
O as centre and OA as radius, draw a semicircle.
Now,
draw BD ⊥ AC, intersecting the semicircle at
D.
Then,
BD = units.
With
B as centre and BD as radius, draw an arc, meeting
AC produced at E.
Then, BE = BD = units.
Represent on the number line.
Draw
a line segment OB = 10.5 units and extend it to C such that BC = 1 unit.
Find
the midpoint D of OC.
With
D as centre and DO as radius, draw a semicircle.
Now,
draw BE ⊥ AC, intersecting the semicircle at
E.
Then,
BE = units.
With
B as centre and BE as radius, draw an arc, meeting
AC produced at F.
Then, BF = BE = units.
Represent _{}geometrically on the number line.
Draw a line segment AB = 7.28 units and extend it to C such that BC = 1 unit.
Find the midpoint O of AC.
With O as centre and OA as radius, draw a semicircle.
Now, draw BD _{}AC, intersecting the semicircle at D.
Then, BD = _{}units.
With D as centre and BD as radius, draw an arc, meeting AC produced at E.
Then, BE = BD = _{}units.
Represent on the number line.
Draw
a line segment OB = 9.5 units and extend it to C such that BC = 1 unit.
Find
the midpoint D of OC.
With
D as centre and DO as radius, draw a semicircle.
Now,
draw BE ⊥ AC, intersecting the semicircle
at E.
Then,
BE = units.
With
B as centre and BE as radius, draw an arc, meeting
AC produced at F.
Then,
BF = BE = units.
Extend
BF to G such that FG = 1 unit.
Then,
BG =
Visualize the representation of
3.765 on the number line using successive magnification.
Chapter 1 – Number Systems Exercise Ex. 1F
Write the rationalising factor of the denominator in .
The rationalising factor of the denominator in is
Simplify
Thus, the given number is rational.
*Question
modified
Rationalise the denominator of following:
_{}
On multiplying the numerator and denominator of the given number by , we get
Rationalise the denominator of following:
_{}
On multiplying the numerator and denominator of the given number by , we get
Rationalise the denominator of following:
_{}
Rationalise the denominator of following:
_{}
Rationalise the denominator of each of the following.
Rationalise the denominator of following:
_{}
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
.
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
find
the value to three places of decimals, of each of the following.
find
the value to three places of decimals, of each of the following.
find
the value to three places of decimals, of each of the following.
Find rational numbers a and b such that
Find rational numbers a and b such that
Find rational numbers a and b such that
Find rational numbers a and b such that
find
to three places of decimals, the value of each of the following.
find
to three places of decimals, the value of each of the following.
find
to three places of decimals, the value of each of the following.
find
to three places of decimals, the value of each of the following.
find
to three places of decimals, the value of each of the following.
find
to three places of decimals, the value of each of the following.
Simplify by rationalising the
denominator.
Simplify by rationalising the
denominator.
Simplify: _{}
Simplify
Simplify
Simplify
Prove that
Prove that
Find the values of a and b if
*Back answer incorrect
Chapter 1 – Number Systems Exercise Ex. 1G
Simplify
Simplify
Simplify:
Simplify
Prove that
Prove that
Prove that
Simplify and express the result in the exponential form of x.
Simplify the product
Simplify
Simplify
Simplify
Find the value of x in each of the following.
Find the value of x in each of the following.
Find the value of x in each of the following.
Find the value of x in each of the following.
5^{x – 3}× 3^{2x – 8} = 225
5^{x – 3} × 3^{2x – 8}
= 225
⇒ 5^{x – 3}× 3^{2x – 8} = 5^{2} × 3^{2}
⇒ x – 3 = 2 and 2x – 8 = 2
⇒ x = 5 and 2x = 10
⇒ x = 5
Find the value of x in each of the following.
Prove that
Prove that
Prove that
Prove that
If x is a positive real number and exponents are rational
numbers, simplify
If prove that m – n = 1.
Write the following in ascending order of magnitude.
Simplify:
_{}
Simplify:
Simplify:
_{}
Simplify:
_{}
Simplify:
Simplify:
Simplify:
_{}
Simplify:
Simplify:
Evaluate:
Evaluate:
Evaluate:
_{}
Evaluate:
Evaluate:
_{}
Evaluate:
If a = 2, b = 3, find the value of (a^{b}
+ b^{a})^{-1}
Given,
a = 2 and b = 3
If a = 2, b = 3, find the value of (a^{a}
+ b^{b})^{-1}
Given,
a = 2 and b = 3
Simplify
Simplify
(14641)^{0.25}
(14641)^{0.25}
Simplify
Simplify
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Evaluate
Chapter 1 – Number Systems Exercise MCQ
Which of the following is a rational number?
(a)
(b) π
(c)
(d) 0
Correct
option: (d)
0
can be written as where p and q are
integers and q ≠ 0.
The decimal expansion that a rational number cannot have
is
(a) 0.25
(b)
(c)
(d) 0.5030030003….
Correct
option: (d)
The decimal expansion of a rational number is either
terminating or non-terminating recurring.
The decimal expansion of 0.5030030003…. is non-terminating
non-recurring, which is not a property of a rational number.
Which of the following is an irrational number?
(a) 3.14
(b) 3.141414….
(c) 3.14444…..
(d) 3.141141114….
Correct
option: (d)
The decimal expansion of an irrational number is
non-terminating non-recurring.
Hence, 3.141141114….. is an
irrational number.
A rational number equivalent to is
(a)
(b)
(c)
(d)
Correct
option: (d)
Choose the rational number which does not lie between
(a)
(b)
(c)
(d)
Correct
option: (b)
Given
two rational numbers are negative and is a positive
rational number.
So,
it does not lie between
Π is
(a) a
rational number
(b) an
integer
(c) an
irrational number
(d) a
whole number
Correct
option: (c)
Π = 3.14159265359…….., which is
non-terminating non-recurring.
Hence,
it is an irrational number.
The decimal expansion of is
(a) finite
decimal
(b) 1.4121
(c) nonterminating
recurring
(d) nonterminating, nonrecurring
Correct
option: (d)
The decimal expansion of , which is non-terminating, non-recurring.
Which of the following is an irrational number?
(a)
(b)
(c) 0.3799
(d)
Correct
option: (a)
The decimal expansion of , which is non-terminating, non-recurring.
Hence, it is an irrational number.
Hoe many digits are there in the repeating block of digits
in the decimal expansion of
(a) 16
(b) 6
(c) 26
(d) 7
Correct
option: (b)
Which of the following numbers is irrational?
(a)
(b)
(c)
(d)
Correct
option: (c)
The decimal expansion of , which is non-terminating, non-recurring.
Hence, it is an irrational number.
The product of two irrational numbers is
(a) always
irrational
(b) always rational
(c) always an
integer
(d)sometimes
rational and sometimes irrational
A rational number between -3 and 3 is
(a) 0
(b) -4.3
(c) -3.4
(d) 1.101100110001….
Correct
option: (a)
On
a number line, 0 is a rational number that lies between -3 and 3.
Which of the following is a true
statement?
(a) The sum of two
irrational numbers is an irrational number
(b) The product of
two irrational numbers is an irrational number
(c) Every real
number is always rational
(d) Every real
number is either rational or irrational
Which of the following is a true
statement?
(a)
(b)
(c)
(d)
A rational number lying between is
(a)
(b)
(c) 1.6
(d) 1.9
Correct option: (c)
Which of the following is a rational number?
(a)
(b) 0.101001000100001…
(c) π
(d) 0.853853853…
Correct
option: (d)
The decimal expansion of a rational number is either
terminating or non-terminating recurring.
Hence, 0.853853853… is a
rational number.
The product of a nonzero rational number with an
irrational number is always a/an
(a) irrational
number
(b) rational
number
(c) whole
number
(d) natural
number
Correct option: (a)
The product of a non-zero rational number with an
irrational number is always an irrational number.
The value of , where p and q are integers and q ≠
0, is
(a)
(b)
(c)
(d)
Correct
option: (b)
Two rational numbers between are
(a)
(b)
(c)
(d)
Correct
option: (c)
Two
rational numbers between
An irrational number
between 5 and 6 is
The sum of
(a)
(b)
(c)
(d)
Correct option: (b)
Let x =
i.e. x = 0.3333…. ….(i)
⇒ 10x =
3.3333…. ….(ii)
On subtracting (i) from (ii), we get
9x = 3
Let y =
i.e. y = 0.4444…. ….(i)
⇒ 10y =
4.4444…. ….(ii)
On subtracting (i) from (ii), we get
9y = 4
The value of
(a)
(b)
(c)
(d)
Correct option: (c)
Let x =
i.e. x = 2.4545…. ….(i)
⇒ 100x = 245.4545……. ….(ii)
On subtracting (i) from (ii), we get
99x = 243
Let y =
i.e. y = 0.3636…. ….(iii)
⇒ 100y = 36.3636…. ….(iv)
On subtracting (iii) from (iv), we get
99y = 36
Which of the following is the value of ?
(a) -4
(b) 4
(c)
(d)
Correct
option: (b)
when simplified is
(a) positive
and irrational
(b) positive
and rational
(c) negative
and irrational
(d) negative
and rational
Correct
option: (b)
Which is positive and rational number.
when simplified is
(a) positive
and irrational
(b) positive
and rational
(c) negative
and irrational
(d) negative
and rational
Correct
option: (b)
Which is positive and rational number.
When is divided by , the quotient is
(a)
(b)
(c)
(d)
Correct
option: (c)
The value of is
(a) 10
(b)
(c)
(d)
Correct
option: (a)
Every point on number line represents
(a) a
rational number
(b) a
natural number
(c) an
irrational number
(d) a
unique number
Correct
option: (d)
Every point on number line represents a unique number.
The value of is
(a)
(b)
(c)
(d)
Correct
option: (b)
= ?
(a)
(b)
(c)
(d) None
of these
Correct
option: (b)
=?
(a)
(b) 2
(c) 4
(d) 8
Correct
option: (b)
(125)^{-1/3} = ?
(a) 5
(b) -5
(c)
(d)
Correct
option: (c)
The value of 7^{1/2}⋅
8^{1/2} is
(a) (28)^{1/2}
(b) (56)^{1/2}
(c) (14)^{1/2}
(d) (42)^{1/2}
Correct
option: (b)
After simplification, is
(a) 13^{2/15}
(b) 13^{8/15}
(c) 13^{1/3}
(d) 13^{-2/15}
Correct
option: (d)
The value of is
(a)
(b)
(c) 8
(d)
Correct
option: (a)
The value of is
(a) 0
(b) 2
(c)
(d)
Correct
option: (b)
The value of (243)^{1/5} is
(a) 3
(b) -3
(c) 5
(d)
Correct
option: (a)
9^{3}
+ (-3)^{3} – 6^{3} = ?
(a) 432
(b) 270
(c) 486
(d) 540
Correct
option: (c)
9^{3} + (-3)^{3} –
6^{3} = 729 – 27 – 216 = 486
Which of the following is a rational
number?
Simplified value of is
(a) 0
(b) 1
(c) 4
(d) 16
Correct
option: (b)
The value of is
(a) 2^{-1/6}
(b) 2^{-6}
(c) 2^{1/6}
(d) 2^{6}
Correct
option: (c)
Simplified value of (25)^{1/3}×
5^{1/3} is
(a) 25
(b) 3
(c) 1
(d) 5
Correct
option: (d)
The value of is
(a) 3
(b) -3
(c) 9
(d)
Correct
option: (a)
There is a number x such that x^{2} is irrational
but x^{4} is rational. Then, x can be
(a)
(b)
(c)
(d)
Correct
option: (d)
If then value of p is
(a)
(b)
(c)
(d)
Correct
option: (b)
The value of is
(a)
(b)
(c)
(d)
Correct
option: (b)
The value of x^{p-q}⋅
x^{q – r}⋅ x^{r – p} is equal to
(a) 0
(b) 1
(c) x
(d) x^{pqr}
Correct
option: (b)
x^{p-q}⋅
x^{q – r}⋅ x^{r – p}
= x^{p – q + q – r + r – p}
= x^{0}
= 1
The value of is
(a) -1
(b) 0
(c) 1
(d) 2
Correct
option: (c)
= ?
(a) 2
(b)
(c)
(d)
Correct
option: (a)
Every rational number is
(a) a natural
number
(b) a whole number
(c) an integer
(d)a real number
If then x = ?
(a) 1
(b) 2
(c) 3
(d) 4
Correct
option: (d)
If (3^{3})^{2} = 9^{x} then 5^{x}
= ?
(a) 1
(b) 5
(c) 25
(d) 125
Correct
option: (d)
(3^{3})^{2} = 9^{x}
⇒
(3^{2})^{3} = (3^{2})^{x}
⇒
x = 3
Then 5^{x} = 5^{3}
= 125
On simplification, the expression equals
(a)
(b)
(c)
(d)
Correct
option: (b)
The simplest rationalisation factor of is
(a)
(b)
(c)
(d)
Correct
option: (d)
Thus, the simplest rationalisation factr of
The simplest rationalisation factor of is
(a)
(b)
(c)
(d)
Correct
option: (b)
The simplest rationalisation factor of is
The rationalisation factor of is
(a)
(b)
(c)
(d)
Correct
option: (d)
Rationalisation of the denominator of gives
(a)
(b)
(c)
(d)
Correct
option: (d)
(a)
(b) 2
(c) 4
(d)
Correct
option: (c)
(a)
(b)
(c)
(d) None
of these
Correct
option: (c)
(a)
(b) 14
(c) 49
(d) 48
Correct option: (b)
Between any two rational numbers there
(a) is no rational
number
(b) is exactly one
rational number
(c) are infinitely
many rational numbers
(d)is no
irrational number
(a) 0.075
(b) 0.75
(c) 0.705
(d) 7.05
Correct
option: (c)
(a) 0.375
(b) 0.378
(c) 0.441
(d) None
of these
Correct
option: (b)
The value of is
(a)
(b)
(c)
(d)
Correct
option: (d)
The value of is
(a)
(b)
(c)
(d)
Correct
option: (c)
(a) 0.207
(b) 2.414
(c) 0.414
(d) 0.621
Correct
option: (c)
= ?
(a) 34
(b) 56
(c) 28
(d) 63
Correct
option: (a)
Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:
(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).
(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).
(c) Assertion (A)
is true and Reason (R) is false.
(d) Assertion (A)
is false and Reason (R) is true.
Assertion (A) | Reason (R) |
A rational number between two |
The correct answer is:
(a)/(b)/(c)/(d).
Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:
(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).
(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).
(c) Assertion (A)
is true and Reason (R) is false.
(d) Assertion (A)
is false and Reason (R) is true.
Assertion (A) | Reason (R) |
Square root of a positive |
The correct answer is:
(a)/(b)/(c)/(d).
Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:
(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).
(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).
(c) Assertion (A)
is true and Reason (R) is false.
(d) Assertion (A)
is false and Reason (R) is true.
Assertion (A) | Reason (R) |
e is an | Π is an |
The correct answer is:
(a)/(b)/(c)/(d).
Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:
(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).
(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).
(c) Assertion (A)
is true and Reason (R) is false.
(d) Assertion (A)
is false and Reason (R) is true.
Assertion (A) | Reason (R) |
The sum of a rational number and |
The correct answer is:
(a)/(b)/(c)/(d).
The decimal representation of a rational
number is
(a) always
terminating
(b) either
terminating or repeating
(c) either
terminating or non-repeating
(d)neither
terminating nor repeating
Match the following columns:
Column I | Column II |
(p) 14 (q) 6 (r) a rational number (s) an irrational number |
The correct answer is:
(a)-…….,
(b)-…….,
(c)-…….,
(d)-…….,
Match the following columns:
Column I | Column II |
The correct answer is:
(a)-…….,
(b)-…….,
(c)-…….,
(d)-…….,
The decimal representation of an
irrational number is
(a) always
terminating
(b) either
terminating or repeating
(c) either terminating
or non-repeating
(d)neither
terminating nor repeating
Chapter 1 – Number Systems Exercise VSAQ
What can you say about the sum of a rational number and an
irrational number?
The sum of a rational number and
an irrational number is irrational.
Example: 5 + is irrational.
Simplify (32)^{1/5} + (-7)^{0} + (64)^{1/2}.
Evaluate .
Simplify .
If a = 1, b = 2 then find the value of (a^{b} + b^{a})^{-1}.
Given,
a = 1 and b = 2
Simplify .
Give an example of two irrational numbers whose sum as
well as product is rational.
Is the product of a rational and irrational numbers always
irrational? Give an example.
Yes, the product of a rational and
irrational numbers is always irrational.
For
example,
Give an example of a number x such that x^{2} is
an irrational number and x^{3} is a rational number.
Write the reciprocal of ().
The reciprocal of ()
Solve .
Simplify
If 10^{x} = 64, find the value of .
Evaluate
Simplify .
The number will terminate after
how many decimal places?
Thus, the given number
will terminate after 3 decimal places.
Find the value of (1296)^{0.17}× (1296)^{0.08}.
(1296)^{0.17}× (1296)^{0.08}
Simplify .
Find an irrational number between 5 and 6.
An irrational number between 5 and
6 =
Find the value of .
Rationalise
Solve for x: .