R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 1  Number Systems
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.
Solve .
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: .
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 ()
Simplify
If 10^{x} = 64, find the value of .
Evaluate
Simplify .
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.
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.
Two rational numbers between are
(a)
(b)
(c)
(d)
Correct option: (c)
Two rational numbers between
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.
Which of the following is a rational number?
Every rational number is
(a) a natural number
(b) a whole number
(c) an integer
(d)a real number
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
The decimal representation of a rational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or nonrepeating
(d)neither terminating nor repeating
The decimal representation of an irrational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or nonrepeating
(d)neither terminating nor repeating
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 nonterminating recurring.
The decimal expansion of 0.5030030003…. is nonterminating nonrecurring, 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 nonterminating nonrecurring.
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 nonterminating nonrecurring.
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 nonterminating, nonrecurring.
Which of the following is an irrational number?
(a)
(b)
(c) 0.3799
(d)
Correct option: (a)
The decimal expansion of , which is nonterminating, nonrecurring.
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 nonterminating, nonrecurring.
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
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 nonterminating 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 nonzero 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)
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)
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
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^{pq}⋅ x^{q  r}⋅ x^{r  p} is equal to
(a) 0
(b) 1
(c) x
(d) x^{pqr}
Correct option: (b)
x^{pq}⋅ 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)
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)
(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 rational numbers p and q is . 
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 integer which is not a perfect square is an irrational number. 
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 irrational number. 
Π is an irrational number. 
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 an irrational number is an irrational number. 
The correct answer is: (a)/(b)/(c)/(d).
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)…….,
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 nonterminating 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 nonterminating recurring decimal expansion.
Write each of the following in decimal form and say what kind of decimal expansion each has.
Hence, it has nonterminating recurring decimal expansion.
Write each of the following in decimal form and say what kind of decimal expansion each has.
Hence, it has nonterminating 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 nonterminating 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 = 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 = 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 = 0.40777..…
⇒ 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.
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 you answer.
is the product of a rational number and an irrational number .
Theorem: The product of a nonzero 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.
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 your answer.
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 nonterminating 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 nonterminating and nonrecurring decimal expansion.
Hence, it is an irrational number.
Classify the following numbers as rational or irrational. Give reasons to support your answer.
3.040040004.....
The given number 3.040040004..... has nonterminating and nonrecurring 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 nonterminating 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 nonterminating 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…….
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.
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
State in each case, whether the given statement is true or false.
_{}is irrational and _{}is rational.
True
Chapter 1  Number Systems Exercise Ex. 1D
Add:
_{}
_{}
We have:
_{}
Add:
_{}
_{}
We have:
_{}
Add:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
_{}
Multiply:
Divide:
_{}
_{}
_{}
Divide:
_{}
_{}
Divide:
_{}
_{}
_{}
Simplify:
_{}
_{}
Simplify:
_{}
_{}
_{}
Simplify:
_{}
_{}
_{}
Simplify
= 9  11
= 2
Simplify
= 9  5
= 4
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. 1G
Simplify:
Simplify
Simplify
Simplify
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
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.
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. 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 cutoff 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.
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 cutoff 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 cutoff 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 cutoff 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 cutoff 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.
Visualize the representation of on the number line up to 4 decimal places.
Chapter 1  Number Systems Exercise Ex. 1F
Write the rationalising factor of the denominator in .
The rationalising factor of the denominator in is
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 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
Simplify
Thus, the given number is rational.
*Question modified
.
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
Rationalise the denominator of each of the following.
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