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Chapters
▶ 1: Rational and Irrational Numbers
UNIT-II: COMMERCIAL MATHEMATICS
2: Compound Interest
UNIT-III: ALGEBRA
3: Expansions
4: Factorisation
5: Simultaneous Linear Equations
6: Indices
7: Logarithms
UNIT-IV: GEOMETRY
8: Triangles
9: Inequalities
10: Mid-point Theorem
11: Pythagoras Theorem
12: Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)
13: Theorems on Area
14: Circles (Chord and Arc Properties)
UNIT-V: STATISTICS
15: Statistics
16: Graphical Representation of Statistical Data
UNIT-VI: MENSURATION
17: Mensuration
18: Surface Area and Volume of Solids
UNIT-VII: TRIGONOMETRY
19: Trigonometry
20: Simple 2-D Problems in Right Triangle
UNIT-VIII: COORDINATE GEOMETRY
21: Coordinate Geometry
![B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 1 - Rational and Irrational Numbers - Shaalaa.com](/images/mathematics-english-class-9-icse_6:a927b361d63845f4b2afea4ec6bbe35a.jpg "B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 1 - Rational and Irrational Numbers")
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Solutions for Chapter 1: Rational and Irrational Numbers
Below listed, you can find solutions for Chapter 1 of CISCE B Nirmala Shastry for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई.
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई 1 Rational and Irrational Numbers EXERCISE 1A [Page 5]
Insert three rational numbers between 7 and 8.
Insert three rational numbers between 9.9 and 10.
Insert three rational numbers between 2 and 2.01.
Insert a rational number between `2/3` and `5/7`.
Insert a rational number between `4/7` and `9/11`.
Insert a rational number between `5/7` and `6/11`.
Insert three rational numbers between `5/9` and `8/11`.
Insert three rational numbers between `3/8` and `10/13`.
Insert three rational numbers between `2/7` and `11/17`.
Without actual division, state whether the following have a terminating decimal.
`17/125`
Without actual division, state whether the following have a terminating decimal.
`19/75`
Without actual division, state whether the following have a terminating decimal.
`41/16`
Without actual division, state whether the following have a terminating decimal.
`37/50`
Without actual division, state whether the following have a terminating decimal.
`5/11`
Without actual division, state whether the following have a terminating decimal.
`23/3125`
Without actual division, state whether the following have a terminating decimal.
`9/14`
Without actual division, state whether the following have a terminating decimal.
`18/35`
Without actual division, state whether the following have a terminating decimal.
`37/80`
Without actual division, state whether the following have a terminating decimal.
`5/12`
Convert the following fraction to a decimal.
`9/5`
Convert the following fraction to a decimal:
`(13)/(25)`
Convert the following fraction to a decimal.
`19/125`
Convert the following fraction to a decimal.
`78/65`
Express the following as a fraction.
0.7
Express the following as a fraction.
`0.bar(39)`
Express the following as a fraction.
`1.bar(42)`
Express the following as a fraction.
`0.bar(213)`
Express the following as a fraction.
`0.bar(285)`
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई 1 Rational and Irrational Numbers EXERCISE 1B [Page 11]
Fill in the blank using rational/irrational/either rational or irrational.
The sum of two rational numbers is ______.
Fill in the blank using rational/irrational/either rational or irrational.
The difference of two irrational numbers is ______.
Fill in the blank using rational/irrational/either rational or irrational.
The product of a rational and an irrational number is ______.
Fill in the blank using rational/irrational/either rational or irrational.
The product of two irrational numbers is ______.
State whether true or false in the following:
`sqrt(4) + sqrt(3) = sqrt(7)`
State whether true or false in the following:
`8sqrt(5) - 2sqrt(5) = 6sqrt(5)`
State whether true or false in the following:
`sqrt(2)sqrt(72) = 12`
State whether true or false in the following:
`sqrt(75)/sqrt(3) = 5`
State whether true or false in the following:
`sqrt(4^2 + 5^2) = 9`
State whether true or false in the following:
`sqrt(9) - sqrt(4) = sqrt(5)`
State whether true or false in the following:
`sqrt(125) + 7 - sqrt(5) - sqrt(80) = 7`
State whether the following is irrational.
`2sqrt(5)`
State whether the following is irrational.
`sqrt(147)/sqrt(3)`
State whether the following is irrational.
`sqrt(4/5)`
State whether the following is irrational.
`(-2)/3`
State whether the following is irrational.
`π/2`
State whether the following is irrational.
`[3/4 sqrt(2)]^3`
State whether the following is irrational.
`sqrt(18) xx sqrt(8)`
State whether the following is irrational.
`sqrt(7) - 1`
State whether the following is irrational.
`(3 + sqrt(2))(3 - sqrt(2))`
State whether the following is irrational.
`(5 + sqrt(3))(4 - sqrt(2))`
Represent `sqrt(10)` and `sqrt(20)` on the number line.
Prove that the following number is irrational.
`2 + sqrt(3)`
Prove that the following number is irrational.
`3 - sqrt(5)`
Prove that the following number is irrational.
`sqrt(2) + sqrt(5)`
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई 1 Rational and Irrational Numbers EXERCISE 1C [Page 15]
State with reason, whether the following is a surd.
`root(3)(81)`
State with reason, whether the following is a surd.
`sqrt(140)`
State with reason, whether the following is a surd.
`root(4)(250) root(4)(40)`
State with reason, whether the following is a surd.
`root(3)(9) root(3)(24)`
State with reason, whether the following is a surd.
`root(3)(2) root(3)(32)`
State with reason, whether the following is a surd.
`sqrt(3 + sqrt(5))`
State with reason, whether the following is a surd.
`sqrt(π + 1)`
State with reason, whether the following is a surd.
`root(5)(243)`
State with reason, whether the following is a surd.
`sqrt(sqrt(3) - sqrt(2))`
State with reason, whether the following is a surd.
`sqrtroot(3)(5)`
State with reason, whether the following is a surd.
`sqrt(-9)`
State with reason, whether the following is a surd.
`sqrt(0.04004000400004...)`
State with reason, whether the following is a surd.
`(root(3)(7))^6`
State with reason, whether the following is a surd.
`root(3)(7/5)`
Express the following surd in simplest form.
`sqrt(75)`
Express the following surd in simplest form.
`sqrt(80)`
Express the following surd in simplest form.
`sqrt(96)`
Express the following surd in simplest form.
`sqrt(112)`
Express the following surd in simplest form.
`sqrt(162)`
Express the following surd in simplest form.
`sqrt(245)`
Write the simplest rationalising factor of `sqrt(32)`.
Write the simplest rationalising factor of `sqrt(48)`.
Write the simplest rationalising factor of `sqrt(72)`.
Write the simplest rationalising factor of `5sqrt(3) + 8`.
Write the simplest rationalising factor of `sqrt(7) - sqrt(5)`.
Write the simplest rationalising factor of `sqrt(6) + sqrt(3)`.
Simplify the following:
`sqrt(75) - sqrt(12) + sqrt(27)`
Simplify the following:
`3sqrt(11) + 8sqrt(99) - sqrt(44)`
Simplify the following:
`3sqrt(63) - 2sqrt(28) - sqrt(112)`
Simplify the following:
`5sqrt(18) + 7sqrt(50) - 10sqrt(8)`
Simplify the following:
`6sqrt(72) - sqrt(18) - 8sqrt(32)`
Simplify the following:
`sqrt(48) - sqrt(12) + sqrt(75) - sqrt(27)`
Simplify the following:
`sqrt(162) - sqrt(98) - sqrt(8) + sqrt(50)`
Expand:
`(2 + sqrt(5))^2`
Expand:
`(7 - sqrt(3))^2`
Expand:
`(2sqrt(3) - 5sqrt(2))^2`
Expand:
`(4sqrt(5) + 3sqrt(7))^2`
Find the product.
`(3 - sqrt(5))(3 + sqrt(5))`
Find the product.
`(5 + 3sqrt(2))(5 - 3sqrt(2))`
Find the product.
`(7sqrt(3) - 4)(5sqrt(3) + 1)`
Find the product.
`(sqrt(2) + 3sqrt(11))(5sqrt(2) - 2sqrt(11))`
Find the product.
`(3sqrt(6) - 5sqrt(2))(4sqrt(6) - 3sqrt(2))`
Find the product.
`(5sqrt(7) + 3)(4 - sqrt(7))`
Rationalise the denominator of `9/sqrt(3)`.
Rationalise the denominator of `4/sqrt(2)`.
Rationalise the denominator of `10/sqrt(5)`.
Rationalise the denominator of `5/(3 - sqrt(2))`.
Rationalise the denominator of `3/(4 - sqrt(7))`.
Rationalise the denominator of `(2 + sqrt(3))/(4 - sqrt(3))`.
Rationalise the denominator of `(4 - sqrt(5))/(3 + sqrt(5))`.
Rationalise the denominator of `(2sqrt(5) - 3)/(3sqrt(2) - 4)`.
Rationalise the denominator of `(3sqrt(3) - 2)/(2sqrt(7) + 1)`.
Rationalise the denominator of `(2 - sqrt(3))/(2 + sqrt(3))`.
Rationalise the denominator of `(2sqrt(5) - sqrt(10))/(2sqrt(5) + sqrt(10))`.
Rationalise the denominator of `(2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2))`.
Rationalise the denominator of `1/(sqrt(2) + sqrt(3) - sqrt(5))`.
Find the value of a and b in the following:
`1/(4 + sqrt(15)) = a + bsqrt(15)`
Find the value of a and b in the following:
`1/(3 + sqrt(7)) = a + bsqrt(7)`
Find the value of a and b in the following:
`(4sqrt(3))/(2 - sqrt(3)) = a + bsqrt(3)`
Find the value of a and b in the following:
`(5sqrt(3) + 3)/(2sqrt(3) - 3) = a + bsqrt(3)`
Find the value of a and b in the following:
`(5 + sqrt(5))/(9 - 4sqrt(5)) = a + bsqrt(5)`
Find the value of a and b in the following:
`(6 + sqrt(3))/(7 - 4sqrt(3)) = a + bsqrt(3)`
Simplify the following:
`(sqrt(7) + sqrt(6))/(sqrt(7) - sqrt(6)) + (sqrt(7) - sqrt(6))/(sqrt(7) + sqrt(6))`
Simplify the following:
`(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) - (sqrt(5) - sqrt(3))/(sqrt(5) + sqrt(3))`
Simplify the following:
`7/(3sqrt(5) - 2) - 2/(3sqrt(5) + 2`
Insert 3 rational numbers between `3/7` and `7/9`.
Insert 3 irrational numbers between 11 and 12.
Insert two irrational numbers between the following:
`sqrt(3)` and `sqrt(6)`
Insert two irrational numbers between the following:
`3sqrt(5)` and `2sqrt(3)`
Insert two irrational numbers between the following:
6 and 7
Write two rational numbers between 2 and `sqrt(3)`.
Write two rational numbers between `sqrt(7)` and `sqrt(8)`.
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई 1 Rational and Irrational Numbers MULTIPLE CHOICE QUESTIONS [Pages 16 - 17]
Which of the following is not a terminating decimal?
`11/80`
`63/35`
`5/11`
`17/1250`
Which of the following is an irrational number?
`sqrt(8) xx sqrt(50)`
`sqrt(48)/sqrt(27)`
`sqrt(75)`
`(sqrt(18) + sqrt(8))^2`
Between two rational numbers:
there is no rational number.
there is no irrational number.
there are many rational numbers but no irrational numbers.
there are infinitely many rational numbers.
A rational number between `sqrt(2)` and `sqrt(3)` is ______.
`(sqrt(2) + sqrt(3))/2`
`(sqrt(2) xx sqrt(3))/2`
1.9
1.6
`sqrt(5) xx sqrt(90)` is ______.
`5sqrt(10)`
`15sqrt(2)`
`10sqrt(2)`
`12sqrt(10)`
`7sqrt(3) + sqrt(3)` is ______.
`8sqrt(6)`
`sqrt(21)`
`8sqrt(3)`
`sqrt(24)`
`(sqrt(3) - sqrt(12))^2` is ______.
irrational
negative integer
not real
natural number
When a and b are natural numbers `(sqrt(a) + sqrt(b)) (sqrt(a) - sqrt(b))` is ______.
rational
irrational
not real
can be rational or irrational
`(3 - sqrt(5))(2 - sqrt(5))` is ______.
rational
irrational
1
11
`sqrt(45) - 4sqrt(5) + sqrt(20)` when simplified is ______.
`-3sqrt(5)`
`5sqrt(5)`
`-sqrt(5)`
`sqrt(5)`
An irrational number between 3 and 4 is ______.
3.1112003
`sqrt(20)`
`sqrt(10)`
3.142
`6/sqrt(2), sqrt(8), 4sqrt(2)` in ascending order is ______.
`4sqrt(2), sqrt(8), 6/sqrt(2)`
`sqrt(8), 6/sqrt(2), 4sqrt(2)`
`6/sqrt(2), 4sqrt(2), sqrt(8)`
`sqrt(8), 4sqrt(2), 6/sqrt(2)`
If `1/(2 + sqrt(3)) = a + bsqrt(3)` then a and b respectively are ______.
2, 3
2, 1
2, –1
2, –3
Which of the following is true?
`sqrt(9 + 5) = 3 + sqrt(5)`
`sqrt(6 - 1) = 5/sqrt(5)`
`sqrt(75)/sqrt(5) = 3`
`(sqrt(11) - sqrt(5))(sqrt(11) - sqrt(5)) = 16`
Which of the following cannot be represented on a number line?
Fractions
Irrational numbers
Imaginary numbers
Non-terminating decimals
`(3sqrt(2) - 2sqrt(3))/(3sqrt(2) + 2sqrt(3))` after rationalisation becomes ______.
`(6sqrt(2) - 4sqrt(3))/(6sqrt(2) + 4sqrt(3))`
`9sqrt(2) - 4sqrt(3)`
`1 + 2sqrt(6)`
`5 - 2sqrt(6)`
`sqrt(2)(sqrt(8) + sqrt(18))` when simplified becomes ______.
`2sqrt(7)`
10
`sqrt(52)`
8
`sqrt(250) + sqrt(160)` is equals to ______.
`sqrt(410)`
200
`9sqrt(10)`
`33sqrt(10)`
Direction for Questions 19 to 25: In each of the following questions, a statement of assertion (A) is given and a statement of Reason (R) given below it choose the correct option for each question.
Assertion: `3/4` is a rational number lying between 2 rational numbers `1/2` and `7/8`.
Reason: A number lying between a and b is `1/2` (ab).
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: Rationalising factor of `5 + 2sqrt(3)` is `5 - 2sqrt(3)`.
Reason: If the product of 2 irrational numbers is rational then each one is the rationalising factor of the other.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: All integers are rational.
Reason: Square root of all positive integers are irrational.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: 0.456 is a terminating decimal.
Reason: A decimal in which a digit or a set of digits is repeated periodically is called a recurring decimal.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: If `a = 5sqrt(2)` and `2a = sqrt(2x)` then x = 100.
Reason: `10sqrt(2) = sqrt(200)`.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: `(sqrt(6) + sqrt(24))^2` is a rational number.
Reason: Product of 2 irrational numbers can be rational or irrational.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: `1/(3 + sqrt(8)) = 3 - sqrt(8)`
Reason: (a + b)(a – b) = a2 – b2
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई 1 Rational and Irrational Numbers MISCELLANEOUS EXERCISE [Pages 17 - 18]
Without actual division, state whether the following is a terminating decimal.
`13/80`
Without actual division, state whether the following is a terminating decimal.
`4/9`
Without actual division, state whether the following is a terminating decimal.
`19/125`
Without actual division, state whether the following is a terminating decimal.
`43/500`
Without actual division, state whether the following is a terminating decimal.
`5/11`
Without actual division, state whether the following is a terminating decimal.
`14/35`
Without actual division, state whether the following is a terminating decimal.
`2/7`
Expand:
`(sqrt(5) + sqrt(2))^2`
Expand:
`(3 - sqrt(8))^2`
Expand:
`(sqrt(3) - sqrt(2))^2`
Which of the following are irrational?
`sqrt(80)/sqrt(5)`
`(sqrt(7) - sqrt(3))^2`
`(sqrt(50) + sqrt(8))^2`
`sqrt(288)/sqrt(2)`
`(3sqrt(80) - 6sqrt(45))^2`
Rationalise the denominator:
`6/(2 + sqrt(3))`
Rationalise the denominator:
`(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))`
Rationalise the denominator:
`(sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2))`
Rationalise the denominator:
`(sqrt(6) - sqrt(5))/(sqrt(6) + sqrt(5))`
Rationalise the denominator:
`(sqrt(7) - sqrt(6))/(sqrt(7) + sqrt(6))`
Rationalise the denominator:
`8/(sqrt(7) + sqrt(3)`
Rationalise the denominator:
`6/(sqrt(5) - sqrt(3))`
Rationalise the denominator:
`9/(sqrt(11) + sqrt(5))`
Rationalise the denominator:
`(sqrt(5) + 2)/(sqrt(5) - 2)`
Find the value of a and b in the following:
`(3 + sqrt(7))/(3 - sqrt(7)) = a + b sqrt(7)`
Find the value of a and b in the following:
`sqrt(5)/(3 - sqrt(5)) = a + bsqrt(5)`
Find the value of a and b in the following:
`(3sqrt(5) + 1)/(2sqrt(5) + 4) = a + bsqrt(5)`
Find the value of a and b in the following:
`sqrt(3)/(3sqrt(2) + 2sqrt(3)) = a + bsqrt(6)`
Find the value of a and b in the following:
`(2 + sqrt(3))/(7 + 4sqrt(3)) = a + bsqrt(3)`
Find the value of a and b in the following:
`(3 + sqrt(2))/(10 - 7sqrt(2)) = a + bsqrt(2)`
Simplify the following:
`sqrt(147) - sqrt(27) + sqrt(75) - sqrt(48)`
Simplify the following:
`8sqrt(3) - 3sqrt(27) + 6/sqrt(3)`
Simplify the following:
`sqrt(175) - sqrt(112) - sqrt(28) + sqrt(63)`
State whether true or false in the following:
`sqrt(8 - 1) = 7/sqrt(7)`
State whether true or false in the following:
`sqrt(75)/sqrt(3) = 5`
State whether true or false in the following:
`sqrt(49 - 5) = 7 - sqrt(5)`
State whether true or false in the following:
`sqrt(9 + 7) = 3 + sqrt(7)`
State whether true or false in the following:
`sqrt(3)sqrt(27) = 9`
State whether true or false in the following:
`(sqrt(11) - sqrt(7))(sqrt(11) + sqrt(7)) = 4`
State whether true or false in the following:
`1/(3 - sqrt(8)) = 3 + 2sqrt(2)`
Rationalise the denominator and simplify to find the value of `4/(sqrt(5) + sqrt(3))`, given that `sqrt(5) = 2.236` and `sqrt(3) = 1.732`
Prove that the following number is irrational.
`2 + sqrt(5)`
Prove that the following number is irrational.
`4 - sqrt(3)`
Prove that the following number is irrational.
`sqrt(3) - sqrt(2)`
Simplify:
`(sqrt(112) - sqrt(80))/(sqrt(63) - sqrt(45))`
Simplify:
`(sqrt(6) + sqrt(10))/(sqrt(27) + sqrt(45))`
Solutions for 1: Rational and Irrational Numbers
B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 1 - Rational and Irrational Numbers
Shaalaa.com has the CISCE Mathematics मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई CISCE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. B Nirmala Shastry solutions for Mathematics मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई CISCE 1 (Rational and Irrational Numbers) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
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Concepts covered in मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 1 Rational and Irrational Numbers are Irrational Numbers and Proof of Irrationality, Rational Numbers, Properties of Rational Numbers, Decimal Representation of Rational Numbers, Concept of Real Numbers, Surds, Rationalisation of Surds, Simplifying an Expression by Rationalization of the Denominator.
Using B Nirmala Shastry मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई solutions Rational and Irrational Numbers exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in B Nirmala Shastry Solutions are essential questions that can be asked in the final exam. Maximum CISCE मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई students prefer B Nirmala Shastry Textbook Solutions to score more in exams.
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