Class 8 Maths Notes for Squares And Square Roots
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SQUARE AND SQUARE ROOTS
Square Numbers
If a natural number m can be expressed as 𝑛2 , where n is also a natural number, then m is a square number.
Example: 1, 4, 9, 16 and 25.
Finding the Square of a Number
If n is a number, then its square is given as 𝑛 × 𝑛 = 𝑛2 .
For example: Square of 5 is equal to 5 × 5 = 25
Properties of Square Numbers
Properties of square numbers are:
If a number has 0, 1, 4, 5, 6 or 9 in the unit‘s place, then it may or may not be a square number. If a
number has 2, 3, 7 or 8 in its units place then it is not a square number.
If a number has 1 or 9 in unit‘s place, then it‘s square ends in 1.
If a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit‘s place.
For example, consider the number 64. The unit digit of 64 is 4 and it is a square number. Because the
square of 8 is 64 and 64 is considered to be a square number.
Consider a number 11 (i.e., the unit‘s place of 11 is 1). Thus, the square of 11 is 121. Hence, square
number 121 also has 1 in the unit‘s place.
Finding square of a number with unit’s place 5
The square of a number N5 is equal to 𝑁 𝑁 + 1 × 100 + 25, where 𝑁 can have one or more than one
digits.
For example: If 𝑁 = 1, then 152 = 1 × 2 × 100 + 25 = 200 + 25 = 225
If 𝑁 = 20, then152 = 20 × 21 × 100 + 25 = 42000 + 25 = 42025
Perfect Squares
A number which is obtained from square of the other number is called perfect squares. For example, 81 is a
perfect square number, which is obtained by taking the square of the number 9.
Numbers between Square Numbers
There are 2𝑛 𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 between squares of the numbers 𝑛 𝑎𝑛𝑑 (𝑛 + 1), where n is
any natural number.
There are two non-perfect square numbers (2, 3) between12 = 1 𝑎𝑛𝑑 22 = 4.
There are four non-perfect square numbers (5, 6, 7, 8) between 22 = 4 𝑎𝑛𝑑 32 = 9.
Square of an odd number as a sum
Square of an odd number n 𝑖. 𝑒 . (𝑛2 ) can be expressed as sum of two consecutive positive integers:
𝑛 −1 2 𝑛+1 2
𝑎𝑛𝑑
2 2
2
2
𝑛 − 1 𝑛+1 2
𝑛 = +
2 2
Product of Two Consecutive Even or Odd Natural Numbers
The product of two even or odd natural number can be calculated as, (𝑎 + 1) × (𝑎 − 1) = (𝑎2 − 1),
where a is a natural number, and 𝑎 − 1, 𝑎 + 1, are the consecutive odd or even numbers.
For example: 11 × 13 = (12 − 1) × (12 + 1) = 122 − 1 = 144 − 1 = 143
Example:
1 + 3 = 4 = 22 , 1 + 3 + 5 = 9 = 32
Pythagorean Triplets: For any natural number m>1, we have 2𝑚 2 + 𝑚2 − 1 2 = 𝑚2 + 1 2 .
23
Square Numbers
If a natural number m can be expressed as 𝑛2 , where n is also a natural number, then m is a square number.
Example: 1, 4, 9, 16 and 25.
Finding the Square of a Number
If n is a number, then its square is given as 𝑛 × 𝑛 = 𝑛2 .
For example: Square of 5 is equal to 5 × 5 = 25
Properties of Square Numbers
Properties of square numbers are:
If a number has 0, 1, 4, 5, 6 or 9 in the unit‘s place, then it may or may not be a square number. If a
number has 2, 3, 7 or 8 in its units place then it is not a square number.
If a number has 1 or 9 in unit‘s place, then it‘s square ends in 1.
If a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit‘s place.
For example, consider the number 64. The unit digit of 64 is 4 and it is a square number. Because the
square of 8 is 64 and 64 is considered to be a square number.
Consider a number 11 (i.e., the unit‘s place of 11 is 1). Thus, the square of 11 is 121. Hence, square
number 121 also has 1 in the unit‘s place.
Finding square of a number with unit’s place 5
The square of a number N5 is equal to 𝑁 𝑁 + 1 × 100 + 25, where 𝑁 can have one or more than one
digits.
For example: If 𝑁 = 1, then 152 = 1 × 2 × 100 + 25 = 200 + 25 = 225
If 𝑁 = 20, then152 = 20 × 21 × 100 + 25 = 42000 + 25 = 42025
Perfect Squares
A number which is obtained from square of the other number is called perfect squares. For example, 81 is a
perfect square number, which is obtained by taking the square of the number 9.
Numbers between Square Numbers
There are 2𝑛 𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 between squares of the numbers 𝑛 𝑎𝑛𝑑 (𝑛 + 1), where n is
any natural number.
There are two non-perfect square numbers (2, 3) between12 = 1 𝑎𝑛𝑑 22 = 4.
There are four non-perfect square numbers (5, 6, 7, 8) between 22 = 4 𝑎𝑛𝑑 32 = 9.
Square of an odd number as a sum
Square of an odd number n 𝑖. 𝑒 . (𝑛2 ) can be expressed as sum of two consecutive positive integers:
𝑛 −1 2 𝑛+1 2
𝑎𝑛𝑑
2 2
2
2
𝑛 − 1 𝑛+1 2
𝑛 = +
2 2
Product of Two Consecutive Even or Odd Natural Numbers
The product of two even or odd natural number can be calculated as, (𝑎 + 1) × (𝑎 − 1) = (𝑎2 − 1),
where a is a natural number, and 𝑎 − 1, 𝑎 + 1, are the consecutive odd or even numbers.
For example: 11 × 13 = (12 − 1) × (12 + 1) = 122 − 1 = 144 − 1 = 143
Example:
1 + 3 = 4 = 22 , 1 + 3 + 5 = 9 = 32
Pythagorean Triplets: For any natural number m>1, we have 2𝑚 2 + 𝑚2 − 1 2 = 𝑚2 + 1 2 .
23
Page 3
2𝑚, (𝑚2 − 1) 𝑎𝑛𝑑 (𝑚2 + 1) forms a Pythagorean triplet.
𝐹𝑜𝑟 𝑚 = 2, 2𝑚 = 4, 𝑚2 − 1 = 3 𝑎𝑛𝑑 𝑚2 + 1 = 5. 𝑆𝑜, 𝟑, 𝟒, 𝟓 𝒊𝒔 𝒕𝒉𝒆 𝒓𝒆𝒒𝒖𝒊𝒓𝒆𝒅 𝑷𝒚𝒕𝒉𝒂𝒈𝒐𝒓𝒆𝒂𝒏 𝒕𝒓𝒊𝒑𝒍𝒆𝒕.
If 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are three numbers such that any one of the following three relations holds
(A) 𝑎2 + 𝑏 2 = 𝑐 2 or (B) 𝑏 2 + 𝑐 2 𝑎2 or (C)𝑐 2 + 𝑎2 = 𝑏 2
then the numbers 𝑎 , 𝑏 𝑐 are said to form a Pythagorean triple.
Q1. Write a Pythagorean triplet whose one number is 6.
6
SOL. Let 2𝑚 = 6, 𝑀 = 2 = 3, Therefore 𝑚2 − 1 = 32 − 1 = 9 − 1 = 8
𝑀2 + 1 = 32 + 1 = 9 + 1 = 10, So a Pythagorean whose one number is 6.
Examples
Q 1. Find the non-perfect square numbers between: (i) 30 and 40
Solution: 30 × 2 = 60
Q 2. Can we say whether the following numbers are perfect squares. How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
Solution: No, as we know that if unit digit of any number is 2,3,7 𝑎𝑛𝑑 8 then such numbers are not perfect
squares.
Q 3. Find whether each of the following numbers, Is a perfect square or not?
(i) 121 (ii) 55 (iii) 81 (iv) 49
2
Solution: (i) Yes, 121 = 11 × 11 = 11 (ii) No, 55 = 5 × 11 , factors are not in pairs (Group of two).
2
(iii) Yes, 81 = 9 × 9 = 9 (iv) No, 222222 = 2 × 3 × 7 × 11 × 13 × 37, factors are not in pairs.
Q4. What will be the unit digit of the following numbers?
(i) 81 (ii)272 (iii)799 (iv) 3853
Solution: (i) Unit digit = 1 (ii) Unit digit = 4 (iii) Unit digit = 1 (iv) Unit digit = 9
IMPORTANT POINTS
(i) Square Roots
The square root of a number ‗a‘ is that number which when multiplied by itself gives that number ‗a‘ as
product. Thus, if b is the square root of 𝑎
Then 𝑏 ×𝑏 = 𝑎 𝑜𝑟 𝑏 2 = 𝑎
Important Points
1. The numbers which can be expressed as the product of the number with itself are called square numbers
or perfect squares.
2. If a natural number 𝑚 can be expressed as𝑛2 , where 𝑛 is also a natural number, then 𝑚 is called a
square number.
For example, 100 is square of 10 , as 100 = 10 × 10 = 102 .So 100 is a square number .
121 is the square of 1, as 121 = 11 × 11 = 112 . So, 121 is a square number.
PROPERTIES OF SQUARE NUMBER
(I) All square numbers end with 0,1,4,5,6 𝑜𝑟 9 only at unit‘s place, i.e., all square numbers have unit‘sdigit
as 0,1,4,5,6or 9 only.
(II) No square number ends with 2,3,7or 8 at its unit‘s place.
(III) Square numbers can have only even number of zeros at the end.
(IV) No square number ends with odd number of zeros at the end.
METHODS TO FIND SQUARE ROOTS
(I) Square root by repeated subtraction method
(II) Square root by prime factorisation method.
(III) Square root by long division method
PRACTICE QUESTIONS:MCQ:
Q1. Which of the following is a perfect square?
A. 1057 B. 625 C. 7928 D. 64000
Q2. Which of the following will have 6 at unit place?
A. 192 B. 112 C. 242 D. 132
24
𝐹𝑜𝑟 𝑚 = 2, 2𝑚 = 4, 𝑚2 − 1 = 3 𝑎𝑛𝑑 𝑚2 + 1 = 5. 𝑆𝑜, 𝟑, 𝟒, 𝟓 𝒊𝒔 𝒕𝒉𝒆 𝒓𝒆𝒒𝒖𝒊𝒓𝒆𝒅 𝑷𝒚𝒕𝒉𝒂𝒈𝒐𝒓𝒆𝒂𝒏 𝒕𝒓𝒊𝒑𝒍𝒆𝒕.
If 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are three numbers such that any one of the following three relations holds
(A) 𝑎2 + 𝑏 2 = 𝑐 2 or (B) 𝑏 2 + 𝑐 2 𝑎2 or (C)𝑐 2 + 𝑎2 = 𝑏 2
then the numbers 𝑎 , 𝑏 𝑐 are said to form a Pythagorean triple.
Q1. Write a Pythagorean triplet whose one number is 6.
6
SOL. Let 2𝑚 = 6, 𝑀 = 2 = 3, Therefore 𝑚2 − 1 = 32 − 1 = 9 − 1 = 8
𝑀2 + 1 = 32 + 1 = 9 + 1 = 10, So a Pythagorean whose one number is 6.
Examples
Q 1. Find the non-perfect square numbers between: (i) 30 and 40
Solution: 30 × 2 = 60
Q 2. Can we say whether the following numbers are perfect squares. How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
Solution: No, as we know that if unit digit of any number is 2,3,7 𝑎𝑛𝑑 8 then such numbers are not perfect
squares.
Q 3. Find whether each of the following numbers, Is a perfect square or not?
(i) 121 (ii) 55 (iii) 81 (iv) 49
2
Solution: (i) Yes, 121 = 11 × 11 = 11 (ii) No, 55 = 5 × 11 , factors are not in pairs (Group of two).
2
(iii) Yes, 81 = 9 × 9 = 9 (iv) No, 222222 = 2 × 3 × 7 × 11 × 13 × 37, factors are not in pairs.
Q4. What will be the unit digit of the following numbers?
(i) 81 (ii)272 (iii)799 (iv) 3853
Solution: (i) Unit digit = 1 (ii) Unit digit = 4 (iii) Unit digit = 1 (iv) Unit digit = 9
IMPORTANT POINTS
(i) Square Roots
The square root of a number ‗a‘ is that number which when multiplied by itself gives that number ‗a‘ as
product. Thus, if b is the square root of 𝑎
Then 𝑏 ×𝑏 = 𝑎 𝑜𝑟 𝑏 2 = 𝑎
Important Points
1. The numbers which can be expressed as the product of the number with itself are called square numbers
or perfect squares.
2. If a natural number 𝑚 can be expressed as𝑛2 , where 𝑛 is also a natural number, then 𝑚 is called a
square number.
For example, 100 is square of 10 , as 100 = 10 × 10 = 102 .So 100 is a square number .
121 is the square of 1, as 121 = 11 × 11 = 112 . So, 121 is a square number.
PROPERTIES OF SQUARE NUMBER
(I) All square numbers end with 0,1,4,5,6 𝑜𝑟 9 only at unit‘s place, i.e., all square numbers have unit‘sdigit
as 0,1,4,5,6or 9 only.
(II) No square number ends with 2,3,7or 8 at its unit‘s place.
(III) Square numbers can have only even number of zeros at the end.
(IV) No square number ends with odd number of zeros at the end.
METHODS TO FIND SQUARE ROOTS
(I) Square root by repeated subtraction method
(II) Square root by prime factorisation method.
(III) Square root by long division method
PRACTICE QUESTIONS:MCQ:
Q1. Which of the following is a perfect square?
A. 1057 B. 625 C. 7928 D. 64000
Q2. Which of the following will have 6 at unit place?
A. 192 B. 112 C. 242 D. 132
24
Page 4
Q3. If 5278 is squared, then what will be at unit place?
A. 8 B. 7 C. 6 D. 4
Q4. What will be the number of zeros in square of 400?
A. 2 B. 3 C. 4 D. 6
2 2
Q5. How many natural numbers lie between 9 and 10 ?
A. 17 B. 18 C. 19 D. 20
Q6. What is the sum of the first four odd natural numbers?
A. 16 B. 17 C. 18 D. 20
2
Q7. The value of 9 – 1 is equal to:
A. 81 B. 80 C. 79 D. None of the above
Q8. The square of 42 is:
A. 1764 B. 1664 C. 1564 D. 1504
Q9. The Pythagorean triples whose smallest number is 8:
A. 8, 16 17 B. 8, 17, 18 C. 8, 15, 17 D. 8, 15, 16
SHORT ANSWER QUESTIONS
Q1. Find the square root of 729 by prime factorisation method.
Q2. Find the length of the side of a square whose area is 441 m2.
Q3. Find the square root of the following number by long division method: 2304
Q4. Find the smallest whole number by which 252 should be multiplied, to get a perfect square.
Q5. Find the square root of 4096 by prime factorisation method.
Q6. Find the square root of 169 by repeated subtraction method.
Q7. Write the Pythagorean triplet whose one number is 16.
Q8. Without doing any calculation, find whether 257 is a perfect square or not.
Q9. Write a greatest two-digit number which is a perfect square.
Q10. Find the value of 17 + 64.
LONG ANSWER QUESTION(LAQ)
Q1. Write a Pythagorean triplet whose one member is 35.
Q2. Find the square root of 225 by repeated subtraction method.
Q3. Use the identity and find the square of 189.
Q4. Find the smallest whole number by which 1008 should be divided, to get a perfect square number. Also,
find the square root of the square number so obtained.
Q5. Find the smallest square number that is divisible by each of the numbers 6, 15 and 20.
Q6. Find the square root of 6241 using long division method.
CASE BASED STUDY QUESTION(CSBQ)
A king wanted to reward his advisor, a wise man of the kingdom. So, he asked the
wiseman to name his own reward. The wiseman thanked the king but said that he
would ask only for some gold coins each day for a month. The coins were to be
counted out in a pattern of one coin for the first day, 3 coins for the second day, 5
coins for the third day and so on for 30 days.
Q.1 Without making calculations, find how many coins will the advisor get in that
month?
ANSWERS: MCQ- Q1. B Q2. C Q3. D Q4.C Q5. C Q6. B Q7. A Q8. B Q9. C
ANSWER SAQ: Q1. 27 Q2. 21 m Q3. 48 Q4. 7 Q5. 48 Q6. 13
Q7. 16, 63, 65 Q8. No, prime factors. Q9. 81 Q10. 5
ANSWER LAQ: Q1. 35, 12, 36 Q2. 15 Q3. 35721 Q4. 7 𝑎𝑛𝑑 144 = 12 Q5. 900
Q6. 79
ANSWER CSBQ: Q1. 900
25
A. 8 B. 7 C. 6 D. 4
Q4. What will be the number of zeros in square of 400?
A. 2 B. 3 C. 4 D. 6
2 2
Q5. How many natural numbers lie between 9 and 10 ?
A. 17 B. 18 C. 19 D. 20
Q6. What is the sum of the first four odd natural numbers?
A. 16 B. 17 C. 18 D. 20
2
Q7. The value of 9 – 1 is equal to:
A. 81 B. 80 C. 79 D. None of the above
Q8. The square of 42 is:
A. 1764 B. 1664 C. 1564 D. 1504
Q9. The Pythagorean triples whose smallest number is 8:
A. 8, 16 17 B. 8, 17, 18 C. 8, 15, 17 D. 8, 15, 16
SHORT ANSWER QUESTIONS
Q1. Find the square root of 729 by prime factorisation method.
Q2. Find the length of the side of a square whose area is 441 m2.
Q3. Find the square root of the following number by long division method: 2304
Q4. Find the smallest whole number by which 252 should be multiplied, to get a perfect square.
Q5. Find the square root of 4096 by prime factorisation method.
Q6. Find the square root of 169 by repeated subtraction method.
Q7. Write the Pythagorean triplet whose one number is 16.
Q8. Without doing any calculation, find whether 257 is a perfect square or not.
Q9. Write a greatest two-digit number which is a perfect square.
Q10. Find the value of 17 + 64.
LONG ANSWER QUESTION(LAQ)
Q1. Write a Pythagorean triplet whose one member is 35.
Q2. Find the square root of 225 by repeated subtraction method.
Q3. Use the identity and find the square of 189.
Q4. Find the smallest whole number by which 1008 should be divided, to get a perfect square number. Also,
find the square root of the square number so obtained.
Q5. Find the smallest square number that is divisible by each of the numbers 6, 15 and 20.
Q6. Find the square root of 6241 using long division method.
CASE BASED STUDY QUESTION(CSBQ)
A king wanted to reward his advisor, a wise man of the kingdom. So, he asked the
wiseman to name his own reward. The wiseman thanked the king but said that he
would ask only for some gold coins each day for a month. The coins were to be
counted out in a pattern of one coin for the first day, 3 coins for the second day, 5
coins for the third day and so on for 30 days.
Q.1 Without making calculations, find how many coins will the advisor get in that
month?
ANSWERS: MCQ- Q1. B Q2. C Q3. D Q4.C Q5. C Q6. B Q7. A Q8. B Q9. C
ANSWER SAQ: Q1. 27 Q2. 21 m Q3. 48 Q4. 7 Q5. 48 Q6. 13
Q7. 16, 63, 65 Q8. No, prime factors. Q9. 81 Q10. 5
ANSWER LAQ: Q1. 35, 12, 36 Q2. 15 Q3. 35721 Q4. 7 𝑎𝑛𝑑 144 = 12 Q5. 900
Q6. 79
ANSWER CSBQ: Q1. 900
25
Page 5
TEST -1 SQUARES AND SQUARE ROOTS MM 20
Q1. Find the perfect square numbers between 30 and 40. (1)
Q2. Write a Pythagorean triplet whose one number is 6 . (2)
2 2
Q3. How many natural numbers lie between 8 𝑎𝑛𝑑 9 . (2)
Q4. Find the square of 105. (2)
Q5. Find the square root of 1764 by prim factorisation method. (2)
Q6. Find the square root of 4489 by Division method. (2)
Q7. Find the square root of 100 by repeated subtraction method. (2)
Q8. Find the number of digits in the square root of 27225. (2)
Q9. Find the least number which must be subtractedfrom1989 to get a perfect square. (2)
Q10. Find the smallest number whole number by which 252 should be multiplied to get a perfect square.
Also find the square root number obtained. (3)
TEST-2: SQUARE AND SQUARE ROOTS MM 30
Q1. The square of which of the following numbers would be an odd number/an even. (2)
Q2. Express49 as the sum of 7 odd numbers. (2)
Q3. Find the square root of 2304 by Division method. (2)
Q4. Find the unit digit of the number 78. (2)
2
Q5. If25 = 625 , then find the square root of 625. (2)
Q6. Find the square of 42 without actual multiplication. (2)
Q7. Write a Pythagorean triplet whose smallest member is 8. (2)
Q8. Find the square root of 6400 . (2)
Q9. Find the square roots of 169 by the method of repeated subtraction. (2)
Q10. Find the greatest 4-digit number which is a perfect square. (2)
Q11. Find the least number that must be added to 1300 to get a perfect square. Also find the square root of
the perfect square. (2)
Q12. Find the square root of 12.2 (2)
2
Q13. Find the length of the side of a square whose area is 441 𝑚 . (3)
Q14. A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the
number of columns remain same. Find the minimum number of plants he needs more for this. (3)
26
Q1. Find the perfect square numbers between 30 and 40. (1)
Q2. Write a Pythagorean triplet whose one number is 6 . (2)
2 2
Q3. How many natural numbers lie between 8 𝑎𝑛𝑑 9 . (2)
Q4. Find the square of 105. (2)
Q5. Find the square root of 1764 by prim factorisation method. (2)
Q6. Find the square root of 4489 by Division method. (2)
Q7. Find the square root of 100 by repeated subtraction method. (2)
Q8. Find the number of digits in the square root of 27225. (2)
Q9. Find the least number which must be subtractedfrom1989 to get a perfect square. (2)
Q10. Find the smallest number whole number by which 252 should be multiplied to get a perfect square.
Also find the square root number obtained. (3)
TEST-2: SQUARE AND SQUARE ROOTS MM 30
Q1. The square of which of the following numbers would be an odd number/an even. (2)
Q2. Express49 as the sum of 7 odd numbers. (2)
Q3. Find the square root of 2304 by Division method. (2)
Q4. Find the unit digit of the number 78. (2)
2
Q5. If25 = 625 , then find the square root of 625. (2)
Q6. Find the square of 42 without actual multiplication. (2)
Q7. Write a Pythagorean triplet whose smallest member is 8. (2)
Q8. Find the square root of 6400 . (2)
Q9. Find the square roots of 169 by the method of repeated subtraction. (2)
Q10. Find the greatest 4-digit number which is a perfect square. (2)
Q11. Find the least number that must be added to 1300 to get a perfect square. Also find the square root of
the perfect square. (2)
Q12. Find the square root of 12.2 (2)
2
Q13. Find the length of the side of a square whose area is 441 𝑚 . (3)
Q14. A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the
number of columns remain same. Find the minimum number of plants he needs more for this. (3)
26
Page 6
Study Materials
Notes
Model Papers Class 6 Notes
Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes
Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats
Maps of India / World Class 12 Notes
Books and Solutions
NCERT Books
NCERT Book Solutions
HC Verma Chapter Wise Solutions
RD Sharma Solutions
CGBSE Solutions
Notes
Model Papers Class 6 Notes
Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes
Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats
Maps of India / World Class 12 Notes
Books and Solutions
NCERT Books
NCERT Book Solutions
HC Verma Chapter Wise Solutions
RD Sharma Solutions
CGBSE Solutions
Document Details
| Board / Org | Aglasem |
| Exam | Class 8 |
| Type | Notes |
| Pages | 6 |
| Language | english |
| Updated | 24 Sep 2026 |