Square and Square Roots Worksheet

Subjective Questions

1. Write 5 numbers whose squares will end with digit 6.
   
   
2. Raju was having fun representing triangular numbers. Now he was able to represent 5 such numbers and then he got stuck. Help Raju to find three more triangular numbers.
   
   
   
3. Reshma is asked to find non-perfect square numbers between the squares of lowest 2-digit even number and lowest 2-digit odd number. Help Reshma to find the answer.
   
   
4. We know that on combining two consecutive triangular number we form a square number. So, which two triangular numbers are to combine to form a square number of 64.
   
   
5. How many successive odd numbers are to be added to get the square number 441.
   
   
6. Write a Pythagorean triplet in which one member is the largest two-digit number.
   
   
7. Find the square root of 144 by repeated subtraction.
   
   
8. Find the Square of 205
   
   
9. Find the square of 1000001
   
   
10. Ram’s school has conducted a tree plantation drive and they have planted 2401 trees in all. If each student has planted as many trees as the number of students participated, then find the total number of student participated.
    
    Multiple Choice Questions
    
    
11. Which of the following numbers will end with 9?
    A. 21²
    B. 22²
    C. 23²
    D. 24²
    
    
12. Which of the following is not a perfect square?
    A. 2304
    B. 2809
    C. 2972
    D. 3721
    
    
13. The square of 7000 will be
    A. 490000000
    B. 49000000
    C. 4900000
    D. 490000
    
    
14. N² = sum of _________
    A. N odd numbers
    B. N even numbers
    C. First N odd numbers
    D. First N even numbers
    
    
15. 432 ___ 3 is a perfect square number (insert appropriate arithmetic operation).
    A. +
    B. ×
    C. ÷
    D. -
    
    
16. 17 × 19 = ( ) ² - 1
    A. 16
    B. 18
    C. 15
    D. 20
    
    
17. Which of the following options is not a perfect square.
    A. 5 × 6 × 6 × 7 × 5 × 7
    B. 9 × 3 × 3 × 2 × 2
    C. 2 × 3 × 6
    D. 16 × 4 × 3
    
    
18. Which of the following options can’t be represented as the sum of consecutive odd numbers starting with 1
    A. 16
    B. 25
    C. 36
    D. 47
    
    
19. 101010101² = 
    A. 102030201
    B. 1020304030201
    C. 10203040504030201
    D. 102030405060504030201
    
    
20. Which of the following is not a Pythagorean triplet
    A. 11, 60, 61
    B. 7, 24, 25
    C. 20, 21, 29
    D. 12, 34, 37

Subjective Questions

1. Write 5 numbers whose squares will end with digit 6.
   Solution:
   Numbers that end with 4 or 6 will have squares that will end with digit 6.
   Some examples include: 14, 16, 24, 26, 36…etc.
   
   
2. Raju was having fun representing triangular numbers. Now he was able to represent 5 such numbers and then he got stuck. Help Raju to find three more triangular numbers.
   
   Solution:
   1
   3 = 1 + 2
   6 = 1 + 2 + 3
   10 = 1 + 2 + 3 + 4
   15 = 1 + 2 + 3 + 4 + 5
   Hence next 3 numbers will be
   1 + 2 + 3 + 4 + 5 + 6 = 21
   1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
   1 + 2 + 3 + 4 + 5 + 6 + 7 + 8  = 36
   
   
   
3. Reshma is asked to find non-perfect square numbers between the squares of lowest 2-digit even number and lowest 2-digit odd number. Help Reshma to find the answer.
   Solution:
   In general, we can say that there are 2n non perfect square numbers between the squares of the numbers n and (n + 1).
   n = lowest 2-digit even number = 10
   n+1 = lowest 2-digit odd number = 11
   Hence, number of non-perfect square numbers between the squares of 10 and 11 is 20.
   
   
4. We know that on combining two consecutive triangular number we form a square number. So, which two triangular numbers are to combine to form a square number of 64.
   Solution:
   64 = 8²
   Hence, we need one triangle with height 8 and other with height 7
   Which are 28 and 36
   
   28 + 36 = 64
   
   
5. How many successive odd numbers are to be added to get the square number 441.
   Solution:
   441 = 21²
   We know, that a perfect square number (n²) can be written as the sum of first n odd natural numbers.
   Hence, we need to add 21 successive odd numbers to get 441.
   
   
6. Write a Pythagorean triplet in which one member is the largest two-digit number.
   Solution:
   Largest two-digit number = 99
   Now, we can get Pythagorean triplets by using general form
   2m, m² – 1, m² + 1
   Let, m² – 1 = 99
   m² = 100
   ∴ m = 10
   Hence, the Pythagorean triplets are 20, 99 and 101.
   
   
7. Find the square root of 144 by repeated subtraction.
   Solution:
   144 – 1 = 143
   143 – 3 = 140
   140 – 5 = 135
   135 – 7 = 128
   128 – 9 = 119
   119 – 11 = 108
   108 – 13 = 95
   95 – 15 = 80
   80 – 17 = 63
   63 – 19 = 44
   44 – 21 = 23
   23 – 23 = 0
   12 steps, hence square root of 144 is 12
   
   
8. Find the Square of 205
   Solution:
   225² = (20×21) hundreds + 25
   =42000+25
   =42,025
   
   
9. Find the square of 1000001
   Solution:
   11² = 121
   101² = 10201
   1001² = 1002001
   Hence,
   1000001² = 1000002000001
   
   
10. Ram’s school has conducted a tree plantation drive and they have planted 2401 trees in all. If each student has planted as many trees as the number of students participated, then find the total number of student participated.
    Solution:
    Let the number of students participated be x
    Then the trees planted by each student = x
    ∴ The total number of trees planted = x²
    ∴ x² = 2401
    
    ∴ x = 49
    Hence, 49 students participated.
    
    Multiple Choice Questions
    
    
11. Which of the following numbers will end with 9?
    A. 21²
    B. 22²
    C. 23²
    D. 24²
    Correct option: C. 23²
    Solution:
    Unit place digit = 3
    3² = 9
    
    
12. Which of the following is not a perfect square?
    A. 2304
    B. 2809
    C. 2972
    D. 3721
    Correct option: C. 2972
    Solution:
    Numbers that end with 2, 3, 7 or 8 at unit’s place can’t be a perfect square
    
    
13. The square of 7000 will be
    A. 490000000
    B. 49000000
    C. 4900000
    D. 490000
    Correct option: B. 49000000
    Solution:
    7² = 49 and we have three zeroes, hence double will be six zeroes.
    Hence,
    7000² = 49000000
    
    
14. N² = sum of _________
    A. N odd numbers
    B. N even numbers
    C. First N odd numbers
    D. First N even numbers
    Correct option: C. First N odd numbers
    Solution:
    Square of any number ‘N’ will be equal to the sum of first N odd numbers.
    
    
15. 432 ___ 3 is a perfect square number (insert appropriate arithmetic operation).
    A. +
    B. ×
    C. ÷
    D. -
    Correct option: C. ÷
    Solution:
    432 ÷ 3 = 144 = 12²
    
    
16. 17 × 19 = ( ) ² - 1
    A. 16
    B. 18
    C. 15
    D. 20
    Correct option: B. 18
    Solution:
    17 × 19 = (18 - 1) (18 + 1) = 18² - 1
    
    
17. Which of the following options is not a perfect square.
    A. 5 × 6 × 6 × 7 × 5 × 7
    B. 9 × 3 × 3 × 2 × 2
    C. 2 × 3 × 6
    D. 16 × 4 × 3
    Correct option: D. 16 × 4 × 3
    Solution:
    16 × 4 × 3 = 4 × 4 × 2 × 2 × 3
    
    
18. Which of the following options can’t be represented as the sum of consecutive odd numbers starting with 1
    A. 16
    B. 25
    C. 36
    D. 47
    Correct option: D. 47
    Solution:
    1 + 3 + 5 + 7 + 9 + 11 = 36
    1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
    
    
19. 101010101² = 
    A. 102030201
    B. 1020304030201
    C. 10203040504030201
    D. 102030405060504030201
    Correct option: C. 10203040504030201
    Solution:
    11² = 121
    101² = 10201
    10101² = 102030201
    101010101² = 10203040504030201
    
    
20. Which of the following is not a Pythagorean triplet
    A. 11, 60, 61
    B. 7, 24, 25
    C. 20, 21, 29
    D. 12, 34, 37
    Correct option: D. 12, 34, 37
    Solution:
    12² + 34² = 1300
    37² = 1369
    As, 12² + 34² ≠ 37²
    Hence, 12, 34, 37 is not a Pythagorean triplet.