CBSE Class 8 Maths Square and Square Roots HOTs

HOTS

1. A square room having one side equal to 12 m is to be paved with square tiles of side 50 cm. What is the number of tiles required to pave the room?
Answer: 576

Question. Give one Pythagorean triplet in which one of the number is 12.
Answer: (5, 12, 13)

Question. Find the least number which when added to 599 to make it a perfect square
Answer: 26

Question. Find (-2/5)² - (1/5)² = ______
Answer: 3/25

Question. Find the greatest number of two digits which is a perfect square.
Answer: 81

Question. How many digits will be there in the square root of 12321?
Answer: 3

Question. The length of a rectangular park is 80m and breadth is 60m. Find the length of its diagonal.
Answer: 100 m

Question. (√4900 + 10)² = __________ .
Answer: 49

Question. (a + b)² - (a - b)² _______ .
Answer: 4ab

Question. If 21x = 441 then x = _______.
Answer: 2

Question. If the area of a square is 38.44 sq. cm. then find the side of the square.
Answer: 6.2 cm

Question. Find 7 + 9 + 11 + 13 + 15 + 17.
[Hint : make pairs (7 + 17) + (9 + 15) + (11 + 13) = 24 × 3 = 72]
Answer: 72

Question. What should be added to 45² to get 46².
Answer: 91

Question. How much is √441/1369 ?
Answer: 21/37

Question. In a cinema hall 729 people are seated in such a way that the number of people in a row is equal to number of rows. Then how many rows of people are there in the hall?
Answer: 27

Question. Neha walks from her house 160 m north and from there 630 m west to visit her friend's house.
While coming back, she walks diagonally from her friend's house to her house. How much distance does she cover while returning?
Answer: 650m

Question. Two buildings are 20 m and 2A5 m high. If they are 12 m apart, find the distance between their tops.
Answer: 13 m

Question. Aditya was flying a kite which was at a height of 40 m just above a tree. The string was 50 m long.
How far away from the tree was Aditya flying the kite?
Answer: 30 m

Question. A square area in front of Hariti's house is converted into a park. She spent ₹ 1,76,400 at the rate of ₹25 per square metre. What is the length of each side of the park?
Answer: 84 m

Question. Complete the grid and find the squares of the following numbers: 

Answer: 

Question. State True or False
a. The sum or difference of two square numbers is a square number.
b. The value of 1/√0.09 x √5.76 is 7
Answer: a. False b. False

CHALLENGES

1. What are the remainders when a perfect square is divided by 4?

2. What are the remainders when a perfect square is divided by 5?

3. What are the remainders when a perfect square is divided by 8?

4. What is the smallest perfect square which is divisible by 21, 36, and 63?

5. Find all perfect squares each of which when divided by 11 gives a prime number as quotient and 4 as remainder.

6. Find all possible remainders when a perfect square is divided by 12.

7. What are the possible remainders when a perfect square is divided by 9?

8. Find all primes ‘p’ such that p + 10 and p + 14 are also primes. (Hint: If p > 3, then p = 6k–1 or 6k+1).

9. Suppose ‘n’ is a sum of two perfect squares; n=a2+b2. Prove that 2n can also be written as the sum of two perfect squares.

10. Find all odd natural numbers ‘n’ for which there is a unique perfect square strictly between n² and 2n².

11. Write 41 as the sum of two perfect squares. Using this, construct a Pythagorean triplet with 41 as hypotenuse.

12. How many 5-digit perfect squares are there?

13. Here each letter represents different digit. Solve TWO²=THREE.

14. Can you find a natural number ‘n’ such that n2 + n is a perfect square? If so, find one. If not, give reasons.

15. Without actually counting, can you find how many perfect squares are there from 1000 to 2000?

16. Find the *s in √*** = ** where the digits 2,4,5,6,7 are used exactly once.

17. Can you find *s in √**** = ** where each of the digits 1,2,3 4,5,6 are used exactly once?

18. The digits of a square from left to right are a, a + 1, a + 2, 3a, a + 3. Find the square.

19. Exploration :

Consider 49.
49 = 7².
Also 4489 = (67)².
Find the number whose square is 444889. Is 44448889 a square of a number? If yes, find the numbers.
If we add the digits 4 and 8 at each stage, do we get a perfect square? Formulate an appropriate result and prove it.

20. Using division method, find the square root of 1522756.

21. Suitably adopting the methods given, find the square root of 2079.36.

22. Find all natural numbers ‘n’ such that n²+n+2 is a perfect square.

23. Find all natural numbers ‘n’ such that n²+n+3 is a perfect square.

24. Using the digits 1,6,9, each exactly once, how many perfect squares can you construct?

25. Prove that a number ending in 425 or 825 cannot be a perfect square.

26. Show that there is no perfect square, having more than one digit and whose digits are all odd.

27. Let n = 10! which is the product of the natural numbers from 1 to 10. Find the smallest number k such that n.k is a perfect square. What is the smallest number by which you have to multiply it to make it a perfect cube?

28. Find all natural numbers such that n+1 divides n2+1.

29. Suppose 3 divides a²+b² for some integers a, b. Prove that 3 divides both a and b.

30. Without actually finding its square root, how do you check that 8573984 and 95450709 are not perfect squares. (You cannot use unit digit test here)

SUMMARY

1. If x is any number, then the square of x = x × x = (x)².

2. A number with 2, 3, 7 or 8 at its units place is never a perfect square.

3. A number ending in odd number of zeros is never a perfect square.

4. The square of an even number is always even and an odd number is always odd.

5. For any natural number n, sum of the first n-odd natural numbers = n².

6. There are no natural numbers a and b such that a² = 2b².

7. For every natural number a(a> 1), there exists a Pythagorean triplet (2a, a² - 1, a²+ 1).

8. For any given number n, if m is its square root, then n = m² or √n = m.

9. A perfect square can be broken up into the product of its prime factors.

10. If x and y are positive numbers, then 
a. x x y = √x x √y  b. √x/y = √x/√y .

ERRORANALYSIS

1. Students make error in using square root symbol.
√x2 = √289
x = √17 → It should be 17.

2. Students make mistake in division while performing prime factorisation. 

3. Students make error in expressing power / exponent. 

4. Students make calculation error while finding the square root using the long division method.

ACTIVITY I

Find the number of factors of following numbers. Is it even or odd?
a. 100,36,81,16
b. 10,27,44,78
What do you observe?
Answer: 100, 36, 81, 16 all have odd number of factors while 10, 27, 44, 78 have even number of factors.
Conclusion : Every perfect square has odd number of factors.

ACTIVITY II 

Square Root Maze 

ACTIVITY III