Exercises and Problem-Solving Strategies

Theory: Numbers ending in 2, 3, 7, or 8 cannot be perfect squares. This helps in fast elimination.

Example:

Question: Identify which among 625, 1296, 937, 1234 are not perfect squares.

Solution: 937 and 1234 end in 7 and 4 respectively.
But \(\sqrt{937} \approx 30.61\) and \(\sqrt{1234} \approx 35.13\) – not perfect squares.
Only 625 (= \(25^2\)) and 1296 (= \(36^2\)) are perfect squares.

Theory: A perfect square must end in 0, 1, 4, 5, 6, or 9. Use this to reject invalid answers.

Example:

Question: Is 7921 a perfect square?

Solution: Ends in 1 → valid. Check \(\sqrt{7921} = 89 \Rightarrow 89^2 = 7921\) ✔

Theory: For square plots or square-shaped objects, if area is given, use square root to find side.

Example:

Question: A square tile has area 2025 cm². What is its side length?

Solution:\(\sqrt{2025} = 45\ \text{cm}\)

Theory: Use square root approximation and squaring method to find the missing digit.

Example:

Question: Find the missing digit in 6□76 so that it becomes a perfect square.

Solution: Try 4: \(6476\), \(\sqrt{6476} \approx 80.45\) → not square
Try 9: \(6976\), \(\sqrt{6976} \approx 83.51\) → not square
Try 0: 6076, etc.
Eventually, find that\(6476 = 804^2\)

Theory: In right-angled triangles:
\(\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2\)

Example:

Question: A triangle has base 9 cm and height 12 cm. Find hypotenuse.

Solution: \(\text{Hypotenuse} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15\ \text{cm}\)

Theory: Prime factorization helps solve puzzles and applied scenarios requiring square root.

Example:

Question: The area of a square field is 2025 m². Find side length using prime factorization.

Solution: \(2025 = 3^4 \times 5^2 \Rightarrow \sqrt{2025} = 3^2 \cdot 5 = 9 \cdot 5 = 45\)

Theory: Use square patterns of consecutive numbers to relate quantities.

Example:

Question: The difference of squares of two consecutive numbers is 17. What are the numbers?

Solution: \((n + 1)^2 – n^2 = 2n + 1 = 17 \Rightarrow 2n = 16 \Rightarrow n = 8\)\
\(\text{Numbers: } 8,\ 9\)

Theory: Use triplets like (3, 4, 5), (5, 12, 13), etc., in construction, ladders, etc.

Example:

Question: A ladder of 13 m reaches a wall 12 m high. How far is the base of the ladder from the wall?

Solution: \(13^2 = 12^2 + x^2 \Rightarrow x^2 = 169 – 144 = 25 \Rightarrow x = 5\ \text{m}\)

Theory: Locate the number between two perfect squares and use approximation.

Example:

Question: Estimate \(\sqrt{180}\)

Solution: \(13^2 = 169,\ 14^2 = 196 \Rightarrow \sqrt{180} \approx 13.4\)

Theory: These require interpreting real-life situations mathematically.

Example:

Question: If area of square mat is 784 cm², what is its side?

Solution: \(\sqrt{784} = 28\ \text{cm}\)

Theory: Helps in deducing perfect squares via count of subtractions.

Example:

Question: Using repeated subtraction, find if 36 is a perfect square.

Solution: 36 – 1 = 35,
35 – 3 = 32,
\(\ldots,\ 1 – 11 = 0 \Rightarrow \text{Count: 9 steps} \Rightarrow \text{Perfect square: } 6^2\)

Theory: In finance, geometry, decimals arise often.

Example:

Question: Find square root of ₹42.25

Solution: \(\sqrt{42.25} = 6.5\)

Theory: Squares of numbers up to 25 and tricks like:
\((a + b)^2 = a^2 + 2ab + b^2\)

Example:

Question: What is \(21^2\)?

Solution: \(20^2 = 400,\ 2ab = 2 \cdot 20 \cdot 1 = 40,\ b^2 = 1 \Rightarrow 441\)

Theory: Used in exams like Olympiads – require logic and square patterns.

Example:

Question: A number has 4 digits and is a square. Its square root ends in 7. Find the number.

Solution: Try \(67^2 = 4489,\ 77^2 = 5929,\ 87^2 = 7569\)… etc.

Theory: Mistakes include wrong pairing, incorrect digit grouping, and ignoring decimal positioning.

Example:

Question: What mistake is in finding \(\sqrt{0.16}\) as 4?

Solution: \(\sqrt{0.16} = 0.4,\ \text{Not 4}\)