Question 1

We can check if a number is a perfect square by looking at its last digit or by finding its square root.

Step 1 — Understanding Perfect Squares

A perfect square is a number you get by multiplying an integer by itself. For example, $25$ is a perfect square because $5 \times 5 = 25$. Also, $100$ is a perfect square because $10 \times 10 = 100$.

Step 2 — Checking Unit Digits of Perfect Squares

The unit digit is the last digit of any number. Let us look at the unit digits of the first few perfect squares: $1^2 = 1$ $2^2 = 4$ $3^2 = 9$ $4^2 = 16$ (ends in 6) $5^2 = 25$ (ends in 5) $6^2 = 36$ (ends in 6) $7^2 = 49$ (ends in 9) $8^2 = 64$ (ends in 4) $9^2 = 81$ (ends in 1) $10^2 = 100$ (ends in 0) We can see that perfect squares can only end in the digits 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8, it cannot be a perfect square.

Step 3 — Applying the Unit Digit Check

Let us check each number given in the question using this rule.

(i) For 2032: The unit digit of 2032 is 2. Since a perfect square cannot end in 2, 2032 is not a perfect square.

(ii) For 2048: The unit digit of 2048 is 8. Since a perfect square cannot end in 8, 2048 is not a perfect square.

(iii) For 1027: The unit digit of 1027 is 7. Since a perfect square cannot end in 7, 1027 is not a perfect square.

(iv) For 1089: The unit digit of 1089 is 9. A number ending in 9 can be a perfect square. So, we need to check this number further to confirm.

Step 4 — Finding the Square Root for 1089

We need to find if there is an integer $x$ such that $x^2 = 1089$. Since 1089 is a 4-digit number, its square root will be a 2-digit number. We know that $30^2 = 900$. We also know that $40^2 = 1600$. So, the square root of 1089 must be between 30 and 40. The unit digit of 1089 is 9. This means the unit digit of its square root must be either 3 (because $3^2=9$) or 7 (because $7^2=49$). So, the possible square roots are 33 or 37. Let us try squaring 33: $33 \times 33$ $= (30 + 3) \times (30 + 3)$ $= 30 \times 30 + 30 \times 3 + 3 \times 30 + 3 \times 3$ $= 900 + 90 + 90 + 9$ $= 1080 + 9$

$\boxed{1089}$ Since $33^2 = 1089$, the number 1089 is a perfect square.

Answer

(i) 2032 is not a perfect square. (ii) 2048 is not a perfect square. (iii) 1027 is not a perfect square.