Squares and square roots | vanisvnps - Free Printable

Let's solve each question one by one from the worksheet and explain the reasoning behind each answer.

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Question 1:

The perfect square number out of 2, 3, 4 and 5 is
(a) 2 (b) 3 (c) 4 (d) 5

Solution:
A perfect square is a number that can be expressed as the square of an integer.

- $ 2 $ → not a perfect square
- $ 3 $ → not a perfect square
- $ 4 = 2^2 $ → Yes, it's a perfect square
- $ 5 $ → not a perfect square

✔ Answer: (c) 4

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Question 2:

Which of $132^2$, $87^2$, $72^2$, and $209^2$ would end with digit 1?
(a) $132^2$ (b) $87^2$ (c) $72^2$ (d) $209^2$

Solution:
To find which square ends in digit 1, we only need to check the last digit of the original number and see what its square ends with.

We know:
- If a number ends in 1 or 9, its square ends in 1
- Because:
- $1^2 = 1$
- $9^2 = 81$

Now check the last digits:
- $132$ → ends in 2 → $2^2 = 4$ → ends in 4
- $87$ → ends in 7 → $7^2 = 49$ → ends in 9
- $72$ → ends in 2 → $2^2 = 4$ → ends in 4
- $209$ → ends in 9 → $9^2 = 81$ → ends in 1

✔ So, $209^2$ ends with 1

Answer: (d) $209^2$

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Question 3:

What will be the number of zeros in the square of the number 100?
(a) 2 (b) 4 (c) 6 (d) 8

Solution:
First, compute $100^2$:
$100^2 = 100 \times 100 = 10,000$

Now count the number of zeros in 10,000 → There are 4 zeros

Alternatively:
$100 = 10^2 = (10)^2$, so $100^2 = (10^2)^2 = 10^4 = 10,000$ → 4 zeros

✔ Answer: (b) 4

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Question 4:

How many natural numbers lie between $12^2$ and $13^2$?
(a) 20 (b) 22 (c) 24 (d) 26

Solution:
First, calculate:
- $12^2 = 144$
- $13^2 = 169$

Now, find how many natural numbers lie between 144 and 169.

That means: numbers from 145 to 168 inclusive.

Number of integers from 145 to 168:
$$
168 - 145 + 1 = 24
$$

So, there are 24 natural numbers between $12^2$ and $13^2$

✔ Answer: (c) 24

> Note: A general rule: Between $n^2$ and $(n+1)^2$, there are $2n$ natural numbers.
> For $n=12$: $2 \times 12 = 24$ → matches.

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Question 5:

Express the square number $5^2$ as the sum of two consecutive integers.
(a) $12 + 13$ (b) $10 + 15$ (c) $9 + 16$ (d) $20 + 5$

Solution:
First, $5^2 = 25$

We need two consecutive integers whose sum is 25.

Let the numbers be $x$ and $x+1$. Then:
$$
x + (x+1) = 25 \\
2x + 1 = 25 \\
2x = 24 \\
x = 12
$$
So the numbers are $12$ and $13$

Check: $12 + 13 = 25$ → correct

Now check options:
- (a) $12 + 13 = 25$ → ✔ Correct
- (b) $10 + 15 = 25$, but not consecutive
- (c) $9 + 16 = 25$, not consecutive
- (d) $20 + 5 = 25$, not consecutive

✔ Answer: (a) $12 + 13$

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✔ Final Answers:

1. (c) 4
2. (d) $209^2$
3. (b) 4
4. (c) 24
5. (a) $12 + 13$

Let me know if you'd like these explained in simpler terms or want practice problems!