DAV Class 8 Maths Chapter 1 Worksheet 1
Squares and Square Roots Worksheet 1
1. Which of the following numbers are perfect squares?
(i) \(11, 16, 32, 36, 50, 64, 75 \)
\[ \begin{align*} (4)^2 &= 16 \\ (6)^2 &= 36 \\ (8)^2 &= 64 \\ \end{align*} \]
Answer \( \color{red}16, 36 \text{ and } 64 \) are perfect squares
2. Which of the following numbers are perfect squares of even numbers?
(i) \( 121, 225, 784, 841, 576, 6561 \)
Square of Even number is always even.
\[ \begin{align*} (28)^2 &= 784 \\ (24)^2 &= 576 \\ \end{align*} \]
Answer \( \color{red} 784 \text{ and } 576 \) are perfect squares of even numbers
3. Which of the following numbers are perfect squares?
(i) \( 100, 205000, 3610000, 212300000 \)
A number ending with an odd number of zeros (one zero, three zeros and so on) is never a perfect square.
\[ \begin{align*} 100 \text{ (No. of Zeros}: 2) &= \text{Perfect square} \\ 205000 \text{ (No. of Zeros}: 3) &= \text{Not a perfect square} \\ 3610000 \text{ (No. of Zeros}: 4) &= \text{Perfect square} \\ 212300000 \text{ (No. of Zeros}: 5) &= \text{Not a perfect square} \\ \end{align*} \]
Answer \( \color{red} 100 \text{ and } 3610000 \) are perfect squares
4. By just observing the digits at ones place, tell which of the following can be perfect squares?
(i) \( 1026, 1022, 1024, 1027 \)
The numbers ending with 2, 3, 7, 8 are not perfect squares.
\[ \begin{align*} 1026 &= \text{Yes, can be perfect squares} \\ 1022 &= \text{ Not a perfect square} \\ 1024 &= \text{Yes, can be perfect squares} \\ 1027 &= \text{ Not a perfect square} \\ \end{align*} \]
Answer \( \color{red} 1024 \text{ and } 1024 \) can be perfect squares.
5. How many non-square numbers lie between the following pairs of numbers?
There are 2n non-perfect square numbers between the square of the numbers, n and (n + 1).
(i) \( 7^2 \text{ and } 8^2 \)
\[ \begin{align*} n &= 7 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 7 \\ &= 14 \end{align*} \]
Answer \( \color{red} 14 \) non-perfect square numbers.
(ii) \( 10^2 \text{ and } 11^2 \)
\[ \begin{align*} n &= 10 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 10 \\ &= 20 \end{align*} \]
Answer \( \color{red} 20 \) non-perfect square numbers.
(iii) \( 40^2 \text{ and } 41^2 \)
\[ \begin{align*} n &= 40 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 40 \\ &= 80 \end{align*} \]
Answer \( \color{red} 80 \) non-perfect square numbers.
(iv) \( 80^2 \text{ and } 81^2 \)
\[ \begin{align*} n &= 80 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 80 \\ &= 160 \end{align*} \]
Answer \( \color{red} 160 \) non-perfect square numbers.
(v) \( 101^2 \text{ and } 102^2 \)
\[ \begin{align*} n &= 101 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 101 \\ &= 202 \end{align*} \]
Answer \( \color{red} 202 \) non-perfect square numbers.
(vi) \( 205^2 \text{ and } 206^2 \)
\[ \begin{align*} n &= 205 \\ \text{Non-perfect square numbers} &= 2n \\ &= 2 \times 205 \\ &= 410 \end{align*} \]
Answer \( \color{red} 410 \) non-perfect square numbers.
6. Write down the correct number in the box:
The difference between the squares of two consecutive natural numbers is equal to their sum.
\[ \begin{align*} \text{Example} \\ 4^2 - 3^2 & = 16 - 9 \\ &= 7 \\ 4^2 - 3^2 & = 4 + 3 \\ &= 7 \end{align*} \](i) \( 100^2 - 99^2 \) = \( 100 + 99 \) = \( 199 \)
(ii) \( 27^2 - 26^2 \) = \( 27 + 26 \) = \( 53 \)
(iii) \( 569^2 - 568^2 \) = \( 569 + 568 \) = \( 1137 \)
7. Observe the pattern in the following and find the missing numbers:
\[ \begin{align*} 1\underline{2}1 &= \frac{(22)^2}{1 + 2 + 1} \\ \\ 12\underline{3}21 &= \frac{(333)^2}{1 + 2 + 3 + 2 + 1} \\ \\ 123\underline{4}321 &= \color{red}\frac{(4444)^2}{1 + 2 + 3 + 4 + 3 + 2 + 1} \\ \\ 1234\underline{5}4321 &= \color{red}\frac{(55555)^2}{1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1} \\ \\ 12345\underline{6}54321 &= \color{red} \frac{(666666)^2}{1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1} \\ \\ \end{align*} \]
8. Which of the following triplets are Pythagorean?
[Hint: Let the smallest even number be 2m and find m from it. Then, find \( (2m, m^2 - 1, m^2 + 1) \) . If you get the triplet, it is Pythagorean.]
(i) \( 3, 4, 5 \)
\[ \begin{align*} \color{green} 2m &= \color{green} 4 \\ \\ m &= \frac{4}{2} \\ \\ m &= 2 \\ \\ m^2 - 1 &= 2^2 - 1 \\ & = 4-1 \\ \color{green} m^2 - 1 &= \color{green} 3 \\ \\ m^2 + 1 &= 2^2 + 1 \\ &= 4 +1 \\ \color{green} m^2 + 1 &= \color{green} 5 \\ \\ (2m, m^2 - 1, m^2 + 1) & = \color{green} (4,3,5) \end{align*} \]
Answer \( \color{red} (3, 4, 5) \) is a Pythagorean triplet.
(ii) \( 6, 7, 8 \)
\[ \begin{align*} \color{green} 2m &= \color{green} 6 \\ \\ m &= \frac{6}{2} \\ \\ m &= 3 \\ \\ m^2 - 1 &= 3^2 - 1 \\ &= 9-1 \\ \color{green} m^2 - 1 &= \color{green} 8 \\ \\ m^2 + 1 &= 3^2 + 1 \\ & = 9 +1 \\ \color{green} m^2 + 1 &= \color{green} 10 \\ \\ (2m, m^2 - 1, m^2 + 1) & = \color{green} (6,8,10) \end{align*} \]
Answer \( \color{red} (6, 7, 8) \) is not a Pythagorean triplet.
(iii) \( 10, 24, 26 \)
\[ \begin{align*} \color{green} 2m &= \color{green} 10 \\ \\ m &= \frac{10}{2} \\ \\ m &= 5 \\ \\ m^2 - 1 &= 5^2 - 1 \\ & = 25-1\\ \color{green} m^2 - 1 &= \color{green} 24 \\ \\ m^2 + 1 &= 5^2 + 1 \\ &= 25 + 1 \\ m^2 + 1 &= \color{green} 26 \\ \\ (2m, m^2 - 1, m^2 + 1) &= \color{green} (10,24,26) \end{align*} \]
Answer \( \color{red} (10, 24, 26) \) is a Pythagorean triplet.
(iv) \( 2, 3, 4 \)
\[ \begin{align*} \color{green} 2m &= \color{green} 2 \\ m &= \frac{2}{2} \\ \\ m &= 1 \\ \\ m^2 - 1 &= 1^2 - 1 \\ &= 1-1 \\ m^2 - 1 & = \color{green} 0 \\ \\ m^2 + 1 &= 1^2 + 1 \\ &= 1 +1 \\ \color{green} m^2 + 1 & = \color{green} 2 \\ \\ (2m, m^2 - 1, m^2 + 1) & = \color{green} (2,0,2) \end{align*} \]
Answer \( \color{red} (2, 3, 4) \) is not a Pythagorean triplet.