Question 1

Q1: Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Accepted Answer

Given: Four numbers: 2032, 2048, 1027, 1089. To Find: Which of the given numbers are not perfect squares. Property: A perfect square cannot end with the digits 2, 3, 7, or 8. Solution: (i) The number 2032 ends with the digit 2. Therefore, it is not a perfect square. (ii) The number 2048 ends with th...

Question 2

Q1: Find the cube roots of 27000 and 10648.

Accepted Answer

Given: The numbers 27000 and 10648. To Find: The cube roots of these numbers. Solution: (i) For 27000: We can write $27000$ as $27 	imes 1000$. $$27000 = 3 	imes 3 	imes 3 	imes 10 	imes 10 	imes 10 = 3^3 	imes 10^3$$ So, the cube root is: $$\sqrt[3]{27000} = \sqrt[3]{3^3 	imes 10^3} = 3 	i...

Question 3

Q2: Which one among $64^2, 108^2, 292^2, 36^2$ has last digit 4?

Accepted Answer

Given: Four squared numbers: $64^2, 108^2, 292^2, 36^2$. To Find: Which of these squares has the last digit 4. Property: The units digit of a square number is determined by the units digit of the original number. Solution: - For $64^2$: The units digit of 64 is 4. The units digit of $4^2$ is 6 (sinc...

Question 4

Q2: What number will you multiply by 1323 to make it a cube number?

Accepted Answer

Given: The number 1323. To Find: The smallest number to multiply 1323 by to make it a perfect cube. Solution: First, we find the prime factorization of 1323. The sum of digits of 1323 is $1+3+2+3=9$, so it is divisible by 3. $$1323 = 3 	imes 441$$ We know that $441 = 21^2 = (3 	imes 7)^2 = 3^2 	i...

Question 5

Q3: State true or false. Explain your reasoning. (i) The cube of any odd number is even. (ii) There is no perfect cube that ends with 8. (iii) The cube of...

Accepted Answer

To Do: State whether the given statements are true or false and provide reasoning. Solution: (i) The cube of any odd number is even. Answer: False. Reason: The product of any number of odd numbers is always odd. Since a cube of an odd number is the product of the number with itself three times (e.g....

Question 6

Q3: Given $125^2=15625$, what is the value of $126^2$? (i) $15625+126$ (ii) $15625+26^2$ (iii) $15625+253$ (iv) $15625+251$ (v) $15625+51^2$

Accepted Answer

Given: $125^2 = 15625$. To Find: The value of $126^2$. Formula: We can use the identity $(n+1)^2 = n^2 + 2n + 1$. The difference between two consecutive squares is $(n+1)^2 - n^2 = 2n+1$. Solution: Let $n = 125$. We need to find $(n+1)^2 = 126^2$. Using the formula for the difference of consecutive...

Question 7

Q4: You are told that 1331 is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of 4913, 12167, a...

Accepted Answer

Given: Four perfect cubes: 1331, 4913, 12167, and 32768. To Find: Their cube roots by estimation. Method: 1. Look at the units digit of the perfect cube to determine the units digit of its cube root.    - Ends in 1 $\rightarrow$ root ends in 1    - Ends in 8 $\rightarrow$ root ends in 2    - Ends in...

Question 8

Q4: Find the length of the side of a square whose area is $441 	ext{ m}^2$.

Accepted Answer

Given: The area of a square is $441 	ext{ m}^2$. To Find: The length of the side of the square. Formula: Area of a square = $(	ext{side})^2$. Solution: Let the length of the side be $s$. Then, $$s^2 = 441$$ To find the side length, we need to find the square root of 441. $$s = \sqrt{441}$$ We know...

Question 9

Q5: Which of the following is the greatest? Explain your reasoning. (i) $67^3-66^3$ (ii) $43^3-42^3$ (iii) $67^2-66^2$ (iv) $43^2-42^2$

Accepted Answer

To Find: The greatest value among the four given expressions. Formulas: - Difference of squares: $a^2 - b^2 = (a-b)(a+b)$. For consecutive integers $n$ and $n-1$, $n^2 - (n-1)^2 = n + (n-1) = 2n-1$. - Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$. For consecutive integers $n$ and $n-1$, $n^3...

Question 10

Q5: Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Accepted Answer

Given: Numbers 4, 9, and 10. To Find: The smallest perfect square that is divisible by 4, 9, and 10. Solution: First, we find the Least Common Multiple (LCM) of 4, 9, and 10. The smallest number divisible by 4, 9, and 10 is their LCM. Prime factorization of the numbers: $4 = 2^2$ $9 = 3^2$ $10 = 2 \...

Question 11

Q6: Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Accepted Answer

Given: The number 9408. To Find: The smallest number to multiply 9408 by to make it a perfect square, and the square root of the resulting number. Solution: First, we find the prime factorization of 9408. $$9408 = 2 	imes 4704$$  $$= 2 	imes 2 	imes 2352$$  $$= 2^2 	imes 2 	imes 1176$$  $$= 2^3...

Question 12

Q7: How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100

Accepted Answer

To Find: The number of non-square integers between the squares of two consecutive numbers. Formula: The number of integers between the squares of two consecutive numbers, $n^2$ and $(n+1)^2$, is given by the formula $2n$. Solution: (i) Between $16^2$ and $17^2$. Here, $n=16$. Number of integers = $2...

Question 13

Q8: In the following pattern, fill in the missing numbers: $$1^2+2^2+2^2=3^2$$ $$2^2+3^2+6^2=7^2$$ $$3^2+4^2+12^2=13^2$$ $$4^2+5^2+20^2=(\quad)^2$$ $$9^2+...

Accepted Answer

Given: A pattern of equations involving sums of squares. To Find: The missing numbers in the last two equations. Observation: The pattern can be described by the relation: $a^2 + (a+1)^2 + (a(a+1))^2 = (a(a+1)+1)^2$. Or more generally, for two numbers $n_1$ and $n_2$, the pattern is $n_1^2 + n_2^2 +...

Question 14

Q9: How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Accepted Answer

Given: A picture of a large square made up of smaller, tiny squares. The large square is a grid. To Find: The total number of tiny squares and its prime factorization. Solution: By observing the picture, we can count the number of tiny squares along one side of the large square. There are 12 tiny sq...

NCERT Solutions