Given , what is the value of ? (i) (ii) (iii) (iv) (v)

Understand the Question We are given the value of and need to find the expression for . We can rewrite as and expand it using the standard algebraic identity: Substituting and allows us to express in terms of . Step 1 · Expand using Write as . Using the identity where and : This matches option (iv). Answer > (iv) Common Mistakes Forgetting the term: Incorrectly assuming , missing the linear cross term . Adding the base instead of the difference: Choosing option (i) by mistakenly adding instead of the actual difference between consecutive squares, .

Rational Numbers | FIO

Question 3

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) $15625 + 51^2$

Solution

Understand the Question

We are given the value of $125^2 = 15625$ and need to find the expression for $126^2$ .
We can rewrite $126$ as $(125 + 1)$ and expand it using the standard algebraic identity: $(a + b)^2 = a^2 + 2ab + b^2$
Substituting $a = 125$ and $b = 1$ allows us to express $126^2$ in terms of $125^2$ .

Step 1 · Expand $126^2$ using $(a + b)^2$

Write $126$ as $(125 + 1)$ .

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$ where $a = 125$ and $b = 1$ :

\begin{aligned} 126^2 &= (125 + 1)^2 \\ &= 125^2 + (2 \times 125 \times 1) + 1^2 \\ &= 15625 + 250 + 1 \\ &= 15625 + 251 \end{aligned}

This matches option (iv).

Answer

(iv) $15625 + 251$

Common Mistakes

Forgetting the $2ab$ term: Incorrectly assuming $(125 + 1)^2 = 125^2 + 1^2 = 15625 + 1$ , missing the linear cross term $2 \times 125 \times 1 = 250$ .
Adding the base instead of the difference: Choosing option (i) by mistakenly adding $126$ instead of the actual difference between consecutive squares, $(n+1)^2 - n^2 = 2n + 1 = 2(125) + 1 = 251$ .

More questions in FIO

Which of the following numbers are not perfect squares?

(i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

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

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) $15625 + 51^2$

Find the length of the side of a square whose area is $441 \text{ m}^2$ .

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

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.

How many numbers lie between the squares of the following numbers?

(i) 16 and 17 (ii) 99 and 100

In the following pattern, fill in the missing numbers:

\begin{aligned} 1^2 + 2^2 + 2^2 &= 3^2 \\ 2^2 + 3^2 + 6^2 &= 7^2 \\ 3^2 + 4^2 + 12^2 &= 13^2 \end{aligned}

(a) $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$

(b) $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

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

Q10

Find the cube roots of $27000$ and $10648$ .

Q11

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

Q12

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 a 2-digit number may be a 3-digit number.

(iv) The cube of a 2-digit number may have seven or more digits.

(v) Cube numbers have an odd number of factors.

Q13

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 $4913$ , $12167$ , and $32768$ .

Q14

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) $43^2 - 42^2$