Question

Assertion (A): If the first number \( k = 5 \) and the second number \( kh = 10 \), then the square of their sum is 225.
Reason (R): This is derived from the algebraic identity: \( (k + kh)^2 = k^2 + 2k \cdot kh + kh^2 \).

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is NOT the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Always check if the math adds up. $15 \times 15 = 225$. The formula $(a+b)^2$ is the standard way to explain how to square a sum of two parts.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Indian mathematics (Bija-ganita) utilized algebraic identities extensively to simplify astronomical calculations. The square of a sum of two numbers is one of the most basic rules, often taught in the Lilavati of Bhaskaracharya.

Step 2: Detailed Explanation of Assertion (A):
Given values:
- \( k = 5 \)
- \( kh = 10 \)
Sum of the numbers: \( k + kh = 5 + 10 = 15 \).
Square of the sum: \( 15^2 = 225 \).
The assertion correctly states the numerical result.

Step 3: Detailed Explanation of Reason (R):
The reason provides the expansion of the binomial:
\[ (k + kh)^2 = k^2 + 2(k)(kh) + kh^2 \]
Substituting the values:
\[ (5 + 10)^2 = 5^2 + 2(5)(10) + 10^2 \]
\[ (5 + 10)^2 = 25 + 100 + 100 \]
\[ (5 + 10)^2 = 225 \]
This algebraic formula is the fundamental method used to derive the result mentioned in the assertion. In Indian texts, this is referred to as 'Varga-parikarma'.

Step 4: Final Answer:
Both are correct and (R) is the correct explanation of (A).