Question

Assertion: (2x – y)³ = 8x³ + 12x²y + 6xy² – y³

Reason: (a – b)³ = a³ – 3a²b + 3ab² – b³

विकल्प

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

• Assertion (A) is true but Reason (R) is false.

• Assertion (A) is false but Reason (R) is true.

Answer 1

Assertion (A) is false but Reason (R) is true.

Explanation:

Step 1: Check the Reason (R) formula

The formula for the cube of a difference is:

(a – b)³ = a³ – 3a²b + 3ab² – b³

Reason (R) matches the known algebraic expansion correctly, so R is true.

Step 2: Check the Assertion (A)

Expand (2x – y)³ using the formula:

(2x – y)³ = (2x)³ – 3(2x)²(y) + 3(2x)(y)² – y³

Calculate each term:

• (2x)³ = 8x³
• 3(2x)²(y) = 3 × 4x² × y = 12x²y
• 3(2x)(y)² = 3 × 2x × y² = 6xy²
• –y³

Thus, (2x – y)³ = 8x³ – 12x²y + 6xy² – y³.

Compare with the assertion statement:

8x³ + 12x²y + 6xy² – y³

Note the sign of the second term in the assertion is positive +12x²y, but from expansion, it should be negative –12x²y.

Therefore, Assertion (A) is false.