Question

Which of the following is/are true?
• [(i)] $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
• [(ii)] $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
• [(iii)] $(x + a)(x + b) = x^2 + (a + b)x + ab$

• A. Both (i) and (ii)
• B. (iii) only
• C. Both (i) and (iii)
• D. (i), (ii) and (iii)

Hint:

Some of the most important algebraic identities to remember are: \[ (a+b)^2=a^2+2ab+b^2 \] \[ (a-b)^2=a^2-2ab+b^2 \] \[ (a+b)^3=a^3+3a^2b+3ab^2+b^3 \] \[ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca \] Memorizing these standard identities saves significant time in algebraic calculations and competitive examinations.

Answer 1

The Correct Option is D

Step 1: Verification of Statement (i)
The given statement is: \[ (a+b)^3=a^3+3a^2b+3ab^2+b^3 \] Let us expand the left-hand side. \[ (a+b)^3=(a+b)(a+b)^2 \] Using the well-known identity: \[ (a+b)^2=a^2+2ab+b^2 \] Substituting: \[ (a+b)^3=(a+b)(a^2+2ab+b^2) \] Applying distributive multiplication: \[ =(a)(a^2+2ab+b^2)+b(a^2+2ab+b^2) \] \[ =a^3+2a^2b+ab^2+a^2b+2ab^2+b^3 \] Combining like terms: \[ =a^3+(2a^2b+a^2b)+(ab^2+2ab^2)+b^3 \] \[ =a^3+3a^2b+3ab^2+b^3 \] This matches exactly with the right-hand side. Therefore, statement (i) is true.

Step 2: Verification of Statement (ii)
The given statement is: \[ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca \] Expanding directly: \[ (a+b+c)^2=(a+b+c)(a+b+c) \] Multiplying term by term: \[ =a(a+b+c)+b(a+b+c)+c(a+b+c) \] \[ =a^2+ab+ac+ab+b^2+bc+ac+bc+c^2 \] Grouping similar terms: \[ =a^2+b^2+c^2+2ab+2ac+2bc \] Rearranging: \[ =a^2+b^2+c^2+2ab+2bc+2ca \] This is exactly the expression given. Hence, statement (ii) is true.

Step 3: Verification of Statement (iii)
The given statement is: \[ (x+a)(x+b)=x^2+(a+b)x+ab \] Expanding the left-hand side: \[ (x+a)(x+b) \] \[ =x(x+b)+a(x+b) \] \[ =x^2+bx+ax+ab \] Combining the middle terms: \[ =x^2+(a+b)x+ab \] This is identical to the right-hand side. Therefore, statement (iii) is also true.

Step 4: Final Conclusion
All three statements have been verified independently and found to be correct algebraic identities. Hence, \[ {\text{Statements (i), (ii) and (iii) are all true}} \] Therefore, the correct answer is: \[ {\text{Option (D)}} \]