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

Concept: An algebraic identity is an equation that remains true for all permissible values of the variables involved. Such identities are frequently used in algebra to simplify expressions and solve equations efficiently. To determine whether the given statements are true, we shall verify each one separately through direct algebraic expansion.

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, \[ \boxed{\text{Statements (i), (ii) and (iii) are all true}} \] Therefore, the correct answer is: \[ \boxed{\text{Option (D)}} \]