Question

If the sum of the coefficients in the expansion of \( (a^2x^2 - 2ax + 1)^{51} \) is zero, then \( a \) is equal to

• A. \( 0 \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( -2 \)
• E. \( 2 \)

Hint:

To find sum of coefficients, always substitute \( x = 1 \).

Answer 1

The Correct Option is B

Concept: The sum of coefficients of a polynomial is obtained by substituting \( x = 1 \).

Step 1: Substitute \( x = 1 \).
\[ (a^2(1)^2 - 2a(1) + 1)^{51} = (a^2 - 2a + 1)^{51} \]

Step 2: Simplify expression.
\[ a^2 - 2a + 1 = (a-1)^2 \]

Step 3: Write full expression.
\[ [(a-1)^2]^{51} = (a-1)^{102} \]

Step 4: Use given condition.
Sum of coefficients = 0: \[ (a-1)^{102} = 0 \]

Step 5: Solve equation.
\[ a - 1 = 0 \Rightarrow a = 1 \]