Multiple choice

Directions: The following question has two statements, (1) and (2). Answer the question using the following options: Is n in the function f(x) = xn + 2x2 + 4 an even? (1) f(- x) = f(x). (2) The equation f(x) = 0 has more than 4 roots.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

  5. Statements (1) and (2) TOGETHER are NOT sufficient.
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A Correct answer
Explanation

f(x) = x^n + 2x^2 + 4. (1) f(-x) = f(x) implies (-x)^n + 2(-x)^2 + 4 = x^n + 2x^2 + 4. This simplifies to (-x)^n = x^n, which is true if n is even. (2) f(x)=0 having more than 4 roots is impossible for a polynomial of degree n unless it is the zero polynomial, which it is not. Statement 1 is sufficient.

AI explanation

For statement 1, if f(-x) equals f(x), then (-x)^n + 2(-x)^2 + 4 equals x^n + 2x^2 + 4, which means x^n is even and is sufficient. For statement 2, the equation f(x) = 0 having more than 4 roots means x^n + 2x^2 + 4 = 0 has many solutions, but since this condition applies to both even and odd polynomials of high degrees, it is not sufficient to determine if n is even. Therefore, statement 1 alone is sufficient, but statement 2 alone is not sufficient.

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