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Section 7 Multiple Choice

While free-response questions are generally preferred, sometimes the nature of a question lends itself to multiple choice.

Note also that the solution to this problem uses an external link.

The number \(\sqrt{2}\)

  • is

  • is not

rational.

Answer

\(\text{is not}\)

Solution

If \(\sqrt{2}\) were rational, then \(\sqrt{2}=\frac{p}{q}\text{,}\) with \(p\) and \(q\) coprime. But then \(2q^2=p^2\text{.}\) By the Fundamental Theorem of Arithmetic, the power of \(2\) dividing the left side is odd, while the power of \(2\) dividing the right side is even. This is a contradiction, so \(\sqrt{2}\) is not rational.

Which of the following suggest that differentiation and integration are inverse processes?

  • The Quadratic Formula

  • The Fundamental Theorem of Calculus

  • The Fundamental Theorem of Arithmetic

  • None of these

Answer

\(\text{The Fundamental ... of Calculus}\)

Solution

The correct answer is The Fundamental ... of Calculus.