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Proof, EXT2 P1 2024 HSC 2 MC

Consider the following statement written in the formal language of proof

\(\forall \theta \in\biggl(\dfrac{\pi}{2}, \pi\biggr) \exists\ \phi \in\biggl(\pi, \dfrac{3 \pi}{2}\biggr) ; \ \sin \theta=-\cos \phi\).

Which of the following best represents this statement?

  1. There exists a \(\theta\) in the second quadrant such that for all \(\phi\) in the third quadrant  \(\sin \theta=-\cos \phi\).
  2. There exists a \(\phi\) in the third quadrant such that for all \(\theta\) in the second quadrant  \(\sin \theta=-\cos \phi\).
  3. For all \(\theta\) in the second quadrant there exists a \(\phi\) in the third quadrant such that  \(\sin \theta=-\cos \phi\).
  4. For all \(\phi\) in the third quadrant there exists a \(\theta\) in the second quadrant such that  \(\sin \theta=-\cos \phi\).
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\(C\)

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\(\Rightarrow C\)

Proof, EXT2 P1 2023 HSC 4 MC

Consider the following statement about real numbers.

"Whichever positive number \(r\) you pick, it is possible to find a number \(x\) greater than 1 such that

\(\dfrac{\ln x}{x^3}<r\). "

When this statement is written in the formal language of proof, which of the following is obtained?

  1. \(\forall x>1 \quad \exists r>0 \quad \dfrac{\ln x}{x^3}<r\)
  2. \(\exists x>1 \quad \forall r>0 \quad \dfrac{\ln x}{x^3}<r\)
  3. \(\forall r>0 \quad \exists x>1 \quad \dfrac{\ln x}{x^3}<r\)
  4. \(\exists r>0 \quad \forall x>1 \quad \dfrac{\ln x}{x^3}<r\)
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\(C\)

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\(\text{Whichever positive number}\ r\ \text{you pick …}\ \forall r>0 \)

\(\text{It is possible to find a number}\ x\ \text{greater then 1 …}\ \exists x>1 \)

\(\text{Such that}\ \ \dfrac{\ln x}{x^3}<r\)

\(\Rightarrow C\)