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Basic Mathematics

# Counter-Examples

If $$n$$ is a prime integer less than or equal to $$10,$$ then $$n^2+2$$ is prime.

How many counter-examples are there to the above claim?

(A) $$\ 0$$
(B) $$\ 1$$
(C) $$\ 2$$
(D) $$\ 3$$
(E) $$\ 4$$

A real number $$x$$ is called nearly-even, if there is an even integer $$n$$ such that

$| x - n | < 0.5.$

If $$X$$ is a nearly-even number, which of the following statements must be true?

$$\begin{array}{r r l} & \text{I.}\ & X\ \text{is an even integer.}\\ & \text{II.}\ & 6X\ \text{is an integer.}\\ & \text{III.}\ & \text{The integer part of}\ 6X\ \text{is even.}\\ \end{array}$$

(A) $$\ \$$ II only
(B) $$\ \$$ I and II only
(C) $$\ \$$ I and III only
(D) $$\ \$$ II and III only
(E) $$\ \$$ None of the statements

If $$n$$ is prime, then $$2n+1$$ is also prime.

Which of the following is a counter-example of the above claim?

(A) $$\ 1$$
(B) $$\ 2$$
(C) $$\ 3$$
(D) $$\ 5$$
(E) $$\ 7$$

A number is called 4-average if it is the average of four positive integers. If $$k$$ is a 4-average number, which of the following statements is true?

$$\begin{array}{r r l} &\text{I.}\ & k\ \text{is an integer.}\\ &\text{II.}\ & 4k\ \text{is an integer.}\\ &\text{III.}\ & k\ \text{is positive.}\\ \end{array}$$

(A)$$\ \$$ II only
(B)$$\ \$$ III only
(C)$$\ \$$ II and III only
(D)$$\ \$$ I, II, and III
(E)$$\ \$$ None of the statements

If $$f\left(f(x)\right)=x,$$ then $$f(x)=x.$$

Which of the following is a counter-example of the above claim?

(A) $$\ f(x)=1, x \in \mathbb{R}$$
(B) $$\ f(x)=x, x \in \mathbb{R}$$
(C) $$\ f(x)=\frac{1}{x}, x > 0$$
(D) $$\ f(x)=x^2, x > 0$$
(E) $$\ f(x)=\sqrt{x}, x > 0$$

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