Basic Mathematics

Reasoning Skills

Counter-Examples

         

If nn is a prime integer less than or equal to 10,10, then n2+2n^2+2 is prime.

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

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

A real number xx is called nearly-even, if there is an even integer nn such that

xn<0.5. | x - n | < 0.5.

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

I. X is an even integer.II. 6X is an integer.III. The integer part of 6X is even.\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 nn is prime, then 2n+12n+1 is also prime.

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

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

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

I. k is an integer.II. 4k is an integer.III. k is positive.\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(f(x))=x,f\left(f(x)\right)=x, then f(x)=x.f(x)=x.

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

(A)  f(x)=1,xR\ f(x)=1, x \in \mathbb{R}
(B)  f(x)=x,xR\ f(x)=x, x \in \mathbb{R}
(C)  f(x)=1x,x>0\ f(x)=\frac{1}{x}, x > 0
(D)  f(x)=x2,x>0\ f(x)=x^2, x > 0
(E)  f(x)=x,x>0\ f(x)=\sqrt{x}, x > 0

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