Quiz about X2 - Proof

Question 1

What is the negation of the statement:

If a triangle has two equal sides, then it has two equal angles.

Accepted Answer

A triangle has two equal sides and it does not have two equal angles.

Answer

If a triangle does not have two equal sides, then it does not have two equal angles.

Answer

If a triangle does not have two equal angles, then it does not have two equal sides.

Answer

A triangle has two equal angles and it does not have two equal sides.

Question 2

Question 2.

Accepted Answer

Line 4

Answer

Line 1

Answer

Line 2

Answer

Line 3

Question 3

Let $$A,\ B,\ P$$ be three points in three-dimensional space with $$A e B.$$

Consider the following statement.

"If $$P$$ is on the line $$AB$$ , then there exists a real number $$\lambda$$ such that $$\overrightarrow{AP}=\lambda\overrightarrow{AB}.$$ "

Which of the following is the contrapositive of this statement?

Accepted Answer

If for all real numbers $$\lambda,\ \ \overrightarrow{AP} e\lambda\overrightarrow{AB},$$ then $$P$$ is not on the line $$AB.$$

Answer

If for all real numbers $$\lambda,\ \ \overrightarrow{AP}=\lambda\overrightarrow{AB},$$ then $$P$$ is on the line $$AB.$$

Answer

If there exists a real number $$\lambda,$$ such that $$\ \overrightarrow{AP}=\lambda\overrightarrow{AB},$$ then $$P$$ is on the line $$AB.$$

Answer

If there exists a real number $$\lambda,$$ such that $$\ \overrightarrow{AP} e\lambda\overrightarrow{AB},$$ then $$P$$ is not on the line $$AB.$$

Question 4

Consider the statement P.

P : For all integers $$n\ge1,$$ if $$n$$ is a real number then $$\frac{n\left(n+1\right)}{2}$$ is a prime number.

Which of the following is true about this statement and its converse?

Accepted Answer

The statement P is false and its converse is true.

Answer

The statement P and its converse are both true.

Answer

The statement P and its converse are both false.

Answer

The statement P is true and its converse is false.

Question 5

Which of the following statements is FALSE?

Accepted Answer

$$\forall\ a,\ b\in R,$$ with $$a,\ b e0,$$ $$a\frac{1}{b}$$

Answer

$$\forall\ a,\ b\in R,$$ $$a

Answer

$$\forall\ a,\ b\in R,$$ $$ae^{-b}$$

Answer

$$\forall\ a,\ b\in\left(0,\ +\infty\right),$$ $$a

Question 6

Consider the proposition:

'If $$2^n-1$$ is not prime, then $$n$$ is not prime.'

Given that each of the following statements is true, which statement disproves the proposition?

Accepted Answer

$$2^{11}-1$$ is divisible by 23

Answer

$$2^5-1$$ is prime

Answer

$$2^6-1$$ is divisible by 9

Answer

$$2^7-1$$ is prime

Question 7

Assume that $$a$$ and $$b$$ are positive real numbers with $$a>b.$$

Which of the following might be false?

Accepted Answer

$$2a>3b$$

Answer

$$\frac{1}{a-b}>0$$

Answer

$$\frac{a}{b}-\frac{b}{a}>0$$

Answer

$$a+b>2b$$

Question 8

Which of the following is the contrapositive of the following statement?

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$x^n+y^n e z^n$$ , then $$n\ge3.$$

Accepted Answer

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n<3,$$ then $$x^n+y^n=z^n$$ .

Answer

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n\ge3,$$ then $$x^n+y^n e z^n$$ .

Answer

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n<3,$$ then $$x^n+y^n e z^n$$ .

Answer

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n\ge3,$$ then $$x^n+y^n=z^n$$ .

Question 9

Question 9.

Accepted Answer

Option C is correct

Answer

A.

Answer

B.

Answer

D.

Question 10

Consider the statement:

"If Bob eats an apple every day, then he does not have to visit a doctor."

What is the converse of this statement?

Accepted Answer

If Bob does not have to visit a doctor, then he eats an apple every day.

Answer

If Bob does not eat an apple every day, then he has to visit a doctor.

Answer

If Bob eats an apple every day, then he has to visit a doctor.

Answer

If Bob has to visit a doctor, then he does not eat an apple every day.

Question 11

Question 11.

Accepted Answer

If the product of three integers is two more than a multiple of 3, then at least one of the three integers must not be one more than a multiple of 3.

Answer

If the product of three integers is one more than a multiple of 3, then the three integers must all be one more than a multiple of 3.

Answer

If the product of three integers is one more than a multiple of 3, then at least one of the three integers must be one more than a multiple of 3.

Answer

If the product of three integers is two more than a multiple of 3, then none of the three integers can be one more than a multiple of 3.

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