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

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

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

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

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

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?

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

Consider the statement: ‘If n is even, then if n is a multiple of 3, then n is a multiple of 6’. Which of the following is the negation of this statement?

n is even and n is a multiple of 3 but not a multiple of 6.

n is odd and n is not a multiple of 3 or 6.

If n is even, then n is not a multiple of 3 and n is not a multiple of 6.

If n is odd, then if n is not a multiple of 3 then n is not a multiple of 6.

Which of the following is the contrapositive of the following statement? $$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$x^n+y^n\ne z^n$$ , then $$n\ge3.$$

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n

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

$$\forall\ x,\ y,\ z,\ n\ \in N$$ : if $$n

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

What is the converse of the following statement? “ If a point is in the first quadrant, then its coordinates are positive ”.

If the coordinates of a point are positive, then the point is in the first quadrant.

If a point is not in the first quadrant, then its coordinates are positive.

If a point is not in the first quadrant, then its coordinates are not positive.

If the coordinates of a point are not positive, then the point is not in the first quadrant.

Consider the statement: “ If my computer is not working, then I cannot finish my homework ”. What is the negation of the above statement?

My computer is not working, and I can finish my homework.

My computer is working, and I can finish my homework.

My computer is working, but I cannot finish my homework.

My computer is not working, but I cannot finish my homework.

What is the contrapositive of the following statement? “If 𝑥 is even, then 5𝑥 − 11 is odd.”

If 5𝑥 − 11 is even, then 𝑥 is odd.

If 𝑥 is odd, then 5𝑥 − 11 is even.

If 𝑥 is odd, then 5𝑥 − 11 is odd.

If 5𝑥 − 11 is odd, then 𝑥 is even.

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?

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

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

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

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

X2 - Proof

Quiz

•

Mathematics

•

12th Grade

•

Practice Problem

•

Medium

•

2022 HSC

Standards-aligned

Grant Rawson

Used 3+ times

FREE Resource

16 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

What is the negation of the statement:
If a triangle has two equal sides, then it has two equal angles.

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

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

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

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

2.

MULTIPLE CHOICE QUESTION

2 mins • 2 pts

Line 1

Line 2

Line 3

Line 4

Answer explanation

From 2022 HSC paper

4.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

The statement P and its converse are both true.

The statement P and its converse are both false.

The statement P is true and its converse is false.

The statement P is false and its converse is true.

Answer explanation

From 2022 HSC paper

5.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Consider the statement:
‘For all integers n, if n is a multiple of 6, then n is a multiple of 2’.
Which of the following is the contrapositive of the statement?

There exists an integer n such that n is a multiple of 6 and not a multiple of 2.

There exists an integer n such that n is a multiple of 2 and not a multiple of 6.

For all integers n, if n is not a multiple of 2, then n is not a multiple of 6.

For all integers n, if n is not a multiple of 6, then n is not a multiple of 2.

Answer explanation

From 2021 HSC paper

6.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Which of the following statements is FALSE?

Answer explanation

From 2021 HSC

7.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Answer explanation

From 2020 HSC paper.

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X2 - Proof

16 questions

What is the negation of the statement: If a triangle has two equal sides, then it has two equal angles.

From 2022 HSC paper

From 2022 HSC paper

From 2022 HSC paper

Consider the statement: ‘For all integers n, if n is a multiple of 6, then n is a multiple of 2’.Which of the following is the contrapositive of the statement?

From 2021 HSC paper

Which of the following statements is FALSE?

From 2021 HSC

From 2020 HSC paper.

Consider the statement: ‘If n is even, then if n is a multiple of 3, then n is a multiple of 6’.Which of the following is the negation of this statement?

2020 HSC Q8

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Discover more resources for Mathematics

What is the negation of the statement:
If a triangle has two equal sides, then it has two equal angles.

Consider the statement:
‘For all integers n, if n is a multiple of 6, then n is a multiple of 2’.
Which of the following is the contrapositive of the statement?

Consider the statement:
‘If n is even, then if n is a multiple of 3, then n is a multiple of 6’.
Which of the following is the negation of this statement?