Let A, B, P be three points in three-dimensional space with A ≠ B - HSC - SSCE Mathematics Extension 2 - Question 3 - 2022 - Paper 1
Question 3
Let A, B, P be three points in three-dimensional space with A ≠ B.
Consider the following statement.
If P is on the line AB, then there exists a real number λ such... show full transcript
Worked Solution & Example Answer:Let A, B, P be three points in three-dimensional space with A ≠ B - HSC - SSCE Mathematics Extension 2 - Question 3 - 2022 - Paper 1
Step 1
Identify the statement and its contrapositive
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The original statement can be identified as 'If P is on the line AB, then there exists a real number λ such that ar{AP} = λ ar{AB}.' The contrapositive of a statement 'If A then B' is 'If not B then not A.' Therefore, the contrapositive of this statement will be 'If there does not exist a real number λ such that ar{AP} = λ ar{AB}, then P is not on the line AB.'
Step 2
Select the correct option
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
When analyzing the provided options, option B states 'If for all real numbers λ, ar{AP} ≠ λ ar{AB}, then P is not on the line AB.' This accurately reflects the contrapositive form of the original statement, confirming that the correct answer is B.
