Which of the following statements is FALSE?
A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2021 - Paper 1
Question 5
Which of the following statements is FALSE?
A. ∀ a, b ∈ ℝ,
a < b ⇒ a³ < b³
B. ∀ a, b ∈ ℝ,
a < b ⇒ eᵃ > eᵇ
C. ∀ a, b ∈ (0, +∞),
a < b ⇒ ln a < ln b
... show full transcript
Worked Solution & Example Answer:Which of the following statements is FALSE?
A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2021 - Paper 1
Step 1
A. ∀ a, b ∈ ℝ,
a < b ⇒ a³ < b³
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
This statement is true. If a is less than b, raising both sides to an odd power (like 3) preserves the inequality.
Step 2
B. ∀ a, b ∈ ℝ,
a < b ⇒ eᵃ > eᵇ
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
This statement is true. The exponential function is increasing, so if a < b, then e raised to a will be less than e raised to b.
Step 3
C. ∀ a, b ∈ (0, +∞),
a < b ⇒ ln a < ln b
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
This statement is true. The natural logarithm is also an increasing function, ensuring that if a < b, ln a will be less than ln b.
Step 4
D. ∀ a, b ∈ ℝ, with a, b ≠ 0,
a < b ⇒ \frac{1}{a} > \frac{1}{b}
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
This statement is FALSE. When a and b are both positive, if a < b, then \frac{1}{a} > \frac{1}{b} is true. However, for negative values of a and b, the reverse inequality holds, making this statement false.
