ATAR Mathematical Reasoning 5 — Questions and Answers
Question 1: Solve: |2x − 5| = 7. What are the solutions?
- x = 6 or x = −1 (Correct answer)
- x = 6 or x = 1
- x = −6 or x = 1
- x = 6 only
Correct answer: x = 6 or x = −1
2x − 5 = 7 gives x = 6, and 2x − 5 = −7 gives x = −1.
Question 2: The graph of y = f(x) is shifted 3 units left and 2 units down. What is the equation of the new graph?
- y = f(x + 3) − 2 (Correct answer)
- y = f(x − 3) − 2
- y = f(x + 3) + 2
- y = f(x − 3) + 2
Correct answer: y = f(x + 3) − 2
A left shift of 3 replaces x with (x+3), and a downward shift of 2 subtracts 2, giving y = f(x+3) − 2.
Question 3: A right-angled triangle has legs of length a and b, and hypotenuse c. If a = 5 and c = 13, what is the value of b?
- 12 (Correct answer)
- 8
- 10
- 11
Correct answer: 12
b = √(c² − a²) = √(169 − 25) = √144 = 12.
Question 4: A function is defined as f(x) = (x² − 4)/(x − 2) for x ≠ 2. What value does f(x) approach as x → 2?
- 4 (Correct answer)
- 2
- 0
- undefined
Correct answer: 4
(x² − 4)/(x − 2) = (x+2)(x−2)/(x−2) = x + 2, which approaches 4 as x → 2.
Question 5: In a class of 30 students, 18 study Mathematics, 15 study Physics, and 10 study both. How many study neither?
- 7 (Correct answer)
- 8
- 5
- 3
Correct answer: 7
By inclusion-exclusion: |M ∪ P| = 18 + 15 − 10 = 23; neither = 30 − 23 = 7.
Question 6: If 8^x = 2^(x+6), what is the value of x?
- 3 (Correct answer)
- 2
- 4
- 6
Correct answer: 3
8^x = (2³)^x = 2^(3x); setting 3x = x + 6 gives 2x = 6, so x = 3.
Question 7: A box contains 4 defective and 6 non-defective items. Three items are selected at random. What is the probability that exactly 1 is defective?
- 1/2 (Correct answer)
- 3/10
- 2/5
- 1/3
Correct answer: 1/2
P = C(4,1)×C(6,2)/C(10,3) = 4×15/120 = 60/120 = 1/2.
ATAR Mathematical Reasoning 5
Solve: |2x − 5| = 7.
What are the solutions?