NYSTCE NYSTCE Mathematics (004) 4 — Questions and Answers
Question 1: A student claims that √(a² + b²) = a + b for all positive values of a and b. Which counterexample best disproves this claim?
- a = 3, b = 4: √(9+16) = 5 ≠ 7 (Correct answer)
- a = 1, b = 1: √(1+1) = √2 ≠ 1
- a = 0, b = 5: √(0+25) = 5 = 5
- a = 2, b = 2: √(4+4) = 2√2 ≠ 2
Correct answer: a = 3, b = 4: √(9+16) = 5 ≠ 7
With a=3, b=4, √(9+16)=5 but a+b=7, clearly disproving the equality.
Question 2: Which of the following best describes a function that is onto (surjective)?
- Every input maps to a unique output
- Every element in the codomain has at least one preimage (Correct answer)
- The function passes the vertical line test
- No two inputs share the same output
Correct answer: Every element in the codomain has at least one preimage
A surjective function maps onto its entire codomain, meaning every output value is achieved by at least one input.
Question 3: A jar contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?
- 5/14 (Correct answer)
- 25/64
- 10/56
- 5/16
Correct answer: 5/14
P(both red) = (5/8) × (4/7) = 20/56 = 5/14.
Question 4: Which transformation maps the graph of y = x² to the graph of y = (x − 3)² + 2?
- Shift left 3 units and up 2 units
- Shift right 3 units and up 2 units (Correct answer)
- Shift right 3 units and down 2 units
- Shift left 3 units and down 2 units
Correct answer: Shift right 3 units and up 2 units
Replacing x with (x−3) shifts right 3 units; adding 2 outside shifts up 2 units.
Question 5: A rectangle has a perimeter of 48 cm and an area of 128 cm². What are the dimensions?
- 8 cm × 16 cm (Correct answer)
- 10 cm × 14 cm
- 12 cm × 12 cm
- 6 cm × 18 cm
Correct answer: 8 cm × 16 cm
Solving 2(l+w)=48 and l×w=128 gives l=16 and w=8.
Question 6: In a normal distribution, approximately what percentage of data falls within two standard deviations of the mean?
- 68%
- 90%
- 95% (Correct answer)
- 99.7%
Correct answer: 95%
The empirical rule states that about 95% of data lies within ±2 standard deviations of the mean.
Question 7: Which of the following is equivalent to log₂(32)?
- 4
- 5 (Correct answer)
- 6
- 16
Correct answer: 5
log₂(32) = log₂(2⁵) = 5.
NYSTCE NYSTCE Mathematics (004) 4
A student claims that √(a² + b²) = a + b for all positive values of a and b.
Which counterexample best disproves this claim?