ISAT Mathematical Problem-Solving 4 — Questions and Answers
Question 1: A function is defined as g(x) = (x² − 9)/(x − 3). What is the value of lim(x→3) g(x)?
- 6 (Correct answer)
- 0
- undefined
- 9
Correct answer: 6
Factor the numerator: (x−3)(x+3)/(x−3) = x+3; as x→3, the limit is 3+3 = 6.
Question 2: How many different 4-letter arrangements can be made from the letters of MATH if no letter is repeated?
- 24 (Correct answer)
- 12
- 16
- 256
Correct answer: 24
This is a permutation: P(4,4) = 4! = 24.
Question 3: The profit P (in dollars) from selling x items is P(x) = −2x² + 40x − 100. How many items should be sold to maximize profit?
- 10 (Correct answer)
- 20
- 5
- 40
Correct answer: 10
The vertex is at x = −40/(2×−2) = 40/4 = 10.
Question 4: If sin θ = 3/5 and θ is in the first quadrant, what is tan θ?
- 3/4 (Correct answer)
- 4/5
- 5/3
- 4/3
Correct answer: 3/4
With sin θ = 3/5, cos θ = 4/5 (Pythagorean theorem), so tan θ = sin/cos = 3/4.
Question 5: A sequence is defined by aₙ = 3n − 1. What is the sum of the first 10 terms?
- 140 (Correct answer)
- 145
- 130
- 150
Correct answer: 140
Sum = Σ(3n−1) from n=1 to 10 = 3(55) − 10 = 165 − 25 = 140.
Question 6: A bag has 4 green, 3 yellow, and 2 purple chips. What is the probability of NOT drawing a yellow chip on one draw?
- 2/3 (Correct answer)
- 1/3
- 3/9
- 2/9
Correct answer: 2/3
P(not yellow) = 1 − 3/9 = 6/9 = 2/3.
Question 7: What is the solution set of 2x² − 8 ≥ 0?
- x ≤ −2 or x ≥ 2 (Correct answer)
- x ≥ 2
- −2 ≤ x ≤ 2
- x ≤ 2
Correct answer: x ≤ −2 or x ≥ 2
2(x²−4) ≥ 0 → x² ≥ 4 → x ≤ −2 or x ≥ 2.
ISAT Mathematical Problem-Solving 4
A function is defined as g(x) = (x² − 9)/(x − 3).
What is the value of lim(x→3) g(x)?