Mathematical Problem-Solving Flashcards

A function is defined as g(x) = (x² − 9)/(x − 3). What is the value of lim(x→3) g(x)?

Answer: 6

Factor the numerator: (x−3)(x+3)/(x−3) = x+3; as x→3, the limit is 3+3 = 6.

How many different 4-letter arrangements can be made from the letters of MATH if no letter is repeated?

Answer: 24

This is a permutation: P(4,4) = 4! = 24.

The profit P (in dollars) from selling x items is P(x) = −2x² + 40x − 100. How many items should be sold to maximize profit?

Answer: 10

The vertex is at x = −40/(2×−2) = 40/4 = 10.

If sin θ = 3/5 and θ is in the first quadrant, what is tan θ?

Answer: 3/4

With sin θ = 3/5, cos θ = 4/5 (Pythagorean theorem), so tan θ = sin/cos = 3/4.

A sequence is defined by aₙ = 3n − 1. What is the sum of the first 10 terms?

Answer: 140

Sum = Σ(3n−1) from n=1 to 10 = 3(55) − 10 = 165 − 25 = 140.

A bag has 4 green, 3 yellow, and 2 purple chips. What is the probability of NOT drawing a yellow chip on one draw?

Answer: 2/3

P(not yellow) = 1 − 3/9 = 6/9 = 2/3.

What is the solution set of 2x² − 8 ≥ 0?

Answer: x ≤ −2 or x ≥ 2

2(x²−4) ≥ 0 → x² ≥ 4 → x ≤ −2 or x ≥ 2.