ISAT Mathematical Problem-Solving 4

Question 1

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

Accepted Answer

6

Answer

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?

Accepted Answer

24

Answer

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?

Accepted Answer

10

Answer

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?

Accepted Answer

3/4

Answer

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?

Accepted Answer

140

Answer

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

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