MTEL Mathematics (Secondary) (63)

1. Question 1 · Number Sense and Operations

   A store buys a jacket for $60 and marks the price up 40%. During a sale the marked price is reduced by 20%, and at the register a loyalty coupon takes an additional 10% off the sale price. Compared with the store's cost, what is the result of selling the jacket with both discounts?
   • A.A profit of $1.20
   • B.A loss of $1.20
   • C.A profit of $6.00
   • D.A profit of $0.48
   Show answer and explanation

   Answer: D. A profit of $0.48

   Successive percent changes multiply: 60 × 1.40 × 0.80 × 0.90 = 60.48, so the store gains 60.48 − 60 = $0.48. Adding and subtracting the percents (40 − 20 − 10 = 10%) wrongly gives a $6.00 profit, and treating the two discounts as one 30% discount gives 60 × 1.40 × 0.70 = 58.80, a $1.20 loss.

2. Question 2 · Relations, Functions, and Algebra

   Let f(x) = sqrt(x − 2) and g(x) = 1/(x − 5). What is the domain of f∘g, where (f∘g)(x) = f(g(x))?
   • A.x ≥ 2 and x ≠ 5
   • B.5 ≤ x ≤ 5.5
   • C.x > 5
   • D.5 < x ≤ 5.5
   Show answer and explanation

   Answer: D. 5 < x ≤ 5.5

   We need x ≠ 5 and g(x) ≥ 2, that is 1/(x − 5) ≥ 2; this forces x − 5 > 0 and then x − 5 ≤ 1/2, so 5 < x ≤ 5.5. Combining the separate domains of f and g gives x ≥ 2 and x ≠ 5, which ignores that the output of g must lie in the domain of f.

3. Question 3 · Relations, Functions, and Algebra

   What is the solution set of (x + 2)(x - 1)^2(x - 4) < 0?
   • A.(-2, 4)
   • B.(-2, 1) ∪ (4, ∞)
   • C.(-∞, -2) ∪ (4, ∞)
   • D.(-2, 1) ∪ (1, 4)
   Show answer and explanation

   Answer: D. (-2, 1) ∪ (1, 4)

   The squared factor is never negative and equals 0 at x = 1, so the sign is set by (x + 2)(x - 4), which is negative on (-2, 4). Because the inequality is strict, x = 1 (where the product is 0) must be removed. Treating x = 1 as a sign change gives the alternating-interval answer. x: 0, 1, 2, 3, 4, 5 y: 1, 2, 15, 52, 125, 246

4. Question 4 · Geometry and Measurement

   From a point P outside a circle, segment PT is tangent to the circle at T and PT = 12. A secant from P passes through the center of the circle and meets the circle first at A and then at B, with PA = 8. What is the radius of the circle?
   • A.4
   • B.10
   • C.9
   • D.5
   Show answer and explanation

   Answer: D. 5

   By the tangent-secant theorem, PT^2 = PA · PB, so 144 = 8 · PB and PB = 18. The chord AB is a diameter of length 18 − 8, or 10, so the radius = 10/2 = 5. Reporting 10 stops at the diameter.

5. Question 5 · Probability, Statistics, Calculus, and Discrete Mathematics

   Two friends each arrive at a café at a random time, uniformly and independently, between 1:00 and 2:00. Each waits 15 minutes for the other and then leaves. What is the probability that they meet?
   • A.7/16
   • B.1/4
   • C.1/2
   • D.9/16
   Show answer and explanation

   Answer: A. 7/16

   Model the arrival times as a point (x, y) in a 60-by-60 square; they meet when |x − y| ≤ 15. The two corner triangles where they miss each have legs of 45, so the probability is 1 − (45/60)^2 = 1 − 9/16 = 7/16. The value 9/16 is the probability that they miss, and 1/4 = 15/60 ignores the two-dimensional model.

6. Question 6 · Number Sense and Operations

   A store sets prices so that its profit on each item is 30% of the item's selling price. What percent markup on cost does this policy represent, to the nearest tenth of a percent?
   • A.30.0%
   • B.70.0%
   • C.23.1%
   • D.42.9%
   Show answer and explanation

   Answer: D. 42.9%

   If the selling price is S, the profit is 0.3S and the cost is 0.7S, so the markup on cost is 0.3S/0.7S ≈ 0.429 = 42.9%. Answering 30% confuses a markup based on selling price with one based on cost, and 23.1% = 0.3/1.3 reverses the relationship.

7. Question 7 · Relations, Functions, and Algebra

   A ball is dropped from a height of 12 feet. Each time it hits the ground it rebounds to 3/4 of its previous height. Assuming it bounces indefinitely, what total vertical distance does the ball travel?
   • A.84 feet
   • B.96 feet
   • C.72 feet
   • D.48 feet
   Show answer and explanation

   Answer: A. 84 feet

   The ball falls 12 feet, then each rebound height is traveled twice (up and down): 12 + 2(9 + 6.75 + ...) = 12 + 2·9/(1 − 3/4) = 12 + 72 = 84 feet. Using 12/(1 − 3/4) = 48 counts each rebound only once, and doubling 48 counts the initial drop twice.

8. Question 8 · Relations, Functions, and Algebra

   If (5x + 1)/((x - 1)(x + 2)) = A/(x - 1) + B/(x + 2) for all x except 1 and -2, what are A and B?
   • A.A = -2, B = 7
   • B.A = 3, B = 2
   • C.A = 2, B = -3
   • D.A = 2, B = 3
   Show answer and explanation

   Answer: D. A = 2, B = 3

   Clear denominators: 5x + 1 = A(x + 2) + B(x - 1). At x = 1, 6 = 3A, so A = 2; at x = -2, -9 = -3B, so B = 3. Substituting each root into the wrong factor swaps the values.

9. Question 9 · Geometry and Measurement

   Quadrilateral ABCD is inscribed in a circle, with vertices in that order. The angle measures are ∠A = (3x + 10)°, ∠B = (4x − 5)°, and ∠C = (2x + 20)°. What is the measure of ∠D?
   • A.100°
   • B.115°
   • C.65°
   • D.80°
   Show answer and explanation

   Answer: C. 65°

   Opposite angles of an inscribed quadrilateral are supplementary, so (3x + 10) + (2x + 20) = 180, giving x = 30. Then ∠B = 115°, and ∠D = 180° − 115° = 65°. Reporting 115° gives ∠B itself, and 100° and 80° are ∠A and ∠C.

10. Question 10 · Probability, Statistics, Calculus, and Discrete Mathematics

    In a promotion, 35% of cereal boxes contain a prize. A teacher wants to use a table of random digits to simulate opening one box. Which assignment gives a valid simulation?
    • A.Read two digits; 00–35 means a prize and 36–99 means no prize.
    • B.Read one digit; 0–3 means a prize and 4–9 means no prize.
    • C.Read two digits; 01–34 means a prize and 35–99 means no prize.
    • D.Read two digits; 00–34 means a prize and 35–99 means no prize.
    Show answer and explanation

    Answer: D. Read two digits; 00–34 means a prize and 35–99 means no prize.

    The 100 equally likely pairs 00–99 must be split so that exactly 35 of them mean a prize, and 00–34 contains 35 pairs. One digit with 0–3 gives 40%, 00–35 contains 36 pairs (36%), and 01–34 contains only 34 pairs.

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