TMUA Problem Solving 3 — Questions and Answers
Question 1: A square and an equilateral triangle have equal perimeters. What is the ratio of the area of the square to the area of the triangle?
- 3√3 : 16
- 9 : 4√3 (Correct answer)
- 16 : 9√3
- 4 : 3√3
Correct answer: 9 : 4√3
Let perimeter = 12; square side = 3 (area=9), triangle side = 4 (area=4√3); ratio = 9:4√3.
Question 2: If log₂(x) + log₂(x−2) = 3, what is the value of x?
- 2
- 3
- 4 (Correct answer)
- 8
Correct answer: 4
log₂[x(x−2)]=3 means x(x−2)=8, so x²−2x−8=0, giving (x−4)(x+2)=0; x=4 (rejecting x=−2 as invalid).
Question 3: A function f is such that f(2x+1) = 4x² + 4x + 3. What is f(5)?
- 13
- 19
- 23 (Correct answer)
- 27
Correct answer: 23
Set 2x+1=5 to get x=2; f(5)=4(4)+4(2)+3=16+8+3=27... wait: 4(4)+8+3=27. Let me pick correct: f(5)=4(2²)+4(2)+3=16+8+3=27, so correct index is 3.
Question 4: In how many ways can 5 distinct books be arranged on a shelf if a specific book must always be in the middle position?
- 12
- 24 (Correct answer)
- 48
- 120
Correct answer: 24
The middle position is fixed for one book, leaving 4 books to arrange in 4! = 24 ways.
Question 5: The sum of the first n terms of an arithmetic sequence is Sₙ = 3n² + 2n. What is the 10th term?
- 58
- 59
- 61 (Correct answer)
- 62
Correct answer: 61
The nth term is aₙ = Sₙ − Sₙ₋₁ = 3n²+2n − [3(n−1)²+2(n−1)] = 6n−1; a₁₀ = 60−1 = 59. Correct index 1.
Question 6: A circle has equation x² + y² − 4x + 6y − 12 = 0. What is its radius?
- 3
- 4
- 5 (Correct answer)
- 6
Correct answer: 5
Completing the square: (x−2)²+(y+3)²=12+4+9=25, so radius = √25 = 5.
Question 7: If 2x + 3y = 12 and xy = 6, what is the value of 4x² + 9y²?
- 72
- 80
- 96 (Correct answer)
- 108
Correct answer: 96
4x²+9y² = (2x+3y)²−12xy = 144−72 = 72. Correct index is 0.
TMUA Problem Solving 3
A square and an equilateral triangle have equal perimeters.
What is the ratio of the area of the square to the area of the triangle?