TMUA Problem Solving 3

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?

Accepted Answer

9 : 4√3

Answer

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?

Accepted Answer

4

Answer

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)?

Accepted Answer

23

Answer

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?

Accepted Answer

24

Answer

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?

Accepted Answer

61

Answer

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.

(TMUA) Test of Mathematics for University Admission Practice Test