TMUA Mathematical Reasoning & Logical Thinking 3

Question 1

Let P = 'x > 0' and Q = 'x² > 0'. Which relationship correctly describes P and Q for real x?

Accepted Answer

P is sufficient but not necessary for Q

Answer

P→Q (if x>0 then x²>0) but Q does not imply P since x=−2 gives x²=4>0; so P is sufficient but not necessary.

Question 2

A function f satisfies f(x + y) = f(x) + f(y) for all real x, y, and f(1) = 3. What is f(7)?

Accepted Answer

21

Answer

By the Cauchy functional equation with f(1)=3, we get f(n)=3n for integers, so f(7)=21.

Question 3

In a group of 30 students, 18 study Maths, 15 study Physics, and 5 study neither. How many study both?

Accepted Answer

8

Answer

By inclusion-exclusion: |M∪P| = 30−5 = 25, so |M∩P| = 18+15−25 = 8.

Question 4

Which of the following correctly negates the statement '∀x ∈ ℝ, x² ≥ 0'?

Accepted Answer

∃x ∈ ℝ such that x² < 0

Answer

Negating a universal statement gives an existential statement with the negated predicate: ∃x such that x² < 0.

Question 5

If the sum of three consecutive even integers is 78, what is the largest of the three?

Accepted Answer

28

Answer

Let the integers be n, n+2, n+4; then 3n+6=78, so n=24 and the largest is 28.

(TMUA) Test of Mathematics for University Admission Practice Test