ATAR Mathematical Reasoning 5

Question 1

Solve: |2x − 5| = 7. What are the solutions?

Accepted Answer

x = 6 or x = −1

Answer

2x − 5 = 7 gives x = 6, and 2x − 5 = −7 gives x = −1.

Question 2

The graph of y = f(x) is shifted 3 units left and 2 units down. What is the equation of the new graph?

Accepted Answer

y = f(x + 3) − 2

Answer

A left shift of 3 replaces x with (x+3), and a downward shift of 2 subtracts 2, giving y = f(x+3) − 2.

Question 3

A right-angled triangle has legs of length a and b, and hypotenuse c. If a = 5 and c = 13, what is the value of b?

Accepted Answer

12

Answer

b = √(c² − a²) = √(169 − 25) = √144 = 12.

Question 4

A function is defined as f(x) = (x² − 4)/(x − 2) for x ≠ 2. What value does f(x) approach as x → 2?

Accepted Answer

4

Answer

(x² − 4)/(x − 2) = (x+2)(x−2)/(x−2) = x + 2, which approaches 4 as x → 2.

Question 5

In a class of 30 students, 18 study Mathematics, 15 study Physics, and 10 study both. How many study neither?

Accepted Answer

7

Answer

By inclusion-exclusion: |M ∪ P| = 18 + 15 − 10 = 23; neither = 30 − 23 = 7.

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