ATAR Mathematical Reasoning 3

Question 1

A bag contains 5 red, 3 blue, and 2 green marbles. Two marbles are drawn without replacement. What is the probability that both are red?

Accepted Answer

2/9

Answer

P(both red) = (5/10) × (4/9) = 20/90 = 2/9.

Question 2

Solve the system: 3x + 2y = 16 and x − y = 2. What is x + y?

Accepted Answer

6

Answer

From x = y + 2: 3(y+2) + 2y = 16 → 5y = 10 → y = 2, x = 4; x + y = 6.

Question 3

A circle has centre (3, −1) and passes through (7, 2). What is its equation?

Accepted Answer

(x−3)² + (y+1)² = 25

Answer

Radius² = (7−3)² + (2−(−1))² = 16 + 9 = 25, giving (x−3)² + (y+1)² = 25.

Question 4

An arithmetic series has first term 7 and common difference 4. Which term equals 103?

Accepted Answer

25th

Answer

aₙ = 7 + (n−1)×4 = 103 → (n−1) = 24 → n = 25.

Question 5

What is the derivative of f(x) = 3x⁴ − 5x² + 2 at x = 1?

Accepted Answer

2

Answer

f′(x) = 12x³ − 10x; at x = 1, f′(1) = 12 − 10 = 2.

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