Advanced Mathematics Practice Questions 1 Flashcards

Factor completely: 6x² + 7x − 5

Answer: (2x − 1)(3x + 5)

Multiply 6 × (−5) = −30 and find two numbers that multiply to −30 and add to 7: those are 10 and −3. Rewrite as 6x² + 10x − 3x − 5, then factor by grouping: 2x(3x + 5) − 1(3x + 5) = (2x − 1)(3x + 5).

If f(x) = 2x + 3 and g(x) = x² − 1, what is f(g(2))?

Answer: 9

First evaluate the inner function: g(2) = 2² − 1 = 3. Then apply f to that result: f(3) = 2(3) + 3 = 9. Composition is always evaluated inside-out.

Solve for x: log₂(x + 3) = 4

Answer: x = 13

Convert the logarithmic equation to exponential form: 2⁴ = x + 3, which gives 16 = x + 3, so x = 13. Always check: log₂(13 + 3) = log₂(16) = 4 ✓

What is the vertex of the parabola y = 2x² − 8x + 5?

Answer: (2, −3)

The x-coordinate of the vertex is x = −b/(2a) = −(−8)/(2·2) = 2. Substitute back: y = 2(4) − 8(2) + 5 = 8 − 16 + 5 = −3. Vertex is (2, −3).

A geometric sequence has a first term of 3 and a common ratio of 2. What is the sum of the first 5 terms?

Answer: 93

The terms are 3, 6, 12, 24, 48. Using the formula Sₙ = a(rⁿ − 1)/(r − 1): S₅ = 3(2⁵ − 1)/(2 − 1) = 3(32 − 1)/1 = 3 × 31 = 93.

When p(x) = x³ − 2x² + 3x − 4 is divided by (x − 2), what is the remainder?

Answer: 2

By the Remainder Theorem, the remainder when dividing by (x − 2) equals p(2). Evaluate: p(2) = 8 − 8 + 6 − 4 = 2. No long division needed.