IB Mathematics: Analysis and Approaches practice questions.

In a proof by mathematical induction for all integers n ≥ 1, after verifying the base case, what is assumed in the inductive step?

• P(k+1) is true
• P(k) holds for some k ≥ 1
• P(n) is true for all n ≥ 1
• P(1) is false

Answer: P(k) holds for some k ≥ 1

The inductive hypothesis assumes truth for one arbitrary k, then shows P(k) ⇒ P(k+1). Assuming it for all n would be circular reasoning.

What is the equation of the vertical asymptote of f(x) = (2x + 1)/(x − 3)?

• x = 3
• x = −3
• y = 2
• x = −1/2

Answer: x = 3

Vertical asymptotes of rational functions occur where the denominator is zero (and the numerator is not): x − 3 = 0 gives x = 3. The horizontal asymptote here would be y = 2.

Which expression equals sin(A + B)?

• sin A cos B + cos A sin B
• sin A cos B − cos A sin B
• cos A cos B − sin A sin B
• sin A sin B + cos A cos B

Answer: sin A cos B + cos A sin B

Compound-angle identity: sin(A+B) = sin A cos B + cos A sin B. For sine the sign matches the sign in the angle; for cosine it flips: cos(A+B) = cos A cos B − sin A sin B.

For a continuous random variable X with pdf f, what is P(X = a) for any single value a?

• f(a)
• 0
• 1
• undefined

Answer: 0

Probabilities for continuous variables are areas under the pdf; a single point has zero width, so P(X = a) = 0. Consequently P(X ≤ a) = P(X < a).

Using the product rule, what is the derivative of y = x²eˣ?

• 2x eˣ
• (x² + 2x)eˣ
• x² eˣ
• (x² − 2x)eˣ

Answer: (x² + 2x)eˣ

Product rule: (uv)′ = u′v + uv′ = 2x·eˣ + x²·eˣ = (x² + 2x)eˣ. Factor out the common eˣ to tidy the answer.