A-Level A-Level Mathematics 2 — Questions and Answers
Question 1: What is the equation of a circle with centre (3, −2) and radius 5?
- (x − 3)² + (y + 2)² = 5
- (x + 3)² + (y − 2)² = 25
- (x − 3)² + (y + 2)² = 25 (Correct answer)
- x² + y² = 25
Correct answer: (x − 3)² + (y + 2)² = 25
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². With centre (3, −2) and radius 5: (x − 3)² + (y − (−2))² = 5², giving (x − 3)² + (y + 2)² = 25.
Question 2: What does it mean for a function to be 'continuous' at a point?
- The function has a maximum at that point
- The limit of the function at that point exists, equals the function value, and there are no breaks or jumps (Correct answer)
- The function is always increasing
- The function equals zero at that point
Correct answer: The limit of the function at that point exists, equals the function value, and there are no breaks or jumps
A function is continuous at point x = a if: (1) f(a) is defined, (2) the limit as x → a exists, and (3) the limit equals f(a). Informally, you can draw the curve without lifting your pen.
Question 3: What is the quotient rule for differentiation?
- d/dx(u/v) = (u'v + uv')/v²
- d/dx(u/v) = (u'v − uv')/v² (Correct answer)
- d/dx(u/v) = u'v'
- d/dx(u/v) = u'/v − uv'/v²
Correct answer: d/dx(u/v) = (u'v − uv')/v²
The quotient rule states: if y = u/v, then dy/dx = (u'v − uv')/v². Remember the order: differentiate the top times the bottom minus the top times the derivative of the bottom, all over the bottom squared.
Question 4: What does 'proof by contradiction' involve?
- Proving a statement by finding one example
- Assuming the statement is true and using algebra
- Assuming the statement is false and showing this leads to an impossibility (Correct answer)
- Using induction to prove the nth case
Correct answer: Assuming the statement is false and showing this leads to an impossibility
Proof by contradiction assumes the negation of the statement is true, then derives a logical contradiction. Since the assumption leads to an impossibility, the original statement must be true.
Question 5: What is the normal distribution and when is it used?
- A skewed distribution for test scores
- A symmetric, bell-shaped probability distribution described by mean (μ) and standard deviation (σ), used to model many natural phenomena (Correct answer)
- A distribution for discrete data only
- A distribution for data with no outliers
Correct answer: A symmetric, bell-shaped probability distribution described by mean (μ) and standard deviation (σ), used to model many natural phenomena
The normal distribution N(μ, σ²) is a continuous, symmetric, bell-shaped distribution. It models many real-world phenomena (heights, IQ, measurement errors). About 68% of values lie within 1σ of the mean; 95% within 2σ.
Question 6: What is a stationary point, and how do you classify it?
- A point where the function equals zero; classified by the sign of f(x)
- A point where the gradient (dy/dx) equals zero; classified as maximum, minimum, or point of inflection using the second derivative (Correct answer)
- A point where x = 0
- A point where the function is discontinuous
Correct answer: A point where the gradient (dy/dx) equals zero; classified as maximum, minimum, or point of inflection using the second derivative
A stationary point occurs where dy/dx = 0. To classify: if d²y/dx² > 0, it's a minimum; if < 0, maximum; if = 0, further investigation needed. Alternatively, examine the sign of dy/dx either side of the point.
A-Level A-Level Mathematics 2
What is the equation of a circle with centre (3, −2) and radius 5?