A-Level Mathematics Flashcards

What is the equation of a circle with centre (3, −2) and radius 5?

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.

What does it mean for a function to be 'continuous' at a point?

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.

What is the quotient rule for differentiation?

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.

What does 'proof by contradiction' involve?

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.

What is the normal distribution and when is it used?

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.

What is a stationary point, and how do you classify it?

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.