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Further Pure 1: lesson checks flashcards

This deck has 10 check-yourself questions and 9 spot-the-error cards. The misconception each lesson corrects, and its check-yourself question with the worked answer. Rate each answer, and the card comes back sooner or later depending on how you rated it.

Further Maths · all 584 maths cards

How the five kinds differ

Standard results ask you to recall a formula or a value. Definitions ask you to state a mathematical term precisely. Recall prompts ask for a rule, condition or method. Check yourself cards are a short question with its answer. Spot the error cards ask you to correct a statement that is wrong.

Every card in this deck

Choose any card to try it in the player above. Each card links to the lesson that teaches it.

• sin θ, cos θ and tan θ are three independent quantities; no single variable can carry all three.Spot the error
• Given t = tan(θ/2) = 1/3, find sin θ and cos θ exactly.Check
• Series expansions only work near zero; away from the origin a function cannot be turned into a polynomial.Spot the error
• Write down the first three terms of the Taylor series of e^x about x = 2.Check
• If numerator and denominator both tend to zero, the fraction must tend to 1, since they shrink together.Spot the error
• Evaluate the limit as x → 0 of sin 3x/x.Check
• To find the fifth derivative of a product you have no choice but to differentiate five times.Spot the error
• Using Leibniz's theorem, write down the third derivative of y = xe^2x.Check
• A differential equation with no closed-form solution gives no information about its solution.Spot the error
• For y' = x² + y with y(0) = 1, find the second and third derivatives of y at x = 0.Check
• The parabola, ellipse and hyperbola are three unrelated curves that happen to share a chapter.Spot the error
• Write down parametric coordinates for a general point on y² = 8x and on xy = 9.Check
• A line either obviously crosses a curve or obviously misses it; tangency needs a picture, not algebra.Spot the error
• Find the gradient of the normal to xy = 9 at the point (3, 3).Check
• Multiplying two vectors can only sensibly produce a number, as the scalar product does.Spot the error
• Vectors u = (2, 0, 1) and v = (4, 0, 2) satisfy u × v = 0. What does this say about them?Check
• A differential equation with no closed-form solution cannot be evaluated at a particular point.Spot the error
• Using the forward difference with h = 0.2, take one step of dy/dx = y − x from y = 1 at x = 0.Check
• Solve (x + 1)/(x − 3) ≥ 0.Check

The lessons these cards come from

• Conic sections
• Inequalities and inequations
• Leibniz's theorem and the Weierstrass substitution
• Limits and L'Hôpital's rule
• Numerical methods for differential equations
• Series solutions of differential equations
• Tangents, normals and loci of conics
• Taylor series
• The t-formulae
• The vector product and the scalar triple product

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