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Complex numbers, matrices and roots: lesson checks flashcards

This deck has 10 check-yourself questions and 10 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.

• A negative number has no square root, so z² = −9 has no solutions.Spot the error
• Solve z² + 4z + 29 = 0, and describe how the two roots sit on the Argand diagram.Check
• The argument of z is tan⁻¹(y/x), whatever the quadrant z is in.Spot the error
• Sketch the locus |z − 3i| = 3, and state its centre and radius. Where does it meet the axes?Check
• To find (1 + i)⁸ there is no alternative to multiplying out the brackets eight times.Spot the error
• Use De Moivre's theorem to evaluate (1 + i)¹⁰, and give the answer in the form r e^iθ.Check
• A number has one cube root, so z³ = 8i has exactly one solution.Spot the error
• Solve z³ = 27, giving all three roots exactly, and state the shape they make on the Argand diagram.Check
• Matrix multiplication, like ordinary multiplication, gives the same answer in either order.Spot the error
• Find the single matrix for reflection in y = x followed by enlargement, scale factor 3, about the origin.Check
• Every matrix has an inverse, just as every non-zero number has a reciprocal.Spot the error
• The matrix with rows (4, k), (2, 3) is singular. Find k, and describe what the transformation does to the plane at that value.Check
• If three planes have no common point, at least two of them must be parallel.Spot the error
• Solve the system 2x + y = 4, x + 3y = 7 by inverse matrix, and verify the answer.Check
• To find the sum of the roots of a cubic you must first solve the cubic.Spot the error
• The equation 2z³ − 4z² + 3z − 7 = 0 has roots α, β, γ. Write down α + β + γ and αβγ.Check
• There is no way to sum 1/(1×2) + 1/(2×3) + … + 1/(n(n+1)) exactly; you can only add the terms up.Spot the error
• Evaluate Σ r² for r from 1 to 10.Check
• A curved function like sin x cannot be approximated by polynomials to any accuracy you like.Spot the error
• Using the series for e^x, write down the series for e^−x up to the x³ term, and hence the series for ½(e^x − e^−x).Check

The lessons these cards come from

• Complex arithmetic and the Argand diagram
• De Moivre's theorem and trigonometric identities
• Determinants and inverses
• Maclaurin series
• Matrix algebra and transformations
• Modulus, argument and loci
• Roots of polynomials
• Roots of unity and complex roots
• Summing series and the method of differences
• Systems of equations and invariance

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