Further Mathematics

Key Facts

• Determinant = ad - bc for [[a,b],[c,d]]
• Matrix multiplication is not commutative.
• Identity matrix I = [[1,0],[0,1]]
• AA^-1 = A^-1A = I
• det(AB) = det(A) × det(B)

Subtopics

Key Facts

• Odd numbers: 2n+1. Even numbers: 2n.
• Consecutive integers: n and n+1.
• Show factorization to prove divisibility.
• One counter-example disproves a conjecture.
• QED means "proof complete".

Subtopics

Key Facts

• Composite: f(g(x)) means do g then f.
• Inverse undoes the function.
• Transformations shift or stretch graphs.
• Domain and range can be restricted.

Subtopics

Key Facts

• Parallel vectors are scalar multiples.
• Collinear points lie on the same straight line.
• Magnitude = √(x² + y²)
• Midpoint of AB = (a + b)/2

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Key Facts

• Differentiate x^n → nx^(n-1)
• Constant terms differentiate to 0.
• Stationary points: dy/dx = 0
• Tangent gradient = dy/dx
• Normal is perpendicular to tangent.

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Key Facts

• Flip inequality when multiplying/dividing by negative.
• Solid line for ≤ or ≥, dashed for < or >.
• Shade the side that satisfies the inequality.
• Intersection of regions satisfies all inequalities.

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Key Facts

• Angle at centre = 2 × angle at circumference.
• Angle in a semicircle = 90°.
• Alternate segment theorem for tangents.
• Opposite angles in cyclic quadrilateral sum to 180°.
• Tangent perpendicular to radius.

Subtopics

Key Facts

• Perpendicular gradients multiply to -1.
• Distance formula: √[(x₂-x₁)² + (y₂-y₁)²]
• Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
• Circle: (x-a)² + (y-b)² = r²

Subtopics