Free QCE Year 11 Mathematical Methods Flashcard Review
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What is a surd?
An irrational number written with a root sign, e.g. √5. √16 = 4 is not a surd.
Simplify √98.
√49 × √2 = 7√2.
√a × √b = ?
√(ab), for a, b ≥ 0.
When can surds be added?
Only when they are like surds (same radicand after simplifying), e.g. 2√3 + 5√3 = 7√3.
(a + √b)(a − √b) = ?
a² − b, which is rational.
How do you rationalise 1/(a + √b)?
Multiply numerator and denominator by the conjugate a − √b to get (a − √b)/(a² − b).
Turning point of y = a(x − h)² + k?
(h, k); axis of symmetry x = h.
Axis of symmetry of y = ax² + bx + c?
x = −b/(2a).
x-intercepts of y = a(x − x₁)(x − x₂)?
x = x₁ and x = x₂; axis halfway at x = (x₁ + x₂)/2.
Quadratic formula?
x = (−b ± √(b² − 4ac))/(2a) for ax² + bx + c = 0.
What does Δ = b² − 4ac tell you?
Δ > 0: two real solutions; Δ = 0: one repeated solution; Δ < 0: no real solutions.
Effect of the sign of a in y = ax² + bx + c?
a > 0: concave up with a minimum; a < 0: concave down with a maximum.
What is a combination?
An unordered selection of r objects from n distinct objects.
Formula for ⁿCᵣ
n! ÷ (r!(n − r)!)
Value of ⁵C₂
10
How is Pascal’s triangle linked to ⁿCᵣ?
Entry r (counting from 0) of row n equals ⁿCᵣ.
Write row 4 of Pascal’s triangle.
1 4 6 4 1
General term of (a + b)ⁿ
ⁿCᵣ aⁿ⁻ʳ bʳ, r = 0, 1, …, n
Expand (x + 1)³
x³ + 3x² + 3x + 1
Degree and leading coefficient of 7 − 2x³ + x
Degree 3, leading coefficient −2