Free TCE Year 11 Maths Methods Foundation Flashcard Review
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Simplify (aᵐ)ⁿ and aᵐ × aⁿ.
(aᵐ)ⁿ = aᵐⁿ and aᵐ × aⁿ = aᵐ⁺ⁿ.
Write a^(m/n) in surd form.
a^(m/n) = (ⁿ√a)ᵐ, the nth root of a, raised to the power m.
Factorise a² − b².
(a − b)(a + b), the difference of two squares.
Completed-square form of x² + bx + c?
(x + b/2)² + c − (b/2)².
State the quadratic formula for ax² + bx + c = 0.
x = (−b ± √(b² − 4ac))/(2a).
What does a positive perfect-square discriminant tell you?
With integer coefficients: two distinct rational solutions; the quadratic factorises over the rationals.
What does a negative discriminant tell you?
No real solutions and no real linear factors; the parabola does not meet the x-axis.
State the null factor law.
If AB = 0 then A = 0 or B = 0 (or both).
State the factor theorem.
If P(a) = 0 then (x − a) is a factor of P(x), and conversely.
Factorise a³ + b³ and a³ − b³.
a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).
Give row 5 of Pascal’s triangle and the formula for ⁿCᵣ.
1 5 10 10 5 1; ⁿCᵣ = n!/(r!(n − r)!).
Write 0.000 47 in scientific notation.
4.7 × 10⁻⁴.
Gradient through (x₁, y₁) and (x₂, y₂)?
m = (y₂ − y₁)/(x₂ − x₁).
How are gradients of perpendicular lines related?
Their product is −1, so m₂ = −1/m₁.
Turning point of y = a(x − b)² + c?
(b, c); a minimum if a is positive, a maximum if a is negative.
Axis of symmetry of y = ax² + bx + c?
x = −b/(2a).
What does the vertical line test check?
Whether a graph is a function: no vertical line may meet it more than once.
Domain versus range?
Domain: the allowed inputs (x-values). Range: the resulting outputs (y-values).
Behaviour of a cubic at a repeated zero (x − a)²?
The graph touches the x-axis at x = a and turns there.
Point of inflection of y = a(x − h)³ + k?
(h, k).