A-LevelMathematicsFunctions9709/12 · Oct/Nov 2013 · Q810 Marks
A function f is defined by f : x → 3 cos x − 2 for 0 ≤ x ≤ 2π.
- (i)Solve the equation f(x) = 0.[3]
- (ii)Find the range of f.[2]
- (iii)Sketch the graph of y = f(x).[2]
- (iv)A function g is defined by g : x → 3 cos x − 2 for 0 ≤ x ≤ k. State the maximum value of k for which g has an inverse.[1]
- (v)Obtain an expression for g⁻¹(x).[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.
- [1] 3cosx − 2 = 0 → cos x = ⅔ Makes cos subject, then cos⁻¹
- [1] x = 0.841
- [1] x = 5.44 (for 2π – 1st answer).
- [1] f(x) ≥ −5
- [1] f(x) ≤ 1
- [1] Graph starts and ends at same point (x=0, y=1 and x=2π, y=1). Starts decreasing. One cycle only.
- [1] Correct shape (cosine curve, not V or U shape).
- [1] k = π or 180°
- [1] Makes x the subject, copes with 'cos'.
- [1] g⁻¹(x) = cos⁻¹ ((x+2)/3) (Needs to be in terms of x)
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