A-LevelMathematicsFunctions9709/12 · May/June 2011 · Q98 Marks
The function f is such that f(x) = 3 – 4 cosᵏx, for 0 ≤ x ≤ π, where k is a constant.
- (i)(a)find the range of f,[2]
- (i)(b)find the exact solutions of the equation f(x) = 1.[3]
- (ii)(a)sketch the graph of y = f(x),[2]
- (ii)(b)state, with a reason, whether f has an inverse.[1]
Mark scheme
Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.
- [2] (i)(a) f(x) = 3 - 4cos²x.; B1 co irrespective of inequalities; B1 co irrespective of inequalities; One limit is -1; Other limit is 3
- [3] (i)(b) 3 - 4cos²x = 1 → cos²x = ½; M1 Makes cos x the subject.; → cosx = ±1/√2; A1 A1√ co (radians). √ for “π – (1st answer)” ("exact" means that decimal answers only earn A0 A1√); → x = ¼π or ¾π
- [2] (ii)(a); B1 Joins (0, -1) to (π, 7), providing increasing function; B1 Not a line, flattens at extremities-needs inflexion.
- [1] (ii)(b) f has an inverse since it is 1:1 or increasing or no turning points.; B1 co independent of part (i)
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