O-LevelAdditional MathematicsFunctions4037/12 · Oct/Nov 2025 · Q410 Marks
Show that 2x²+5x+3 can be written in the form 2(x+a)²+b, where a and b are constants to be found.
- (a)Show that 2x²+5x+3 can be written in the form 2(x+a)²+b, where a and b are constants to be found.[2]
- (b)Hence write down the coordinates of the stationary point on the curve y = 2x² +5x+3.[2]
- (c)(i)A function f is such that f(x) = 2x²+5x+3, for x ≥p, where p is a constant. It is given that f⁻¹ exists. Write down the least possible value of p.[1]
- (c)(ii)Using your value of p, sketch the graphs of y = f(x) and y = f⁻¹(x). Label each graph. State the intercepts of each of the graphs with the axes.[5]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- [2] B1 for 2(x+5/4)²
- B1 for -1/8
- [2] B1FT on their – a
- B1FT on their b
- B0 if calculus used as question says 'Hence'
- [1] B1 FT on their – a from part (a). Allow if from calculus
- [5] B1 for y = f(x) drawn as a one-one function in the first, second and third quadrants or first and second quadrants. Must have correct curvature
- Dep B1 for y=f¯¹(x) as a reflection of their y = f(x) in the line y=x. Maybe implied by intercepts, but must have the correct curvature
- B1 for y = f(x) clearly passing through x=-1 correctly soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature
- B1 for y = f(x) passing through y=3 only on the y-axis soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature
- B1 for -1 and 3 marked correctly for y=f¯¹(x) allow for unrestricted range. Must have correct curvature.
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