| Date | May 2012 | Marks available | 3 | Reference code | 12M.1.sl.TZ2.11 |
| Level | SL only | Paper | 1 | Time zone | TZ2 |
| Command term | Find | Question number | 11 | Adapted from | N/A |
Question
Part of the graph of the quadratic function f is given in the diagram below.
On this graph one of the x-intercepts is the point (5, 0). The x-coordinate of the maximum point is 3.
The function f is given by \( f (x) = -x^2 + bx + c \), where \(b,c \in \mathbb{Z}\)
Find the value of
(i) b ;
(ii) c .
The domain of f is 0 ≤ x ≤ 6.
Find the range of f .
Markscheme
(i) \(3 = \frac{{-b}}{{-2}}\) (M1)
Note: Award (M1) for correct substitution in formula.
OR
\(-1 + b + c = 0\)
\(-25+5b + c = 0\)
\(-24 + 4b = 0\) (M1)
Note: Award (M1) for setting up 2 correct simultaneous equations.
OR
\(-2x + b = 0\) (M1)
Note: Award (M1) for correct derivative of \(f (x)\) equated to zero.
\(b = 6\) (A1) (C2)
(ii) \(0 = -(5)^2 + 6 \times 5 + c\)
\(c =-5\) (A1)(ft) (C1)
Note: Follow through from their value for b.
Note: Alternatively candidates may answer part (a) using the method below, and not as two separate parts.
\( (x - 5)(-x +1)\) (M1)
\(-x^2 +6x - 5\) (A1)
\(b = 6{\text{ }} c = -5\) (A1) (C3)
[3 marks]
–5 ≤ y ≤ 4 (A1)(ft)(A1)(ft)(A1) (C3)
Notes: Accept [–5, 4]. Award (A1)(ft) for –5, (A1)(ft) for 4. (A1) for inequalities in the correct direction or brackets with values in the correct order or a clear word statement of the range. Follow through from their part (a).
[3 marks]
Examiners report
Question 11 proved to be the most problematic of the whole paper. Many candidates attempted this question but were not able to set up a system of equations to find the value of b or use the formula \(x = \frac{-b}{2a}\). From the working seen, many candidates did not understand the non-standard notation for the domain, with a number believing it to be a coordinate pair. This was taken into careful consideration by the senior examiners when setting the grade boundaries for this paper.
Question 11 proved to be the most problematic of the whole paper. Many candidates attempted this question but were not able to set up a system of equations to find the value of b or use the formula \(x = \frac{-b}{2a}\). From the working seen, many candidates did not understand the non-standard notation for the domain, with a number believing it to be a coordinate pair. This was taken into careful consideration by the senior examiners when setting the grade boundaries for this paper.
