June 2023 Paper 2 Q1
1
(a)
(i) Express \(x^2 - 8x + 11\) in the form \((x - a)^2 + b\) where \(a\) and \(b\) are constants. [2]
(ii) Hence write down the minimum value of \(x^2 - 8x + 11\). [1]
(b) Determine the value of the constant \(k\) for which the equation \(x^2 - 8x + 11 = k\) has two equal roots. [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) \((x - 4)^2 - 5\) | B1 B1 | 1.1 1.1 |
| [2] | ||
| (ii) \(-5\) | B1FT | 1.1 |
| [1] |
Notes
(a)(i)
B1 B1: B1 for each constant.
For ‘4’ (Allow \(a = 4\))
For ‘\(-5\)’ (Allow \(b = -5\))
ISW
(a)(ii)
B1FT: FT their \(b\)
Accept \((4, -5)\) as the coordinates of the minimum point but must be correct for their \(a\) and \(b\).
| Scheme | Marks | AO |
|---|---|---|
| \((-8)^2 - 4 \times (11 - k) = 0\) or \(x^2 - 8x + 11 - k \equiv (x - 4)^2\) | M1 | 1.1 |
| \(k = -5\) | A1 | 1.1 |
| [2] |
Notes
M1: Write \(b^2 - 4ac = 0\) (accept \(\gt 0\) for this mark only) or equate with completed square form.
Accept a sketch or equivalent reasoning.
A1: Final answer must be given as \(k = -5\)
SC B1: correct answer without working (max [1/2])