October 2021 Paper 1 Q1
1 Determine the set of values of \(k\) such that the equation \(x^2 + 4x + (k + 3) = 0\) has two distinct real roots. [4]
| Scheme | Marks | AO |
|---|---|---|
| \(16 - 4(k + 3)\) | M1* | 1.1 |
| \(-4k - 12 + 16 \gt 0\) | A1 | 2.3 |
| \(4k - 4 \lt 0\) | M1dep* | 1.1 |
| \(k \lt 1\) | A1 | 1.1 |
| [4] |
Notes
M1*: Attempt discriminant
Allow \(b^2 + 4ac\) for M1, but nothing else
A1: Obtain correct inequality
Not necessarily expanded
M1dep*: Attempt to solve their inequality or equation for \(k\)
A1: Obtain \(k \lt 1\)
OR (completing the square or differentiating)
M1* – attempt to complete the square, or differentiate, and link minimum point to 0
A1 – obtain \((k + 3) - 4 \lt 0\)
M1d* – solve their inequality or equation
A1 – obtain \(k \lt 1\)
OR (using perfect square)
M1* – link \(k + 3\) to 4
A1 – obtain \(k + 3 \lt 4\)
M1d* – solve their inequality or equation
A1 – obtain \(k \lt 1\)