June 2019 Paper 2 Q26
26 \(\mathrm{f}(x)\) is a function with domain \(\quad\) all values of \(x\).
\(\mathrm{f}(x) = \sqrt{x^2 + 6x - a} \qquad\) where \(a\) is a constant.
Work out the possible values of \(a\).
Give your answer as an inequality. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \((x + 3)^2\ldots\) | M1 | |
| \((x + 3)^2 - 3^2 - a\) or \((x + 3)^2 - 3^2 \geqslant a\) or \((x + 3)^2 \geqslant a + 3^2\) | M1dep | oe expression or inequality eg \((x + 3)^2 \geqslant 9 + a\) allow \(\geqslant\) to be any inequality symbol or \(=\) eg allow \((x + 3)^2 - 9 = a\) implies M2 |
| \(-3^2 - a \geqslant 0\) or \(-3^2 - a \gt 0\) | M1dep | oe inequality eg \(-9 - a \geqslant 0\) or \(-9 - a \gt 0\) or \(a \lt -9\) implies M3 |
| \(a \leqslant -9\) or \(-9 \geqslant a\) | A1 | SC1 \(x^2 + 6x - a \geqslant 0\) oe inequality (may be seen in working lines) |
| Alternative method 2 | ||
| \(2x + 6 = 0\) | M1 | must have \(= 0\) |
| (minimum at) \(x = -3\) | M1dep | implies M2 \(x = -3\) must be the only value or be clearly chosen |
| \((-3)^2 + 6 \times (-3) - a \geqslant 0\) or \((-3)^2 + 6 \times (-3) - a \gt 0\) | M1dep | oe inequality eg \(9 - 18 - a \geqslant 0\) or \(9 - 18 - a \gt 0\) or \(a \lt -9\) implies M3 |
| \(a \leqslant -9\) or \(-9 \geqslant a\) | A1 | SC1 \(x^2 + 6x - a \geqslant 0\) oe inequality (may be seen in working lines) |
| Alternative method 3 | ||
| \(6^2 - 4 \times 1 \times -a\) | M1 | \(b^2 - 4ac\) must be selected if seen in quadratic formula |
| \(6^2 - 4 \times 1 \times -a \leqslant 0\) or \(6^2 - 4 \times 1 \times -a \lt 0\) | M1dep | oe inequality implies M2 |
| \(36 + 4a \leqslant 0\) or \(36 + 4a \lt 0\) | M1dep | oe inequality eg \(4a \leqslant -36\) implies M3 |
| \(a \leqslant -9\) or \(-9 \geqslant a\) | A1 | SC1 \(x^2 + 6x - a \geqslant 0\) oe inequality (may be seen in working lines) |
Additional guidance
| Alt 1 2nd M1 Any inequality symbol or \(=\) allowed 3rd M1 Only the inequality symbols shown are allowed (do not allow \(=\)) | |
| Allow \((x + 3)(x + 3)\) for \((x + 3)^2\) |