4MA0 Higher January 2016 Paper 3 Q15
15 The diagram shows the graph of \(y = \mathrm{f}(x)\) for \(-3.5 \leqslant x \leqslant 1.5\)
(a) Find \(\mathrm{f}(0)\) (1)
(b) For which values of \(k\) does the equation \(\mathrm{f}(x) = k\) have only one solution? (2)
(c) Find an estimate for the gradient of the curve at the point where \(x = -2.5\) (3)
\(\mathrm{g}(x) = \dfrac{1}{2 + x}\)
(d) State which value of \(x\) must be excluded from any domain of g (1)
(e) Find \(\mathrm{fg}(-3)\) (2)
| Scheme | Marks |
|---|---|
| 5 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| (\(-7 \leqslant\) ) \(k \lt 5\), \(13 \lt k\) (\(\leqslant 25\)) | B2 |
| (2) |
Notes
B2: accept \(k \leqslant 5\), \(k \geqslant 13\)
If not B2 then B1 for 5 and 13
or \(k \lt 5\) or \(k \gt 13\)
| Scheme | Marks |
|---|---|
| tangent drawn at \(x = -2.5\) | M1 |
| M1 | |
| 7.5 | A1 |
| (3) |
Notes
M1: complete method to find gradient
A1: accept answers in range 7 - 8 with working seen
| Scheme | Marks |
|---|---|
| −2 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2 + -3}\) or −1 or \(\mathrm{f}(-1)\) | M1 |
| 9 | A1 |
| (2) |