| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question on completing the square and inverse functions. Part (i) is routine algebraic manipulation, part (ii) requires understanding that a quadratic is one-one when restricted to one side of its vertex, parts (iii-iv) are textbook applications of finding range and inverse. All techniques are straightforward with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks |
|---|---|
| (i) \(2(x-3)^2 - 5\) or \(a = 2, b = -3, c = -5\) | B1B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(3\) | B1√ | ft on – their \(b\). Allow \(k \geq 3\) or \(x \geq 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) \((y) \geq 27\) | B1 | Allow \(>\), Allow \(27 \leq y \leq \infty\) etc. OR (\(y/y\) interchange as 1st operation) |
| Answer | Marks | Guidance |
|---|---|---|
| (iv) \(2(x-3)^2 = (y+5)\) | M1 | \(x = 2(y-3)^2 - 5\) |
| \(x - 3 = (\pm)\sqrt{\frac{1}{2}(y+5)}\) | M1 | \((y-3)^2 = \frac{1}{2}(x+5)\) |
| \(x = 3 + \pm\sqrt{\frac{1}{2}(y+5)}\) | A1√ | \(y - 3 = (\pm)\sqrt{\frac{1}{2}(x+5)}\) |
| \((f^{-1}(x)) = 3 + \sqrt{\frac{1}{2}(x+5)}\) for \(x \geq 27\) | A1B1√ | ft on their 27 from (iii) |
(i) $2(x-3)^2 - 5$ or $a = 2, b = -3, c = -5$ | B1B1B1 |
[3]
(ii) $3$ | B1√ | ft on – their $b$. Allow $k \geq 3$ or $x \geq 3$
[1]
(iii) $(y) \geq 27$ | B1 | Allow $>$, Allow $27 \leq y \leq \infty$ etc. OR ($y/y$ interchange as 1st operation)
[1]
(iv) $2(x-3)^2 = (y+5)$ | M1 | $x = 2(y-3)^2 - 5$
$x - 3 = (\pm)\sqrt{\frac{1}{2}(y+5)}$ | M1 | $(y-3)^2 = \frac{1}{2}(x+5)$
$x = 3 + \pm\sqrt{\frac{1}{2}(y+5)}$ | A1√ | $y - 3 = (\pm)\sqrt{\frac{1}{2}(x+5)}$
$(f^{-1}(x)) = 3 + \sqrt{\frac{1}{2}(x+5)}$ for $x \geq 27$ | A1B1√ | ft on their 27 from (iii)
[5]8 (i) Express $2 x ^ { 2 } - 12 x + 13$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
(ii) The function f is defined by $\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 13$ for $x \geqslant k$, where $k$ is a constant. It is given that f is a one-one function. State the smallest possible value of $k$.
The value of $k$ is now given to be 7 .\\
(iii) Find the range of f .\\
(iv) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2013 Q8 [10]}}