| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Complete the square technique |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard completing the square technique, basic function properties, and solving a quadratic inequality. All parts follow routine procedures with no novel insight required—easier than the typical A-level question which would involve more problem-solving or integration of multiple concepts. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x^2 - 12x + 7 = 2(x-3)^2 - 11\) | \(3 \times\) B1 [3] | B1 for each value – accept if \(a\), \(b\), \(c\) not specifically quoted |
| Answer | Marks | Guidance |
|---|---|---|
| Range of \(f \geqslant -11\) | B1\(\sqrt{}\) [1] | \(\sqrt{}\) to his "\(c\)". allow \(>\) or \(\geqslant\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x^2 - 12x + 7 < 21 \rightarrow 2x^2 - 12x - 14\ \text{or}\ 2(x-3)^2 < 32\) | M1 | 3-term quadratic to 0 or \(2(x-3)^2 < 32\) |
| \(\rightarrow\) end-values of 7 or \(-1\); \(\rightarrow -1 < x < 7\) | A1, A1 [3] | Correct end-values; co |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{gf}(x) = 2(2x^2 - 12x + 7) + k = 0\) | M1 A1 | Puts \(f\) into \(g\). co |
| Use of \(b^2 - 4ac\); \(\rightarrow 24^2 - 16(14 + k)\); \(\rightarrow k = 22\) | M1, A1 [4] | Used correctly with quadratic; co |
## Question 9:
**Part (i):**
$2x^2 - 12x + 7 = 2(x-3)^2 - 11$ | $3 \times$ B1 [3] | B1 for each value – accept if $a$, $b$, $c$ not specifically quoted
**Part (ii):**
Range of $f \geqslant -11$ | B1$\sqrt{}$ [1] | $\sqrt{}$ to his "$c$". allow $>$ or $\geqslant$
**Part (iii):**
$2x^2 - 12x + 7 < 21 \rightarrow 2x^2 - 12x - 14\ \text{or}\ 2(x-3)^2 < 32$ | M1 | 3-term quadratic to 0 or $2(x-3)^2 < 32$
$\rightarrow$ end-values of 7 or $-1$; $\rightarrow -1 < x < 7$ | A1, A1 [3] | Correct end-values; co
**Part (iv):**
$\text{gf}(x) = 2(2x^2 - 12x + 7) + k = 0$ | M1 A1 | Puts $f$ into $g$. co
Use of $b^2 - 4ac$; $\rightarrow 24^2 - 16(14 + k)$; $\rightarrow k = 22$ | M1, A1 [4] | Used correctly with quadratic; co
---9 The function f is defined by $\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7$ for $x \in \mathbb { R }$.\\
(i) Express $\mathrm { f } ( x )$ in the form $a ( x - b ) ^ { 2 } - c$.\\
(ii) State the range of f .\\
(iii) Find the set of values of $x$ for which $\mathrm { f } ( x ) < 21$.
The function g is defined by $\mathrm { g } : x \mapsto 2 x + k$ for $x \in \mathbb { R }$.\\
(iv) Find the value of the constant $k$ for which the equation $\operatorname { gf } ( x ) = 0$ has two equal roots.
\hfill \mbox{\textit{CAIE P1 2010 Q9 [11]}}