| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Complete the square technique |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise followed by a standard discriminant application. Part (i) is routine algebraic manipulation with a direct read-off of the minimum value. Part (ii) requires rearranging to standard form and applying b²-4ac < 0, which is a standard technique taught early in P1. The question is easier than average as it involves well-practiced procedures with no problem-solving insight required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x^2 - 10x + 8 \rightarrow a(x+b)^2 + c\) | ||
| (i) \(a = 2, b = -\frac{1}{2}, c = -4\frac{1}{2}\) | 3 × B1 | Or \(2\left[x - 2\frac{1}{2}\right]^2 - 4\frac{1}{2}\) |
| \(\rightarrow\) min value is \(-4\frac{1}{2}\) Allow \(\left(2\frac{1}{2}, -4\frac{1}{2}\right)\) | B1✓ | Can score by sub \(x = 2\frac{1}{2}\) into original but not by differentiation |
| [4] | ||
| (ii) \(2x^2 - 10x + 8 - kx = 0\) | M1 | Sets equation to 0 and uses discriminant correctly |
| Use of "\(b^2 - 4ac\)" | M1 | |
| \((-10-k)^2 - 64 < 0\) or \(k^2 + 20k + 36 < 0\) | A1 | Realises discriminant \(< 0\). Allow \(\leq\) co Dep on 1st M1 only |
| \(\rightarrow k = -18\) or \(-2\) | A1 | co |
| \(-18 < k < -2\) | A1 | |
| [4] |
$2x^2 - 10x + 8 \rightarrow a(x+b)^2 + c$ | | |
(i) $a = 2, b = -\frac{1}{2}, c = -4\frac{1}{2}$ | 3 × B1 | Or $2\left[x - 2\frac{1}{2}\right]^2 - 4\frac{1}{2}$ |
$\rightarrow$ min value is $-4\frac{1}{2}$ Allow $\left(2\frac{1}{2}, -4\frac{1}{2}\right)$ | B1✓ | Can score by sub $x = 2\frac{1}{2}$ into original but not by differentiation |
| | [4] |
(ii) $2x^2 - 10x + 8 - kx = 0$ | M1 | Sets equation to 0 and uses discriminant correctly |
Use of "$b^2 - 4ac$" | M1 | |
$(-10-k)^2 - 64 < 0$ or $k^2 + 20k + 36 < 0$ | A1 | Realises discriminant $< 0$. Allow $\leq$ co Dep on 1st M1 only |
$\rightarrow k = -18$ or $-2$ | A1 | co |
$-18 < k < -2$ | A1 | |
| | [4] |8 (i) Express $2 x ^ { 2 } - 10 x + 8$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants, and use your answer to state the minimum value of $2 x ^ { 2 } - 10 x + 8$.\\
(ii) Find the set of values of $k$ for which the equation $2 x ^ { 2 } - 10 x + 8 = k x$ has no real roots.
\hfill \mbox{\textit{CAIE P1 2014 Q8 [8]}}