| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2006 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve quadratic inequality |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question covering standard P1 techniques: solving a quadratic inequality (routine factorization), completing the square (direct application of formula), stating range from completed square form, and checking invertibility (standard criterion). Part (v) involves a substitution u=√x which is a common textbook exercise. All parts are procedural with no problem-solving insight required, making this easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^2 - 3x - 4 \rightarrow -1\) and 4 \(\rightarrow x < -1\) and \(x > 4\) | M1 A1 A1 [3] | Solving Quadratic = 0. Correct values. co. – allow ≤ and/or ≥. 4<x<-1 ok |
| (ii) \(x^2 - 3x = (x - \frac{3}{2})^2 - \frac{9}{4}\) | B1 B1 [2] | B1 for \(\frac{3}{2}\). B1 for \(\frac{9}{4}\). |
| (iii) \(f(x)\) (or \(y\)) \(\geq -\frac{7}{4}\) | B1∨ [1] | ∨ for \(f(x) \geq -b\). |
| (iv) No inverse – not 1 : 1. | B1 [1] | Independent of previous working. |
| (v) Quadratic in \(\sqrt{x}\). Solution \(\rightarrow \sqrt{x} = 5\) or \(-2\) \(\rightarrow x = 25\) | M1 DM1 A1 [3] | Recognition of "Quadratic in \(\sqrt{x}\)". Method of solution. co. Loses this mark if other answers given. Nb ans only full marks. |
(i) $x^2 - 3x - 4 \rightarrow -1$ and 4 $\rightarrow x < -1$ and $x > 4$ | M1 A1 A1 [3] | Solving Quadratic = 0. Correct values. co. – allow ≤ and/or ≥. 4<x<-1 ok
(ii) $x^2 - 3x = (x - \frac{3}{2})^2 - \frac{9}{4}$ | B1 B1 [2] | B1 for $\frac{3}{2}$. B1 for $\frac{9}{4}$.
(iii) $f(x)$ (or $y$) $\geq -\frac{7}{4}$ | B1∨ [1] | ∨ for $f(x) \geq -b$.
(iv) No inverse – not 1 : 1. | B1 [1] | Independent of previous working.
(v) Quadratic in $\sqrt{x}$. Solution $\rightarrow \sqrt{x} = 5$ or $-2$ $\rightarrow x = 25$ | M1 DM1 A1 [3] | Recognition of "Quadratic in $\sqrt{x}$". Method of solution. co. Loses this mark if other answers given. Nb ans only full marks.10 The function f is defined by $\mathrm { f } : x \mapsto x ^ { 2 } - 3 x$ for $x \in \mathbb { R }$.\\
(i) Find the set of values of $x$ for which $\mathrm { f } ( x ) > 4$.\\
(ii) Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } - b$, stating the values of $a$ and $b$.\\
(iii) Write down the range of f .\\
(iv) State, with a reason, whether f has an inverse.
The function g is defined by $\mathrm { g } : x \mapsto x - 3 \sqrt { } x$ for $x \geqslant 0$.\\
(v) Solve the equation $\mathrm { g } ( x ) = 10$.
\hfill \mbox{\textit{CAIE P1 2006 Q10 [10]}}