| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constant then solve inequality or further work |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution to find 'a', then polynomial division or inspection to factorise, followed by a routine inequality solution. All steps are standard textbook procedures with no novel insight required, making it easier than average but not trivial due to the multi-part nature. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute 2 for \(x\) and equate to zero, or divide by \(x – 2\) and equate remainder to zero | M1 | |
| Obtain answer \(a = -3\) | A1 | Total: 2 marks |
| (ii) Attempt to find quadratic factor by division or inspection | M1 | |
| State quadratic factor \(2x^2 + x + 2\) | A1 | [The M1 is earned if division reaches a partial quotient of \(2x^2 + kx\), or if inspection has an unknown factor of \(2x^2 + bx + c\) and an equation in \(b\) and/or \(c\), or if two coefficients with the correct moduli are stated without working.] |
| Total: 2 marks | ||
| (iii) State answer \(x > 2\) (and nothing else) | B1* | |
| Make a correct justification e.g. \(2x^2 + x + 2\) (has no zeros and) is always positive | B1(dep*) | [SR: The answer is \(x \geq 2\) gets B0, but in this case allow the second B mark if the remaining work is correct.] |
| Total: 2 marks |
**(i)** Substitute 2 for $x$ and equate to zero, or divide by $x – 2$ and equate remainder to zero | M1 | |
Obtain answer $a = -3$ | A1 | **Total: 2 marks** |
**(ii)** Attempt to find quadratic factor by division or inspection | M1 | |
State quadratic factor $2x^2 + x + 2$ | A1 | [The M1 is earned if division reaches a partial quotient of $2x^2 + kx$, or if inspection has an unknown factor of $2x^2 + bx + c$ and an equation in $b$ and/or $c$, or if two coefficients with the correct moduli are stated without working.] |
| | | **Total: 2 marks** |
**(iii)** State answer $x > 2$ (and nothing else) | B1* | |
Make a correct justification e.g. $2x^2 + x + 2$ (has no zeros and) is always positive | B1(dep*) | [SR: The answer is $x \geq 2$ gets B0, but in this case allow the second B mark if the remaining work is correct.] |
| | | **Total: 2 marks** |3 The polynomial $2 x ^ { 3 } + a x ^ { 2 } - 4$ is denoted by $\mathrm { p } ( x )$. It is given that ( $x - 2$ ) is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$.
When $a$ has this value,\\
(ii) factorise $\mathrm { p } ( x )$,\\
(iii) solve the inequality $\mathrm { p } ( x ) > 0$, justifying your answer.
\hfill \mbox{\textit{CAIE P3 2004 Q3 [6]}}