| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants with divisibility condition |
| Difficulty | Standard +0.3 This is a straightforward application of the factor theorem requiring polynomial division to find a constant, followed by a routine cubic inequality. The divisibility condition gives a clear method (equate remainder to zero), and part (ii) is standard sign analysis once factored. Slightly above average due to the two-part structure and algebraic manipulation required, but no novel insight needed. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Attempt division by \(2x^2 - 3x + 3\) and state partial quotient \(2x\) | B1 | |
| Complete division and form an equation for \(a\) | M1 | |
| Obtain \(a = 3\) | A1 | |
| OR1: By inspection or using an unknown factor \(bx + c\), obtain \(b = 2\) | B1 | |
| Complete the factorisation and obtain \(a\) | M1 | |
| Obtain \(a = 3\) | A1 | |
| OR2: Find a complex root of \(2x^2 - 3x + 3 = 0\) and substitute it in \(p(x)\) | M1 | |
| Equate a correct expression to zero | A1 | |
| Obtain \(a = 3\) | A1 | |
| OR3: Use \(2x^2 = 3x - 3\) in \(p(x)\) at least once | B1 | |
| Reduce the expression to the form \(a + c = 0\), or equivalent | M1 | |
| Obtain \(a = 3\) | A1 | [3] |
| (ii) State answer \(x < -\frac{1}{2}\) only | B1 | |
| Carry out a complete method for showing \(2x^2 - 3x + 3\) is never zero | M1 | |
| Complete the justification of the answer by showing that \(2x^2 - 3x + 3 > 0\) for all \(x\) | A1 | [3] |
| [These last two marks are independent of the B mark, so B0M1A1 is possible. Alternative methods include: (a) Complete the square M1 and use a correct completion to justify the answer A1; (b) Draw a recognizable graph of \(y = 2x^2 + 3x - 3\) or \(p(x)\) M1 and use a correct graph to justify the answer A1; (c) Find the x-coordinate of the stationary point of \(y = 2x^2 + 3x - 3\) and either find its y-coordinate or determine its nature M1, then use minimum point with correct coordinates to justify the answer A1.] |
**(i)** **EITHER:** Attempt division by $2x^2 - 3x + 3$ and state partial quotient $2x$ | B1 |
Complete division and form an equation for $a$ | M1 |
Obtain $a = 3$ | A1 |
**OR1:** By inspection or using an unknown factor $bx + c$, obtain $b = 2$ | B1 |
Complete the factorisation and obtain $a$ | M1 |
Obtain $a = 3$ | A1 |
**OR2:** Find a complex root of $2x^2 - 3x + 3 = 0$ and substitute it in $p(x)$ | M1 |
Equate a correct expression to zero | A1 |
Obtain $a = 3$ | A1 |
**OR3:** Use $2x^2 = 3x - 3$ in $p(x)$ at least once | B1 |
Reduce the expression to the form $a + c = 0$, or equivalent | M1 |
Obtain $a = 3$ | A1 | [3]
**(ii)** State answer $x < -\frac{1}{2}$ only | B1 |
Carry out a complete method for showing $2x^2 - 3x + 3$ is never zero | M1 |
Complete the justification of the answer by showing that $2x^2 - 3x + 3 > 0$ for all $x$ | A1 | [3] |
[These last two marks are independent of the B mark, so B0M1A1 is possible. Alternative methods include: (a) Complete the square M1 and use a correct completion to justify the answer A1; (b) Draw a recognizable graph of $y = 2x^2 + 3x - 3$ or $p(x)$ M1 and use a correct graph to justify the answer A1; (c) Find the x-coordinate of the stationary point of $y = 2x^2 + 3x - 3$ and either find its y-coordinate or determine its nature M1, then use minimum point with correct coordinates to justify the answer A1.] |5 The polynomial $4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that $\mathrm { p } ( x )$ is divisible by $2 x ^ { 2 } - 3 x + 3$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, solve the inequality $\mathrm { p } ( x ) < 0$, justifying your answer.
\hfill \mbox{\textit{CAIE P3 2008 Q5 [6]}}