| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve inequality involving polynomial |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring routine application of the factor theorem to find a constant, then solving a cubic inequality. Part (a) involves substituting x = -3/2 and solving a linear equation. Part (b) requires factorizing the cubic (one factor given) and analyzing sign changes, which is standard A-level technique with no novel insight needed. Slightly above average difficulty due to the cubic inequality in part (b), but still a textbook-style question. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x = -\frac{3}{2}\) and equate result to zero | M1 | Or divide by \(2x+3\) and set constant remainder equal to zero. Or state \((2x^3 - x^2 + a) = (2x+3)(x^2 + px + q)\), compare coefficients and solve for \(p\) or \(q\) |
| Obtain \(a = 9\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Commence division by \((2x+3)\) reaching a partial quotient \(x^2 + kx\) | *M1 | M1 earned if inspection reaches unknown factor \(x^2 + Bx + C\) and equation in \(B\) and/or \(C\), or unknown factor \(Ax^2 + Bx + 3\) and equation in \(A\) and/or \(B\) |
| Obtain factorisation \((2x+3)(x^2 - 2x + 3)\) | A1 | Allow if correct quotient seen. Correct factors seen in (a) and quoted or used here scores M1A1 |
| Show that \(x^2 - 2x + 3\) is always positive, or \(2x^3 - x^2 + 9\) only intersects the \(x\)-axis once | DM1 | Must use their quadratic factor. SC If M0, allow B1 if state \(x < -\frac{3}{2}\) and no error seen |
| State final answer \(x < -\frac{3}{2}\) from correct work | A1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x = -\frac{3}{2}$ and equate result to zero | M1 | Or divide by $2x+3$ and set constant remainder equal to zero. Or state $(2x^3 - x^2 + a) = (2x+3)(x^2 + px + q)$, compare coefficients and solve for $p$ or $q$ |
| Obtain $a = 9$ | A1 | |
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## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Commence division by $(2x+3)$ reaching a partial quotient $x^2 + kx$ | *M1 | M1 earned if inspection reaches unknown factor $x^2 + Bx + C$ and equation in $B$ and/or $C$, or unknown factor $Ax^2 + Bx + 3$ and equation in $A$ and/or $B$ |
| Obtain factorisation $(2x+3)(x^2 - 2x + 3)$ | A1 | Allow if correct quotient seen. Correct factors seen in (a) and quoted or used here scores M1A1 |
| Show that $x^2 - 2x + 3$ is always positive, or $2x^3 - x^2 + 9$ only intersects the $x$-axis once | DM1 | Must use their quadratic factor. SC If M0, allow B1 if state $x < -\frac{3}{2}$ and no error seen |
| State **final** answer $x < -\frac{3}{2}$ from correct work | A1 | |
---2 The polynomial $2 x ^ { 3 } - x ^ { 2 } + a$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that ( $2 x + 3$ ) is a factor of $\mathrm { p } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item When $a$ has this value, solve the inequality $\mathrm { p } ( x ) < 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q2 [6]}}