| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two polynomials, shared factor or separate conditions |
| Difficulty | Moderate -0.3 Part (i) is straightforward application of the factor theorem to find two constants by substituting x=-3 into both polynomials. Part (ii) requires forming a quadratic from q(x)-p(x) and using the discriminant, which is standard technique. The curve/tangent part involves basic differentiation and solving a transcendental equation. All steps are routine A-level procedures with no novel insight required, making this slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -3\) into either \(p(x)\) or \(q(x)\) and equate to zero (may be implied) | M1 | Allow long division, but the remainder needs to be independent of \(x\) |
| Obtain \(a = -11\) | A1 | |
| Obtain \(b = -8\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Divide \(x+3\) into expression for \(q(x) - p(x)\) (may be a four term cubic equation), or obtain a 3 term cubic equation by subtraction | \*M1 | Allow \*M1 for their \(x^3 + 3x + 36\), but must have integer values for \(a\) and \(b\) |
| Obtain \(x^2 - 3x + 12\) or \(x^2 - 2x - 5\) and \(2x^2 - 5x + 7\) | A1 | |
| Apply discriminant to quadratic factor of \(q(x) - p(x)\) or equivalent | DM1 | dep on \*M |
| Obtain \(-39\) or equivalent and conclude appropriately | A1 | |
| Total | 4 |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -3$ into either $p(x)$ or $q(x)$ and equate to zero (may be implied) | M1 | Allow long division, but the remainder needs to be independent of $x$ |
| Obtain $a = -11$ | A1 | |
| Obtain $b = -8$ | A1 | |
| **Total** | **3** | |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide $x+3$ into expression for $q(x) - p(x)$ (may be a four term cubic equation), or obtain a 3 term cubic equation by subtraction | \*M1 | Allow \*M1 for their $x^3 + 3x + 36$, but must have integer values for $a$ and $b$ |
| Obtain $x^2 - 3x + 12$ or $x^2 - 2x - 5$ and $2x^2 - 5x + 7$ | A1 | |
| Apply discriminant to quadratic factor of $q(x) - p(x)$ or equivalent | DM1 | dep on \*M |
| Obtain $-39$ or equivalent and conclude appropriately | A1 | |
| **Total** | **4** | |
---4 The polynomials $\mathrm { p } ( x )$ and $\mathrm { q } ( x )$ are defined by
$$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$
where $a$ and $b$ are constants. It is given that $( x + 3 )$ is a factor of $\mathrm { p } ( x )$ and also of $\mathrm { q } ( x )$.\\
(i) Find the values of $a$ and $b$.\\
(ii) Show that the equation $\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0$ has only one real root.\\
\includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749}
The diagram shows the curve $y = 4 e ^ { - 2 x }$ and a straight line. The curve crosses the $y$-axis at the point $P$. The straight line crosses the $y$-axis at the point $( 0,9 )$ and its gradient is equal to the gradient of the curve at $P$. The straight line meets the curve at two points, one of which is $Q$ as shown.\\
\hfill \mbox{\textit{CAIE P2 2017 Q4 [7]}}