| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then solve related equation |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution of x=-2, followed by routine polynomial division and factorization. Part (ii) involves a simple substitution y=1/x to relate the roots. All steps are standard textbook procedures with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate expression when \(x = -2\) | M1 | |
| Obtain 0 with all necessary detail present | A1 | Use of \(f(x) = (x+2)(ax^2+bx+c)\) to find \(a\), \(b\) and \(c\), allow M1 A0. Use of \(f(x) = (x+2)(ax^2+bx+c)+d\) to find \(a\), \(b\) and \(c\), and show \(d=0\), allow M1 A1 |
| Carry out division, or equivalent, at least as far as \(x^2\) and \(x\) terms in quotient | M1 | |
| Obtain \(6x^2 + x - 35\) | A1 | |
| Obtain factorised expression \((x+2)(2x+5)(3x-7)\) | A1 | |
| Total: | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply substitution \(x = \frac{1}{y}\) or equivalent | M1 | |
| Obtain \(-\frac{1}{2}\), \(-\frac{2}{5}\), \(\frac{3}{7}\) | A1 | |
| Total: | 2 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate expression when $x = -2$ | M1 | |
| Obtain 0 with all necessary detail present | A1 | Use of $f(x) = (x+2)(ax^2+bx+c)$ to find $a$, $b$ and $c$, allow **M1 A0**. Use of $f(x) = (x+2)(ax^2+bx+c)+d$ to find $a$, $b$ and $c$, and show $d=0$, allow **M1 A1** |
| Carry out division, or equivalent, at least as far as $x^2$ and $x$ terms in quotient | M1 | |
| Obtain $6x^2 + x - 35$ | A1 | |
| Obtain factorised expression $(x+2)(2x+5)(3x-7)$ | A1 | |
| **Total:** | **5** | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply substitution $x = \frac{1}{y}$ or equivalent | M1 | |
| Obtain $-\frac{1}{2}$, $-\frac{2}{5}$, $\frac{3}{7}$ | A1 | |
| **Total:** | **2** | |6 (i) Use the factor theorem to show that ( $x + 2$ ) is a factor of the expression
$$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$
and hence factorise the expression completely.\\
(ii) Deduce the roots of the equation
$$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$
\hfill \mbox{\textit{CAIE P2 2017 Q6 [7]}}