| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Show equation reduces to polynomial |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard logarithm laws (part i), factor theorem with simple integer testing (part ii), and checking domain restrictions (part iii). While it involves multiple steps, each technique is routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use law for logarithm for product or quotient or exponentiation AND for power | M1 | |
| Obtain \((4x - 5)^2(x + 1) = 27\) | B1 | |
| Obtain given equation correctly \(16x^3 - 24x^2 - 15x - 2 = 0\) | A1 | [3] |
| (ii) Obtain \(x = 2\) is root or \((x - 2)\) is a factor, or likewise with \(x = -\frac{1}{4}\) | B1 | |
| Divide by \((x - 2)\) to reach a quotient of the form \(16x^2 + kx\) | M1 | |
| Obtain quotient \(16x^2 + 8x + 1\) | A1 | |
| Obtain \((x - 2)(4x + 1)^2\) or \((x - 2), (4x + 1), (4x + 1)\) | A1 | [4] |
| (iii) State \(x = 2\) only | A1 | [1] |
(i) Use law for logarithm for product or quotient or exponentiation AND for power | M1 |
Obtain $(4x - 5)^2(x + 1) = 27$ | B1 |
Obtain given equation correctly $16x^3 - 24x^2 - 15x - 2 = 0$ | A1 | [3]
(ii) Obtain $x = 2$ is root or $(x - 2)$ is a factor, or likewise with $x = -\frac{1}{4}$ | B1 |
Divide by $(x - 2)$ to reach a quotient of the form $16x^2 + kx$ | M1 |
Obtain quotient $16x^2 + 8x + 1$ | A1 |
Obtain $(x - 2)(4x + 1)^2$ or $(x - 2), (4x + 1), (4x + 1)$ | A1 | [4]
(iii) State $x = 2$ only | A1 | [1]6 It is given that $2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3$.\\
(i) Show that $16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0$.\\
(ii) By first using the factor theorem, factorise $16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2$ completely.\\
(iii) Hence solve the equation $2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3$.
\hfill \mbox{\textit{CAIE P3 2014 Q6 [8]}}