| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Standard +0.3 This is a straightforward application of the factor theorem followed by a substitution problem. Part (i) involves routine verification and polynomial factorization (likely by inspection or algebraic division after finding one factor). Part (ii) requires recognizing that substituting x = 3^y transforms the exponential equation into f(x) = 0, then using logarithms to find y. While it requires connecting two ideas, both steps are standard techniques with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Verify that \(-96 + 100 + 8 - 12 = 0\) | B1 | |
| Attempt to find quadratic factor by division by \((x+2)\), reaching partial quotient \(12x^2 + kx\), inspection or use of identity | M1 | |
| Obtain \(12x^2 + x - 6\) | A1 | |
| State \((x+2)(4x+3)(3x-2)\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(3^y = \frac{2}{3}\) and no other value | B1 | |
| Use correct method for finding \(y\) from equation of form \(3^y = k\), where \(k > 0\) | M1 | |
| Obtain \(-0.369\) and no other value | A1 | [3] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Verify that $-96 + 100 + 8 - 12 = 0$ | B1 | |
| Attempt to find quadratic factor by division by $(x+2)$, reaching partial quotient $12x^2 + kx$, inspection or use of identity | M1 | |
| Obtain $12x^2 + x - 6$ | A1 | |
| State $(x+2)(4x+3)(3x-2)$ | A1 | [4] |
**Note:** M1 can be earned if inspection has unknown factor $Ax^2 + Bx - 6$ and an equation in $A$ and/or $B$ or equation $12x^2 + Bx + C$ and an equation in $B$ and/or $C$.
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $3^y = \frac{2}{3}$ and no other value | B1 | |
| Use correct method for finding $y$ from equation of form $3^y = k$, where $k > 0$ | M1 | |
| Obtain $-0.369$ and no other value | A1 | [3] |
---4 The polynomial $\mathrm { f } ( x )$ is defined by
$$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$
(i) Show that $\mathrm { f } ( - 2 ) = 0$ and factorise $\mathrm { f } ( x )$ completely.\\
(ii) Given that
$$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$
state the value of $3 ^ { y }$ and hence find $y$ correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2011 Q4 [7]}}