| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants with divisibility condition |
| Difficulty | Standard +0.3 This is a straightforward application of the factor theorem requiring students to recognize that x²-4x+4=(x-2)² means (x-2) is a repeated root, then use f(2)=0 to find a=8. Part (ii) involves factorizing and analyzing signs, which is routine once a is found. The question requires multiple steps but uses standard techniques without novel insight. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER State or imply that \(x - 2\) is a factor of \(f(x)\) | B1 | Substitute 2 for x and equate to zero |
| OR Commence division by \(x^2 - 4x + 4\) and obtain partial quotient \(x^2 + 2x\) | B1 | Complete the division and equate the remainder to zero |
| OR Commence inspection and obtain unknown factor \(x^2 + 2x + c\) | B1 | Obtain \(4c = a\) and an equation in c |
| (ii) EITHER Substitute \(a = 8\) and find other factor \(x^2 + 2x + 2\) by inspection or division | B1 | State that \(x^2 - 4x + 4 \geq 0\) for all x (condone \(>\) for \(\geq\)) |
| OR Equate derivative to zero and attempt to solve for x | M1 | Obtain \(x = -\frac{1}{2}\) and 2 |
**(i)** **EITHER** State or imply that $x - 2$ is a factor of $f(x)$ | B1 | Substitute 2 for x and equate to zero | M1 | Obtain answer $a = 8$ | A1 |
**OR** Commence division by $x^2 - 4x + 4$ and obtain partial quotient $x^2 + 2x$ | B1 | Complete the division and equate the remainder to zero | M1 | Obtain answer $a = 8$ | A1 |
**OR** Commence inspection and obtain unknown factor $x^2 + 2x + c$ | B1 | Obtain $4c = a$ and an equation in c | M1 | Obtain answer $a = 8$ | A1 | **[3]**
**(ii)** **EITHER** Substitute $a = 8$ and find other factor $x^2 + 2x + 2$ by inspection or division | B1 | State that $x^2 - 4x + 4 \geq 0$ for all x (condone $>$ for $\geq$) | B1 | Attempt to establish sign of the other factor | M1 | Show that $x^2 + 2x + 2 > 0$ for all x and complete the proof | A1 |
**OR** Equate derivative to zero and attempt to solve for x | M1 | Obtain $x = -\frac{1}{2}$ and 2 | A1 | Show correctly that $f(x)$ has a minimum at each of these values | A1 | Having also obtained and considered $x = 0$, complete the proof | A1 | **[4]**4 The polynomial $x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a$ is denoted by $\mathrm { f } ( x )$. It is given that $\mathrm { f } ( x )$ is divisible by $x ^ { 2 } - 4 x + 4$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, show that $\mathrm { f } ( x )$ is never negative.
\hfill \mbox{\textit{CAIE P3 2003 Q4 [7]}}