| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Show equation reduces to polynomial |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring routine application of the factor theorem, algebraic manipulation using the double angle formula (cos 2θ = 2cos²θ - 1), and solving a cubic equation using the factor from part (i). While it involves multiple steps, each step uses standard A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -\frac{3}{2}\) and simplify | M1 | Allow use of identity assuming a factor of \(2x+3\) to obtain a quadratic factor. Need to see use of 4 equations to verify quadratic for M1, A1 for conclusion. Allow verification by expansion. Allow use of long division, must reach a remainder of zero for M1 |
| Obtain \(-27+9+15+3\) or equivalent, hence zero and conclude; may have explanation at start of working | A1 | Need powers of \(-\frac{3}{2}\) evaluating for A1. AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\cos 2\theta = 2\cos^2\theta - 1\) | B1 | |
| Simplify \(a\cos^2\theta + b = \frac{6\cos\theta - 5}{2\cos\theta + 1}\) to polynomial form | M1 | |
| Obtain \(8\cos^3\theta + 4\cos^2\theta - 10\cos\theta + 3 = 0\) | A1 | AG; necessary detail needed, must be completely correct with no poor use of brackets for A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt either division by \(2x+3\) and reach partial quotient \(x^2 + kx\) or use of identity or inspection | \*M1 | Or equivalent using \(\cos\theta\) or \(c\) |
| Obtain quotient \(4x^2 - 4x + 1\) | A1 | Or equivalent |
| Obtain factorised form \((2x+3)(2x-1)^2\) | A1 | Or equivalent, may be implied by later work |
| Solve for \(\cos\theta = k\) to find at least one value between 0 and 360 | M1 | Dependent \*M |
| Obtain 60 and 300 and no others | A1 | SC1: Equation solver used to obtain 60 and 300 and no others, then 5/5. SC2: Equation solver used to obtain 60 then 4/5. SC3: \(\cos\theta = 0.5\), \((\cos\theta = -1.5)\) seen implies first 3 marks |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -\frac{3}{2}$ and simplify | **M1** | Allow use of identity assuming a factor of $2x+3$ to obtain a quadratic factor. Need to see use of 4 equations to verify quadratic for M1, A1 for conclusion. Allow verification by expansion. Allow use of long division, must reach a remainder of zero for M1 |
| Obtain $-27+9+15+3$ or equivalent, hence zero and conclude; may have explanation at start of working | **A1** | Need powers of $-\frac{3}{2}$ evaluating for A1. AG; necessary detail needed |
**Total: 2 marks**
---
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\cos 2\theta = 2\cos^2\theta - 1$ | **B1** | |
| Simplify $a\cos^2\theta + b = \frac{6\cos\theta - 5}{2\cos\theta + 1}$ to polynomial form | **M1** | |
| Obtain $8\cos^3\theta + 4\cos^2\theta - 10\cos\theta + 3 = 0$ | **A1** | AG; necessary detail needed, must be completely correct with no poor use of brackets for A1 |
**Total: 3 marks**
---
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt either division by $2x+3$ and reach partial quotient $x^2 + kx$ or use of identity or inspection | **\*M1** | Or equivalent using $\cos\theta$ or $c$ |
| Obtain quotient $4x^2 - 4x + 1$ | **A1** | Or equivalent |
| Obtain factorised form $(2x+3)(2x-1)^2$ | **A1** | Or equivalent, may be implied by later work |
| Solve for $\cos\theta = k$ to find at least one value between 0 and 360 | **M1** | Dependent \*M |
| Obtain 60 and 300 and no others | **A1** | SC1: Equation solver used to obtain 60 and 300 and no others, then 5/5. SC2: Equation solver used to obtain 60 then 4/5. SC3: $\cos\theta = 0.5$, $(\cos\theta = -1.5)$ seen implies first 3 marks |
**Total: 5 marks**7 (i) Use the factor theorem to show that ( $2 x + 3$ ) is a factor of
$$8 x ^ { 3 } + 4 x ^ { 2 } - 10 x + 3$$
(ii) Show that the equation $2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }$ can be expressed as
$$8 \cos ^ { 3 } \theta + 4 \cos ^ { 2 } \theta - 10 \cos \theta + 3 = 0 .$$
(iii) Solve the equation $2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P2 2018 Q7 [10]}}