| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.3 This is a standard A-level question testing routine application of factor and remainder theorems. Part (i) involves setting up two simultaneous equations using p(-2)=0 and p(2)=28, which is straightforward. Part (ii) requires factorising a cubic once coefficients are known. Part (iii) adds a minor twist with the substitution 2^y but only requires recognizing that 2^y is always positive. All techniques are standard textbook exercises with no novel problem-solving required, 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 theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -2\) and equate to zero | M1 | |
| Substitute \(x = 2\) and equate to \(28\) | M1 | |
| Obtain \(-9a + 4b + 34 = 0\) and \(7a + 4b - 62 = 0\) or equivalents | A1 | |
| Solve a relevant pair of simultaneous equations for \(a\) or \(b\) | M1 | |
| Obtain \(a = 6\), \(b = 5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Divide by \(x + 2\), or equivalent, at least as far as \(k_1x^2 + k_2x\) | M1 | |
| Obtain \(6x^2 - 7x - 3\) | A1 | |
| Obtain \((x+2)(3x+1)(3x-3)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Refer to, or clearly imply, fact that \(2^y\) is positive | M1 | |
| State one | A1\(\checkmark\) | following 3 linear factors from part (ii) |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -2$ and equate to zero | M1 | |
| Substitute $x = 2$ and equate to $28$ | M1 | |
| Obtain $-9a + 4b + 34 = 0$ and $7a + 4b - 62 = 0$ or equivalents | A1 | |
| Solve a relevant pair of simultaneous equations for $a$ or $b$ | M1 | |
| Obtain $a = 6$, $b = 5$ | A1 | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide by $x + 2$, or equivalent, at least as far as $k_1x^2 + k_2x$ | M1 | |
| Obtain $6x^2 - 7x - 3$ | A1 | |
| Obtain $(x+2)(3x+1)(3x-3)$ | A1 | |
## Question 6(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Refer to, or clearly imply, fact that $2^y$ is positive | M1 | |
| State one | A1$\checkmark$ | following 3 linear factors from part (ii) |6 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$
where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is 28 when $\mathrm { p } ( x )$ is divided by $( x - 2 )$.\\
(i) Find the values of $a$ and $b$.\\
(ii) Hence factorise $\mathrm { p } ( x )$ completely.\\
(iii) State the number of roots of the equation $\mathrm { p } \left( 2 ^ { y } \right) = 0$, justifying your answer.\\
\includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799}
The diagram shows part of the curve
$$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$
The shaded region is bounded by the curve and the two axes.\\
(i) Show that $2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$ can be expressed in the form
$$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$
where the values of the constants $k _ { 1 }$ and $k _ { 2 }$ are to be determined.\\
(ii) Find the exact area of the shaded region.\\
\hfill \mbox{\textit{CAIE P2 2017 Q6 [10]}}