| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find function constants from given conditions |
| Difficulty | Moderate -0.8 This is a straightforward two-equation system using exact trig values (cos(π/3)=1/2, cos(π)=-1). Part (i) requires simple simultaneous equations; part (ii) asks for range exclusion from a cosine function, which is standard bookwork. Both parts are routine applications with no problem-solving insight required. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.05a Sine, cosine, tangent: definitions for all arguments1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + \frac{1}{2}b = 5\) | B1 | Alternatively these marks can be awarded when \(\frac{1}{2}\) and \(-1\) appear after \(a\) or \(b\) has been eliminated |
| \(a - b = 11\) | B1 | |
| \(\rightarrow a = 7\) and \(b = -4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + b\) or \(their\ a + their\ b\ (3)\) | B1 | Not enough to be seen in a table of values – must be selected. Graph from their values can get both marks. Note: Use of \(b^2 - 4ac\) scores 0/3 |
| \(a - b\) or \(their\ a - their\ b\ (11)\) | B1 | |
| \(\rightarrow k < 3,\ k > 11\) | B1 | Both inequalities correct. Allow combined statement as long as correct inequalities if taken separately. Both answers correct from T & I or guesswork 3/3 otherwise 0/3 |
## Question 4(i):
$a + \frac{1}{2}b = 5$ | B1 | Alternatively these marks can be awarded when $\frac{1}{2}$ and $-1$ appear after $a$ or $b$ has been eliminated
$a - b = 11$ | B1 |
$\rightarrow a = 7$ and $b = -4$ | B1 |
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## Question 4(ii):
$a + b$ or $their\ a + their\ b\ (3)$ | B1 | Not enough to be seen in a table of values – must be selected. Graph from their values can get both marks. **Note: Use of $b^2 - 4ac$ scores 0/3**
$a - b$ or $their\ a - their\ b\ (11)$ | B1 |
$\rightarrow k < 3,\ k > 11$ | B1 | Both inequalities correct. Allow combined statement as long as correct inequalities if taken separately. Both answers correct from T & I or guesswork 3/3 otherwise 0/3
---4 The function f is such that $\mathrm { f } ( x ) = a + b \cos x$ for $0 \leqslant x \leqslant 2 \pi$. It is given that $\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5$ and $\mathrm { f } ( \pi ) = 11$.\\
(i) Find the values of the constants $a$ and $b$.\\
\includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}\\
(ii) Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = k$ has no solution.\\
\includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632}
The diagram shows a three-dimensional shape. The base $O A B$ is a horizontal triangle in which angle $A O B$ is $90 ^ { \circ }$. The side $O B C D$ is a rectangle and the side $O A D$ lies in a vertical plane. Unit vectors $\mathbf { i }$ and $\mathbf { j }$ are parallel to $O A$ and $O B$ respectively and the unit vector $\mathbf { k }$ is vertical. The position vectors of $A , B$ and $D$ are given by $\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }$ and $\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }$.\\
\hfill \mbox{\textit{CAIE P1 2018 Q4 [6]}}