| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with routine application of R-cos(x-α) method, followed by straightforward equation solving and Simpson's rule application. All three parts use well-practiced techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(b\cos c = 1,\ b\sin c = 3\) | M1 | |
| \(b = \sqrt{1^2 + 3^2} = \sqrt{10}\) | A1 | |
| \(\tan c = 3,\ c = 1.25\) (3sf) | A1 | |
| \(\therefore f(x) = 2 + \sqrt{10}\cos(x - 1.25)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(2 + \sqrt{10}\cos(x - 1.249) = 0,\quad \cos(x-1.249) = -\frac{2}{\sqrt{10}}\) | M1 | |
| \(x - 1.249 = \pi - 0.8861,\ \pi + 0.8861 = 2.256, 4.028\) | M1 | |
| \(x = 3.50, 5.28\) (3sf) | A2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| Table of values: \(f(0)=3,\ f(0.5)=4.3159,\ f(1)=5.0647,\ f(1.5)=5.0632,\ f(2)=4.3117\) | M1 | |
| \(I \approx \frac{1}{3} \times 0.5 \times [3 + 4.3117 + 4(4.3159 + 5.0632) + 2(5.0647)]\) | M1 | |
| \(= 9.16\) (3sf) | A1 | (10) |
# Question 7:
## Part (i)
| Working/Answer | Mark | Notes |
|---|---|---|
| $b\cos c = 1,\ b\sin c = 3$ | M1 | |
| $b = \sqrt{1^2 + 3^2} = \sqrt{10}$ | A1 | |
| $\tan c = 3,\ c = 1.25$ (3sf) | A1 | |
| $\therefore f(x) = 2 + \sqrt{10}\cos(x - 1.25)$ | | |
## Part (ii)
| Working/Answer | Mark | Notes |
|---|---|---|
| $2 + \sqrt{10}\cos(x - 1.249) = 0,\quad \cos(x-1.249) = -\frac{2}{\sqrt{10}}$ | M1 | |
| $x - 1.249 = \pi - 0.8861,\ \pi + 0.8861 = 2.256, 4.028$ | M1 | |
| $x = 3.50, 5.28$ (3sf) | A2 | |
## Part (iii)
| Working/Answer | Mark | Notes |
|---|---|---|
| Table of values: $f(0)=3,\ f(0.5)=4.3159,\ f(1)=5.0647,\ f(1.5)=5.0632,\ f(2)=4.3117$ | M1 | |
| $I \approx \frac{1}{3} \times 0.5 \times [3 + 4.3117 + 4(4.3159 + 5.0632) + 2(5.0647)]$ | M1 | |
| $= 9.16$ (3sf) | A1 | **(10)** |
---7
$$f ( x ) = 2 + \cos x + 3 \sin x$$
(i) Express $\mathrm { f } ( x )$ in the form
$$\mathrm { f } ( x ) = a + b \cos ( x - c )$$
where $a , b$ and $c$ are constants, $b > 0$ and $0 < c < \frac { \pi } { 2 }$.\\
(ii) Solve the equation $\mathrm { f } ( x ) = 0$ for $x$ in the interval $0 \leq x \leq 2 \pi$.\\
(iii) Use Simpson's rule with four strips, each of width 0.5 , to find an approximate value for
$$\int _ { 0 } ^ { 2 } f ( x ) d x$$
\hfill \mbox{\textit{OCR C3 Q7 [10]}}