Standard +0.8 This question requires using the double angle formula to express cos x in terms of cos(x/2), leading to a quadratic equation in cos(x/2). Students must then solve the quadratic, apply inverse cosine carefully to the half-angle, and find all solutions in the given range. The half-angle manipulation and ensuring all solutions are found makes this moderately harder than standard trigonometric equations.
Use correct double angle formula to obtain an equation in \(\cos\left(\frac{x}{2}\right)\) only
*M1
e.g. \(2\left(2\cos^2\left(\frac{x}{2}\right)-1\right) - \cos\left(\frac{x}{2}\right) = 1\)
Obtain a 3 term quadratic in \(\cos\left(\frac{x}{2}\right)\)
A1
e.g. \(4\cos^2\left(\frac{x}{2}\right) - \cos\left(\frac{x}{2}\right) - 3 = 0\). Allow \(4\cos^2 u - \cos u - 3 = 0\), condone \(\frac{x}{2} = x\)
Obtain \(\cos\left(\frac{x}{2}\right) = -\frac{3}{4}\) and \(\cos\left(\frac{x}{2}\right) = 1\)
A1
Allow answer in \(u\) e.g. \((4\cos u + 3)(\cos u - 1)\) and condone \(\frac{x}{2} = x\)
Solve for the original \(x\)
DM1
Must see evidence of doubling, not halving
Obtain \(x = 0\) and \(4.84\) and no others in the interval
A1
Ignore any answers outside interval. Accept AWRT 4.84. Accept \(1.54\pi\). Must be in radians. 277.2 indicates M1 but is A0
Alternative Method:
Answer
Marks
Guidance
Answer
Mark
Guidance
Use correct double angle formula to obtain an equation in \(\cos x\) only
*M1
e.g. \(2\cos x - 1 = \sqrt{\frac{\cos x + 1}{2}}\)
Obtain a 3 term quadratic in \(\cos x\)
A1
e.g. \(8\cos^2 x - 9\cos x + 1 = 0\)
Obtain \(\cos x = \frac{1}{8}\) and \(\cos x = 1\)
A1
Solve for \(x\)
DM1
Obtain answers \(x = 0\) and \(4.84\) and no others in the interval
A1
Ignore any answers outside interval. Accept AWRT 4.84. Must be in radians. 277.2 is A0
Total: 5
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct double angle formula to obtain an equation in $\cos\left(\frac{x}{2}\right)$ only | *M1 | e.g. $2\left(2\cos^2\left(\frac{x}{2}\right)-1\right) - \cos\left(\frac{x}{2}\right) = 1$ |
| Obtain a 3 term quadratic in $\cos\left(\frac{x}{2}\right)$ | A1 | e.g. $4\cos^2\left(\frac{x}{2}\right) - \cos\left(\frac{x}{2}\right) - 3 = 0$. Allow $4\cos^2 u - \cos u - 3 = 0$, condone $\frac{x}{2} = x$ |
| Obtain $\cos\left(\frac{x}{2}\right) = -\frac{3}{4}$ and $\cos\left(\frac{x}{2}\right) = 1$ | A1 | Allow answer in $u$ e.g. $(4\cos u + 3)(\cos u - 1)$ and condone $\frac{x}{2} = x$ |
| Solve for the **original** $x$ | DM1 | Must see evidence of doubling, not halving |
| Obtain $x = 0$ and $4.84$ and no others in the interval | A1 | Ignore any answers outside interval. Accept AWRT 4.84. Accept $1.54\pi$. Must be in radians. 277.2 indicates M1 but is A0 |
**Alternative Method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct double angle formula to obtain an equation in $\cos x$ only | *M1 | e.g. $2\cos x - 1 = \sqrt{\frac{\cos x + 1}{2}}$ |
| Obtain a 3 term quadratic in $\cos x$ | A1 | e.g. $8\cos^2 x - 9\cos x + 1 = 0$ |
| Obtain $\cos x = \frac{1}{8}$ and $\cos x = 1$ | A1 | |
| Solve for $x$ | DM1 | |
| Obtain answers $x = 0$ and $4.84$ and no others in the interval | A1 | Ignore any answers outside interval. Accept AWRT 4.84. Must be in radians. 277.2 is A0 |
| **Total: 5** | | |
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