Standard +0.8 This question requires finding where dr/dθ and r satisfy the condition for a tangent parallel to the initial line (dy/dx = 0), which involves the formula dy/dx = (dr/dθ sin θ + r cos θ)/(dr/dθ cos θ - r sin θ). Students must derive and solve a trigonometric equation, then calculate the distance OP. This goes beyond routine polar coordinate problems and requires understanding the geometric relationship between polar derivatives and Cartesian tangents, making it moderately challenging for Further Maths.
2. The curve \(C\) has polar equation
$$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.
Given that \(O\) is the pole, find the exact length of the line \(O P\).
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I appreciate you sharing this, but the content you've provided appears to be severely corrupted or damaged during extraction. The text shows:
- Fragmented numbering (1, 2, 3, 4, 5, 6, 7, 8)
- Incomplete marking codes (M1, A1, B1, DM1)
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- No coherent mathematical content or clear marking points
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**Could you please provide:**
1. The original mark scheme file (PDF, image, or text)
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2. The curve $C$ has polar equation
$$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
At the point $P$ on $C$, the tangent to $C$ is parallel to the initial line.\\
Given that $O$ is the pole, find the exact length of the line $O P$.\\
\hfill \mbox{\textit{Edexcel FP2 2012 Q2 [7]}}