1 Find the cartesian equation corresponding to the polar equation \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\).
Sketch the the graph of \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\), indicating clearly the polar coordinates of the intersection with the initial line.
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Question 1:
Answer Marks
Guidance
Working/Answer Mark
Guidance
\(r\cos\left(\theta - \frac{\pi}{4}\right) = \sqrt{2}\) M1
Re-writes equation
\(r\left(\frac{\cos\theta}{\sqrt{2}} + \frac{\sin\theta}{\sqrt{2}}\right) = \sqrt{2}\) A1
Uses compound angle formula
\(r\cos\theta + r\sin\theta = 2 \Rightarrow x + y = 2\) or \(y = 2-x\) A1
Changes to Cartesian; part total 3
Straight line at \(-\frac{\pi}{4}\) to the initial line B1
Sketches graph
Point \((2,0)\) clearly indicated B1
Part total 2; Total [5]
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## Question 1:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $r\cos\left(\theta - \frac{\pi}{4}\right) = \sqrt{2}$ | M1 | Re-writes equation |
| $r\left(\frac{\cos\theta}{\sqrt{2}} + \frac{\sin\theta}{\sqrt{2}}\right) = \sqrt{2}$ | A1 | Uses compound angle formula |
| $r\cos\theta + r\sin\theta = 2 \Rightarrow x + y = 2$ or $y = 2-x$ | A1 | Changes to Cartesian; part total 3 |
| Straight line at $-\frac{\pi}{4}$ to the initial line | B1 | Sketches graph |
| Point $(2,0)$ clearly indicated | B1 | Part total 2; **Total [5]** |
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1 Find the cartesian equation corresponding to the polar equation $r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)$.
Sketch the the graph of $r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)$, for $- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi$, indicating clearly the polar coordinates of the intersection with the initial line.
\hfill \mbox{\textit{CAIE FP1 2012 Q1 [5]}}