OCR A-Level Further Maths A polar and hyperbolic overview quiz quiz

the angle from the initial line

the distance from the pole

the $x$-coordinate

the area swept

$y = r\cos\theta$

$y = \frac{r}{\theta}$

$y = r\sin\theta$

$y = r^2$

differentiate both sides with respect to $\theta$

square the angle $\theta$ throughout

divide both sides through by $\theta$

multiply both sides by $r$

$\frac{1}{2}\int r^2\,d\theta$

$\frac{1}{2}\int r^2\theta\,d\theta$

$\pi\int r^2\,d\theta$

$\frac{1}{2}\int r\,d\theta$

the two consecutive values of $\theta$ where $r$ is a maximum

consecutive values of $\theta$ where $r = 0$

$\theta = 0$ and $\theta = 2\pi$ in every case

wherever the polar curve crosses the $y$-axis

$\frac{e^x - e^{-x}}{2}$

$\frac{e^x + e^{-x}}{e^x - e^{-x}}$

$\frac{e^x + e^{-x}}{2}$

$e^x + e^{-x}$

$\cosh^2 x + \sinh^2 x = 1$

$\sinh^2 x - \cosh^2 x = 1$

$\tanh^2 x + 1 = \cosh^2 x$

$\cosh^2 x - \sinh^2 x = 1$

even

odd

neither

periodic

$-\sinh x$

$\sinh x$

$\cosh x$

$\text{sech}^2 x$

$\ln(x + \sqrt{x^2 - 1})$

$\frac{1}{2}\ln\frac{1+x}{1-x}$

$\ln(x + \sqrt{x^2 + 1})$

$\ln x$

$x = a\cosh u$

$x = a\tanh u$

$x = a\sin u$

$x = a\sinh u$

$\sinh x + C$

$-\sinh x + C$

$\cosh x + C$

$\text{sech}^2 x + C$