Show that $\cos \theta  = \frac{3}{4}$.

Given that $\tan \theta  > 0$, find $\tan \theta$.

Let $y = \frac{1}{{\cos x}}$, for $0 < x < \frac{\pi }{2}$. The graph of $y$between $x = \theta$ and $x = \frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

evidence of summing to 1     (M1)

eg$\,\,\,\,\,$$\sum {p = 1}$

correct equation     A1

eg$\,\,\,\,\,$$\cos \theta  + 2\cos 2\theta  = 1$

correct equation in $\cos \theta$     A1

eg$\,\,\,\,\,$$\cos \theta  + 2(2{\cos ^2}\theta  - 1) = 1,{\text{ }}4{\cos ^2}\theta  + \cos \theta  - 3 = 0$

evidence of valid approach to solve quadratic     (M1)

eg$\,\,\,\,\,$factorizing equation set equal to $0,{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4 \times 4 \times ( - 3)} }}{8}$

correct working, clearly leading to required answer     A1

eg$\,\,\,\,\,$$(4\cos \theta  - 3)(\cos \theta  + 1),{\text{ }}\frac{{ - 1 \pm 7}}{8}$

correct reason for rejecting $\cos \theta  \ne  - 1$     R1

eg$\,\,\,\,\,$$\cos \theta$ is a probability (value must lie between 0 and 1), $\cos \theta  > 0$

Note:     Award R0 for $\cos \theta  \ne  - 1$ without a reason.

$\cos \theta  = \frac{3}{4}$    AG  N0

valid approach     (M1)

eg$\,\,\,\,\,$sketch of right triangle with sides 3 and 4, ${\sin ^2}x + {\cos ^2}x = 1$

correct working     

(A1)

eg$\,\,\,\,\,$missing side $= \sqrt 7 ,{\text{ }}\frac{{\frac{{\sqrt 7 }}{4}}}{{\frac{3}{4}}}$

$\tan \theta  = \frac{{\sqrt 7 }}{3}$     A1     N2

[3 marks]

attempt to substitute either limits or the function into formula involving ${f^2}$     (M1)

eg$\,\,\,\,\,$$\pi \int_\theta ^{\frac{\pi }{4}} {{f^2},{\text{ }}\int {{{\left( {\frac{1}{{\cos x}}} \right)}^2}} }$

correct substitution of both limits and function     (A1)

eg$\,\,\,\,\,$$\pi \int_\theta ^{\frac{\pi }{4}} {{{\left( {\frac{1}{{\cos x}}} \right)}^2}{\text{d}}x}$

correct integration     (A1)

eg$\,\,\,\,\,$$\tan x$

substituting their limits into their integrated function and subtracting     (M1)

eg$\,\,\,\,\,$$\tan \frac{\pi }{4} - \tan \theta$

Note:     Award M0 if they substitute into original or differentiated function.

$\tan \frac{\pi }{4} = 1$    (A1)

eg$\,\,\,\,\,$$1 - \tan \theta$

$V = \pi  - \frac{{\pi \sqrt 7 }}{3}$     A1     N3

[6 marks]