# STEP II - 91*4023334091 Sixth Term Examination Papers 9470...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Section A: Pure Mathematics 1 Let P be a given point on a given curve C . The osculating circle to C at P is deﬁned to be the circle that satisﬁes the following two conditions at P : it touches C ; and the rate of change of its gradient is equal to the rate of change of the gradient of C . Find the centre and radius of the osculating circle to the curve y = 1 - x + tan x at the point on the curve with x -coordinate 1 4 π . 2 Prove that cos 3 x = 4 cos 3 x - 3 cos x . Find and prove a similar result for sin 3 x in terms of sin x . (i) Let I( α ) = Z α 0 ( 7 sin x - 8 sin 3 x ) d x . Show that I( α ) = - 8 3 c 3 + c + 5 3 , where c = cos α . Write down one value of c for which I( α ) = 0. (ii) Useless Eustace believes that Z sin n x d x = sin n +1 x n + 1 for n = 1 , 2 , 3 , . . . . Show that Eustace would obtain the correct value of I( β ) , where cos β = - 1 6 . Find all values of α for which he would obtain the correct value of I( α ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

STEP II - 91*4023334091 Sixth Term Examination Papers 9470...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online