Created by T Madas Question 206 (****) y y = x3 + 3x − 11x − O Q P R x The figure above shows the curve with equation y = x3 + x − 11x − The curve crosses the x axis at the points P , Q and R ( 2,0 ) The tangent to the curve at R is the straight line L1 a) Find an equation of L1 The normal to the curve at P is the straight line L2 The point S is the point of intersection between L1 and L2 b) Show that PSR = 90° SYN-M , y = 25 x − 50 Created by T Madas Created by T Madas Question 207 (****) A 4r D O θ R 3r C B The figure above shows a circular sector OAB of radius 4r subtending an angle θ radians at the centre O Another circular sector OCD of radius 3r also subtending an angle θ radians at the centre O is removed from the first sector leaving the shaded region R It is given that R has an area of 50 square units a) Show that the perimeter P , of the region R , is given by P = 2r + 100 r b) Given that the value of r can vary, … i … find an exact value of r for which P is stationary ii … show clearly that the value of r found above gives the minimum iii value for P c) Calculate the minimum value of P C2L , r = ≈ 7.07 , Pmin = 20 ≈ 28.28 Created by T Madas Created by T Madas Question 208 (****) y O x2 y= − x P x The figure above shows the curve C with equation y= x2 − , x ≠ x The curve crosses the x axis at the point P The straight line L is the normal to C at P a) Find … i … the coordinates of P ii … an equation of L b) Show that L does not meet C again C1Y , P ( 2,0 ) , x + y = Created by T Madas