Step by step solutions.

Problem 30: Integer solutions of cubics ( vVV ) (i) Show that, if in is an integer such that (m - 3)' + m = (m + 3) then m' is a factor of 54. Deduce that there is no integer m which satisfies the equation (.). (ii) Show that, if n is an integer such that (n - 6) +n = (n+6), (.. ) then n is even. By writing n = 2m deduce that there is no integer n which satisfies the equation (..). (iii) Show that, if n is an integer such that (n - a) ) +n' = (n+a). where a is a non-zero integer, then n is even and a is even. Deduce that there is no integer in which satisfies the equation (. . .).Problem 60: Newton's cradle ( vV ) N particles P, Pr, Py. ..., Py with masses m, qm, q' m, ..., q -m, respectively, are at rest at distinct points along a straight line in gravity-free space. The particle P, is set in motion towards A with velocity V and in every subsequent impact the coefficient of restitution is e, where 0 off and that the shell hits the target, and let 8 be the (positive) angle through which the target rotates between the time at which the shell is fired and the time of impact. Show that 8 satisfies the equation 9'8" - 40'V38" + AR-W' =0. Deduce that there are exactly two possible values of 8. (iii) Let 8, and 8, be the possible values of S and let Pi and P, be the corresponding points of impact. By considering the quantities (8; + 8:) and 878; , or otherwise, show that the linear distance between P and P, is 2Rsin - VV2 - Rg