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By Eva Shipton, Michael Kenwood, C.T. Moss, Charles Plumpton

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If 1 is wrong, 2 is wrong, key must be E; there is no need to check that 3 is correct. If 1 is wrong, 2 is correct, key must be C; there is no need to check that 3 is correct. A good diagram often supplies all that is needed to answer a question. R, 1 has only positive roots 2 has an infinite number of roots 3 has two roots in the interval (0, n) For this type of equation, which has no easy method of solution, a graphical method provides a great deal of information about the roots. In this case the roots are shown graphically as the values of x at the intersection of the curves y = e-x and y = sin x.

Although the values of tX are not needed). The question is based on the fact that lsin xl ~ 1, giving 1 and 2 both incorrect, and so 3 is correct, giving key E. 30 The equation of a plane given in parametric form must have two parameters, so 1 is incorrect while 2 is correct. 3 is also correct as it is a correct perpendicular form of the equation of a plane. There is no need to identify the vectors, simply to see whether the equations are feasible as the equations of a plane. Key C. 4 Paper 2 (multiple-choice) Time allowed: 1!

2! = 360. 42 Examinations in Mathematics 00 7 L <-lY<*Y = - * + <*>2 - + (- *>2 + (- w+ . . If you do not recognize the series immediately, the first two or three terms are a help to see that it is a geometrical progression. _i s S 00 4 = 1- (-*>=-g-. 8 cos oo =cos 310°. The general solution is required. r--- that so• cos 310° =cos (- 50°) =cos 50° so cos - oo = cos 50° (} = 360n ± 50 (the general solution should be known). 9 Care is needed with this one. Use => dy = dy;dx dx dt dt dy dx 2t 2 = 3t2 = 3t" Then or 11 d 3 -d (ln4+3lnx]=-.

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