physicsandmathstutor.com 4724/01 Mark Scheme June 2010

Size: px

Start display at page:

Download "physicsandmathstutor.com 4724/01 Mark Scheme June 2010"

Charlene Cook
5 years ago
Views:

1 phsicsandmathstut.com 7/0 Mark Scheme June 00 First terms in epansion = B (simp to this, now later). rd term shown as 8 M 8 can be can be 9 = + 0 A. 8. th term shown as. M can be can be 7 = 0 0 ISW A Accept ISW 0 8 N.B. If 0, SR B to be awarded f 0 9. Do not mark as a MR. Attempt quotient rule M [ Show fraction with denom sin & num / sin / sin / cos / cos ] Numerat = sin. sin cos. cos A terms in an der { Product smbols must be clear implied b further wk } Reduce crect numerat to sin B sin sin cos Simplif to sin ISW A Accept sin A B C M F crect fmat A B C A A B A (B if cover-up rule used) C A (B if cover-up rule used) [NB: Partial fractions need not be written out; crect fmat + crect values sufficient. NB: Having obtained B & C b cover-up rule, candidates ma substitute into general epression & algebraicall manipulate; the M & A are then available if deserved.] These special cases using different fmats are the onl other ones to be considered Ma A B C D A B C ; M M; A0 f an values of A, B & C, A B f D ; M0 M; A f M A and B, A B f C

2 phsicsandmathstut.com 7/0 Mark Scheme June 00 d u Att b diff to connect & du find (not =du) M no accurac; not b parts d u du = u du A u u Indefinite integral u ( ) du {If relevant, cancel u/u and} attempt to square out {dep ki du where k A M Ma be implied later and I = u u u } Att to change limits if wking with f(u) after integration M re-subst into integral attempt and use & 7 Indef integ = 8 u / u 8u u / u u A u / u u 0 7 ISW but no +c A d d s.o.i. B Implied b e.g., d d B Diff eqn(=0 can be implied)(solve f 7 and ) put 0 M Produce onl 0 (though acceptable ) *A without an err seen Eliminate from curve eqn & eqn(s) just produced M Produce either 9 dep* A Disregard other solutions,, as the onl answer ISW dep* A Sign aspect must be clear (i) State/impl scalar product of an two vects = 0 M Scalar product of crect two vects = + a A ( a 0 MA) a A (ii) (a) Attempt to produce at least two relevant equations M e.g. t s.. Solve two not containing a f s and t 7 M Obtain at least one of s, t A Substitute in third equation & produce a A (b) Method f finding magnitude of an vect M possibl involving a Using a.b cos θ f the pair of direction vects M possibl involving a a b 07, 08 (07.8) 7, 7, 7., 7. (7.) c.a.o. A.87,.88 (.87707). 0

3 phsicsandmathstut.com 7/0 Mark Scheme June 00 7 (i) Differentiate as a quotient, t t t t v du u dv v u dv v du v M product clearl defined t A WWW t B M quoted/implied and used t t t t State squares +ve t & t ve (dep st marks) *A igne ref t, t t +ve dep*a ve t. Igne 0 (ii) Attempt to obtain t from either the equation M No accurac required t t A Substitute in the equation not et used in this part M equate the values of t Use crect meth to eliminate ( double-decker ) fractions M Obtain ISW A but not involving fractions 8 (i) Long division method Identit method Evidence of division process as far as st stage incl sub M Q R (Quotient = ) A Q (Remainder =) ISW A R ; N.B. might be B (ii) (a) Separate variables; M ma be implied later Change into their (Quotient + Rem ) M ln = (integration of their previous result) (+c)isw A f.t. if using Quot + (ii) (b) Substitute = 7, = 8 into their eqn containing c M & attempt c (., ln Substitute = and their value of c M & attempt to find =.00 (.009) Also + 0 e A Accept,.0, 9 Beware: an wrong wking anwhere A0 even if answer is one of the acceptable ones. 0 Rem ) 9

4 phsicsandmathstut.com 7/0 Mark Scheme June 00 9 (i) Attempt to multipl out cos M Min of crect terms Finding cos Use u, dv cos M st stage sin sin A cos sin cos A Finding cos st stage f()+/ g Change to k / / cos M where k, Crect version cos cos sin B seen anwhere in this part Result = A sin A 8 (i) ans = sin cos sin (+ c) A 9 Full crect 8 π cos M (ii) V = π 0 Use limits 0 & π (i) crect value = crectl on their (i) answer M π π A Final answer = A c.a.o. No follow-through Alternative methods cos If is changed into sin cos, award sin M f clear use of the product rule (though possibl trig differentiation inaccurate) A f cos sin sin B f reducing to a fraction with sin sin sin cos in the numerat A f crect final answer of sin sin If M cos cos sin, award sin f clear use of the product rule (though possibl trig differentiation inaccurate) is changed into A f cos sin sin. sin

5 phsicsandmathstut.com 7/0 Mark Scheme June 00 B f reducing to a fraction with sin sin sin cos in the numerat A f crect final answer of sin sin (ii)(a) If candidates use some long drawn-out method to find a instead of the direct route, allow M M as befe, f producing the equations f an satisfact method which will/does produce a, however involved A f a 7(ii) Marks f obtaining this Cartesian equation are not available in part (i). If part (ii) is done first and then part (i) is attempted using the Cartesian equation, award marks as follow: Method where candidates differentiate implicitl M f attempt at implicit differentiation A f M f substituting parametric values of and t A f simplifing to t A f finish as in iginal method Method where candidates manipulate the Cartesian equation to find = = M f attempt to re-arrange so that either = f() = g() A f crect = = M f differentiating as a quotient A f obtaining A f finish as in iginal method 8(ii)(b) If definite integrals are used, then 7 8 M f equivalent M f 8 7 equivalent A f,.0,.00 (.009) with caveat as in main scheme dep M

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com 7 Mark Scheme June 00 (Quotient =) ² + + (Remainder =) 0 Allow without working B M A A For correct leading term ² in quotient For evidence of division/identity process For correct