Cambridge Assessment International Education Cambridge Ordinary Level. Published

Size: px

Start display at page:

Download "Cambridge Assessment International Education Cambridge Ordinary Level. Published"

Elisabeth Norton
5 years ago
Views:

1 Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/ Paper 07 MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the eamination. It shows the basis on which Eaminers were instructed to award marks. It does not indicate the details of the discussions that took place at an Eaminers meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Eaminer Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the 07 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is a registered trademark. This document consists of 8 printed pages. UCLES 07 [Turn over

2 4037/ Cambridge O Level Mark Scheme 07 The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M A B Method marks, awarded for a valid method applied to the problem. Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. Mark for a correct result or statement independent of Method marks. When a part of a question has two or more method steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation dep is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt cao dep FT isw nfww oe rot SC soi answers which round to correct answer only dependent follow through after error ignore subsequent working not from wrong working or equivalent rounded or truncated Special Case seen or implied UCLES 07 Page of 8

3 4037/ Cambridge O Level Mark Scheme 07 z = Accept ( ) b( ) a = + 3 Equate both 3 terms and constant terms to obtain two equations in a and b. 7a+ b= 4a+ b= Attempt to solve a pair of linear simultaneous eqns to a = or b = Both correct. Accept equation with a multiple of 3 dep a = and b= ( ) + = + Attempt to multiply by or or divide by = + 5 or = or = or = Simplified numerical powers 5+ 6= 0 dep obtain a three term quadratic. Allow errors in signs and coefficients but not powers ( )( ) 3 = 0 Solve a three term quadratic = 3 or only 3 Correctly obtain a value of = Inequality not required Correctly obtain a value of > and < = Inequality not required dep mark final answer(s). Allow < < UCLES 07 Page 3 of 8

4 4037/ Cambridge O Level Mark Scheme = y 7y = 6 7y 6+ 4= y y 7y+ ( y 3)( y 4)( = 0) or ( 5)( )( = 0) allow 4 for6 Attempt to eliminate or y to obtain a three term quadratic. Solve a three term quadratic dep y= 3, = 5 or y= 4 = Allow for values seen even if correct pairs not clear. 5(i) 0 C 4 = 0 5(ii) Mystery others = 5 C 5 C=00 3 Mystery other = 5 C 5 3 C= 50 4 Mystery = 5 C 4 =5 Total 55 Alternative Method B3 for one combination, unsimplifiied for second combination, unsimplifiied for third combination, unsimplifiied and total All 0 Mystery Mystery All minus 0 or or both = 0 C4 C C dep Mystery and 0 mystery 3 unsimplified = = 55 dep final answer 5(iii) CR = 5 C 3 C C = 60 MCR = 5 C 3 C C = C C C = CR = 5 Total 05 B3 for one combination, unsimplifiied for second combination, unsimplifiied for third combination, unsimplifiied and total 6(i) 500 Ignore units Condone rfor π h= 500 h= π 6(ii) A = π + π h Correct epression for A and insert for their h. 500 = π + π = π 000 π + Answer given Condone rfor. UCLES 07 Page 4 of 8

5 4037/ Cambridge O Level Mark Scheme 07 6(iii) Differentiate: at least one power reduced by da 000 = 4π d da 000 = = = d 4π ( ( )) 0 3 isw or A = π = 349 cm 4.3 ( ) awrt 349 d A 000 = 4 π + ( > 0 ) or a positive value 3 d ( min) 7(i) dy 3 (Gradient or ) = d = 3 3 ( ) y = + C Correct second differential (need not be evaluated) and conclusion. or Eamine correct gradient either side of = 4.3 and conclusion Gradient = Negative reciprocal. Can be implied. ± One correct term at least fractional power increased by. 0 = + C C = 0 one term correct with simplified coefficients 3 y= 0 For C from correct working. 7(ii) = 4 y = 6 4 0= dy = 6 = 5.5 d Eqn with their grad and point (4,...) Eqn of tangent: y = 5.5 y = oe Must be in the form y = m + c but accept y = 40 8(i) A= ( ) 6 A = B 6 for 8 4 for 8 UCLES 07 Page 5 of 8

6 4037/ Cambridge O Level Mark Scheme 07 8(ii) 4+ y = y = 9 5 Pre multiply 9 by a matri. Allow recovery 6 5 = y Pre multiply their 9 answer to (i) by their.5 = = y A for value for y value oe Allow both unsimplified 9(i) d ( ln) = + ln isw d Product rule. One correct term + another term. Allow unsimplified. 9(ii) +lnd= ln Correct use of (i) and must be dealing with terms. soi ( ) lnd = ln + C Correct answer with no working is fine. 9(iii) k ln d = k ln k k k ln k k k = k(lnk lnk ) [ ] [ ] ( ln ( ) ln ) = k k k Insert limits and subtract correctly using their result from (ii) which must contain an ln function Uses nlna= lna n somewhere oe 4k = k ln k a Uses ln a ln b= ln b or lna+ lnb= ln ab somewhere ( ln4k ) = k Answer given Correct completion. UCLES 07 Page 6 of 8

7 4037/ Cambridge O Level Mark Scheme 07 0(i) 3 c 6() 7() 0 ( c ) = + = is a factor. Or correct division. Finding or using one correct factor. Attempt to factorise or use long division to obtain 6c ± or 6c ± c respectively ( c )( c c ) 6 = 0 ( c )( c )( 3c ) + =0 c =,, 3 FT From three different linear factors 0(ii) dy sec 6cos d = + B for each term 0(iii) 6cos 7 cos + = dep Replaces sec by cos 3 6cos 7cos + = 0 dep Answer given so all steps must be correct. 0(iv) cos =,,. 3 π = 0,.05 or 3,.9 A for values awrt for third value and no others in range. No credit for answers in degrees (i) y ( )( ) = = 0 Solve A is ( 4,0) nfww Indication somewhere that = 4 when y = 0 (ii) 4+ 3 = m+ 8 ( ) + m 3 + 4= 0 Eliminate y. b 4ac( = 0) ( m 3) = 6 dep Use of discriminant m = Do not award if m = 7 is not discarded (iii) Obtain quadratic ( ) their m and attempt to solve. + m 3 + 4= 0 using Working must be seen for any marks to be awarded. Must not be awarded if m is not obtained correctly Point B (, 6) UCLES 07 Page 7 of 8

8 4037/ Cambridge O Level Mark Scheme 07 (iv) 4 Area under curve = ( + ) 4 3 d Integrate powers increased in at least terms 3 3 = = = 7 3 dep Insert limits of their and 4 and subtract in correct order. May be implied by 8 3 Intercept is (8,0) so area of triangle = 6 6 Area of triangle using = 8 their B = ( their 8 ) B yb or Attempt to find other suitable areas to result in a complete method. Shaded area = 8 7 = Accept 0.7. Must not be awarded if point B is not obtained correctly. UCLES 07 Page 8 of 8

Cambridge Assessment International Education Cambridge Ordinary Level. Published

Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 407/ Paper 07 MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers and