Transcription of Jan 2006 - 6665 Core C3 - Mark scheme
1 January 2006 6665 Core Mathematics C3 Mark scheme 1 Question Number scheme Marks 1. (a) y Shape unchanged B1 (2, 7) Point B1 (2) O x (b) y (2, 4)
2 Shape B1 Point B1 (2) O x (c) y (2 Shape (2,4) , 4)B1 (2, 4) B1 (2,4) B1 (3) [7] O x January 2006 6665 Core Mathematics C3 Mark scheme 2 Question Number scheme Marks 2.
3 222xxx x 1 At any stage B1 2232323 2 23 22xxxxxxxxx x B1 221623623 222 1xxxxxx xx xx M1 2621xxxx A1 3221xxxx M1 A1 31xx A1 (7) [7] Alternative method 222xxx x 1 At any stage B1 23x appearing as a factor of the numerator at any stage B1 2223 162323623 22 123 2 1xxxxxxxx xxxxx M1 322591823 2 1xxxxxx can be implied A1 2222 99236326or or 23 2 123 2 123 2 1xxxxxxxxxxxxxxxxxx 2 M1 Any one linear factor quadratic
4 23 2 323 2 1xxxxxx Complete factors A1 31xx A1 (7) January 2006 6665 Core Mathematics C3 Mark scheme 3 Question Number scheme Marks 3. ddy1xx accept 33x M1 A1 At x = 3, d13d3ymx Use of 1mm M1 ln133yx M1 3yx9 Accept 93y xA1 (5) [5] dd3y1xx leading to is a maximum of M1 A0 M1 M1 A0 =3/5 92yx 7 4.
5 (a) (i) 323223de3eor 3eed xxx x At any stage B1 23232d3e2edxxyxxx Or equivalent M1 A1+A1 (4) (ii) 32dcos 26sin 2d 3xxxx At any stage M1 A1 33218 sin 23 cos 2dd93xxxyxx M1 A1 (4) Alternatively using the product rule for second M1 A1 133cos2yxx 2132d33cos 26 3sin 2dy3xxxxxx Accept equivalent unsimplified forms (b) dd18cos2 6or 8cos2 6ddyxyxy y M1 d1d8cos26yxy M1 A1 2d11d2168 cos arcsin4yxxx M1 A1 (5)
6 [13] January 2006 6665 Core Mathematics C3 Mark scheme 4 Question Number scheme Marks 5. (a) 2421xx0 Dividing equation by x M1 21422xx Obtaining 2x .. M1 212xx cso A1 (3) (b) , , x B1, B1, B1 If answers given to more than 2 dp, penalise first time then accept awrt above.
7 (3) (c) Choosing or a tighter interval , M1 3f 10 , f 103 Both, awrt A1 Change of sign (and continuity) , to 3 decimal places cso A1 (3) [9] 6. (a) cos12,sin4RR Accept if just written down, awrt 22124160R M1 A1 4tan, awrt M1, A1(4) (b) 7cos their their xR M1 x + their awrt 56 A1 =.
8 , 360their principal value M1 Ignore solutions out of range , A1, A1 (5) If answers given to more than 1 dp, penalise first time then accept awrt above. (c)(i) minimum value is 160 ft their R B1ft (ii) their 1x cos M1 cao A1 (3) [12] January 2006 6665 Core Mathematics C3 Mark scheme 5 Question Number scheme Marks 7.
9 (a) (i) Use of 2cos 2cossin2xx x in an attempt to prove the identity. M1 22cossincossincos 2cossincossincossincossincossinxx xxxxxxxxx xxxx cso A1 (2) (ii) Use of 2cos 22 cos1xx in an attempt to prove the identity. M1 Use of sin22 sin cosxx x in an attempt to prove the identity. M1 2211cos 2sin 22 cos1 2 sin coscoscos sin22xxxxx xxx 12 cso A1 (3) (b) 1coscossin2 Using (a)(i) M1 21coscos sin02 1cos 2sin 202 Using (a)(ii) M1 cos 2sin 2 A1 (3) (c)
10 Tan 21 M1 59132,,,4444 2 any one correct value of A1 5913,,,88 8 8 Obtaining at least 2 solutions in range M1 The 4 correct solutions A1 (4) If decimals or degrees , , , , , , are [12] given, but all 4 solutions are found, penalise one A mark only. Ignore solutions out of range. January 2006 6665 Core Mathematics C3 Mark scheme 6 Question Number scheme Marks 8.
