Mark Scheme (Results) Summer 2007 - Edexcel

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Question Scheme Marks number 2 5. C.F. m + 3m + 2 = 0 m = −1 and m = −2 M1 y = Ae − x + Be −2x A1 (2) P.I. 2 y = cx + dx + e B1 d 2 2 y d y 2 = 2cx + d, = 2c 2c + 3(2cx + d) + 2( cx + dx + e) ≡ 2x + 6x M1 2 dx dx 2 c = 2 c = 1 (One correct value) A1 6 c + 2d = 6 d = 0 2c + 3d + 2e = 0 e = −1 (Other two correct values) A1 − x −2x 2 General soln: y = Ae + Be + x −1 (Their C.F. + their P.I.) A1ft (5) ( A + 2) x = 0 , y = 1: 1 = A + B −1 B = M1 d −x − 2 x y = −Ae dx − 2Be dy + 2x, x = 0, = 1: 1 = −A − 2B M1 dx Solving simultaneously: A = 5 and B = −3 M1 A1 − x −2x 2 Solution: y = 5e − 3e + x −1 A1 (5) 1 st M: Attempt to solve auxiliary equation. 12 2 nd 2 dy d y M: Substitute their and 2 into the D.E. to form an identity in x with dx dx unknown constants. . 3 rd M: Using y = 1 at x = 0 in their general solution to find an equation in A and B. 4 th M: Differentiating their general solution (condone ‘slips’, but the powers of d y each term must be correct) and using = 1 at x = 0 to find an equation dx in A and B. 5 th M: Solving simultaneous equations to find both a value of A and a value of B.
Question Scheme Marks number * 6. (a) z = 3 + i B1 z z * = ( 3 − i)( 3 − i) ( 3 + i)( 3 − i) 3 − 2 3i −1 = , 3 + 1 2 z ⎛ 1 ⎞ ⎛ ± 3 ⎞ (b) = ⎜ ⎟ , = 1 * ⎜ ⎟ + z ⎝ 2 ⎠ 2 ⎝ ⎠ 2 = 1 2 − 3 2 ⎡ ⎢Or : ⎢ ⎣ i z * z z 3 + 1 ⎤ = = , = 1⎥ * z 3 + 1 ⎥ ⎦ (c) arg ( w ) = arctan⎜ real( w) ⎟ or arg ( ) = arctan⎜ ⎠ imag( w) ⎟ ⎠ (*) M1, A1cso (3) M1, A1 (2) ⎛ imag ( w) ⎞ ⎜ ± w ⎛ ⎜ ± real ( w) ⎞ , M1 ⎝ ⎝ where w is z or * z z or * z ⎛ − 3 ⎞ ⎛ z ⎞ ⎜ ⎟ ⎜ ⎟ = 2 π arg arctan⎜ ⎟ = − * ⎝ z ⎠ ⎜ 1 ⎟ 3 ⎝ 2 ⎠ A1 ⎛ −1 ⎞ π ⎛ 1 ⎞ π * arctan⎜ ⎟ = − and arctan⎜ ⎟ = (Ignore interchanged z and z ) A1 ⎝ 3 ⎠ 6 ⎝ 3 ⎠ 6 * π π π ⎛ z ⎞ arg z − arg z = − − = − = arg⎜ * ⎟ 6 6 3 ⎝ z ⎠ A1 (4) (d) 2 y 1 * z z and * z (Correct quadrants, approx. symmetrical) B1 − 1 2 3 − (e) ( − ( 3 − i )( x − ( 3 + i ) − x z * z z (Strictly inside the triangle shown here) B1 (2) x M1 ⎛ − b ⎞ ⎞ Or: Use sum of roots ⎜= ⎟ and product of roots ⎜ ⎟ ⎝ a ⎠ ⎝ ⎠ 2 x − 2 3x + 4 A1 (2) 13 (a) M: Multiplying both numerator and denominator by 3 − i , and multiplying out brackets with some use of i 2 = −1. (b) Answer 1 with no working scores both marks. (c) Allow work in degrees: −60°, −30° and 30° 5π 11π π Allow arg between 0 and 2π : , and (or 300°, 330° and 30°). 3 6 6 Decimals: Allow marks for awrt −1.05 (A1), −0.524 and 0.524 (A1), but then A0 for final mark. (Similarly for 5.24 (A1), 5.76 and 0.524 (A1)). (d) Condone wrong labelling (or lack of labelling), if the intention is clear.
Page 1 and 2: Mark Scheme (Results) Summer 2007 G
Page 3 and 4: Question Scheme Marks number 2. Int
Page 5: 20 10 −2 −1 1 2 −10 −20 y x

Mark Scheme (Results) Summer 2007 - Edexcel

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