Mark (Results) Summer 009 GCE GCE Mathematics (6668/0)
June 009 6668 Further Pure Mathematics FP (new) Mark Q (a) = rr ( + ) r ( r+ ) r ( r+ ) B aef () (b) n n r = r = = rr ( + ) r r+ = + +...... + + n n + n n + List the first two terms and the last two terms Includes the first two underlined terms and includes the final two = + ; underlined terms. n + n + + n + n + = n + n + = = ( n + )( n + ) ( n + ) ( n + ) ( n + )( n + ) n + 9n + 6 n n ( n + )( n + ) Attempt to combine to an at least term fraction to a single fraction and an attempt to take out the brackets from their numerator. = n + 5n ( n + )( n + ) = n(n + 5) ( n + )( n + ) Correct Result cso AG (5) [6] 6668/0 GCE Mathematics June 009
Q (a) z = i, < θ O α arg z (, ) r = ( ) + ( ) = + = 6 = 8 θ = tan ( ) = ( ( ) isin ( ) ) z = 8cos + ( ) So, z = 8 cos + isin A valid attempt to find the modulus and argument of i. Taking the cube root of the modulus and dividing the argument b. z = cos ( ( ) + isin( )) cos ( ( ) isin( )) + Also, ( ( 7 ) ( 7 z = 8cos + isin )) or ( ( 9 ) ( 9 z = 8cos + isin )) Adding or subtracting to the argument for z in order to find other roots. 7 7 ( isin ) = + z cos ( ( ) ( )) An one of the final two roots and z = cos + isin Both of the final two roots. Special Case : Award SC: A0A0 for ALL three of ( + ) ( isin ) cos 7 7 + and cos( ) isin( ) ( ) +. cos isin, [6] Special Case : If r is incorrect (and not equal to 8) and candidate states the brackets ( )correctl then give the first accurac mark ONLY where this is applicable. 6668/0 GCE Mathematics June 009
Q d sin cos sin sin = d cos sin sin = sin sin An attempt to divide ever term in the differential equation b sin. Can be implied. d cos = sin sin Integrating factor cos ± ( ) sin cos ± e or e d sin ln sin = e = e ln sin e or their P( ) ( ) ln cosec e d aef = sin sin or (sin ) or cosec aef d cos sin = sin d sin sin d = sin sin sin d ( ) their I.F. = sin their I.F d = sin cos d = cos or sin cos ( ) sin = cos sin = sin K sin = + A credible attempt to integrate the RHS with/without + K ddd = sin + Ksin sin Ksin = + cao [8] 6668/0 GCE Mathematics June 009
Q A = + 0 ( a cosθ ) dθ with Applies r ( dθ) 0 correct limits. Ignore dθ. B ( a + cos ) = a + 6acos + 9cos θ θ θ + cosθ = a + 6acosθ + 9 ± ± cosθ cos θ = Correct underlined epression. 9 9 = + 6 cosθ + + cosθ dθ A a a 0 Integrated epression with at least out of terms of the form ± Aθ ± Bsinθ ± Cθ ± Dsin θ. * 9 9 Ignore the θ 6 sinθ θ sinθ. Ignore limits. 0 a θ + 6asinθ + correct ft integration. ft Ignore the. Ignore limits. = a + a + + = + + + ( a 0 9 0) ( 0) = a + 9 a + 9 Hence, 9 07 a + = Integrated epression equal to 07. 9 07 a + = d* a = 9 As a > 0, a = 7 a = 7 Some candidates ma achieve a = 7 from incorrect working. Such candidates will not get full marks cso [8] 6668/0 GCE Mathematics June 009 5
Q5 = sec = (sec ) (a) d d (sec ) (sec tan ) sec tan = = Either (sec ) (sec tan ) or sec tan B aef Appl product rule: u = sec v = tan du dv = sec tan = sec d = sec tan + sec Two terms added with one of either Asec tan or Bsec in the correct form. Correct differentiation = sec (sec ) + sec (b) d Hence, = 6sec sec d = ( ) =, ( ) d = () = Applies tan = sec leading to the correct result. Both d = and d = AG B () d ( ) ( ) = 6 = 8= 6 Attempts to substitute = into both terms in the epression for d. d = sec (sec tan ) 8sec (sec tan ) Two terms differentiated with either sec tan or 8sec tan being correct = sec tan 8sec tan ( ) ( ) d d = () 8 () = 96 6 = 80 = 80 B 6 80 ( ) ( ) ( ) sec + + + +... 6 { ( ) ( ) ( ) } 0 sec + + 8 + +... Applies a Talor epansion with at least out of terms ft correctl. Correct Talor series epansion. (6) [0] 6668/0 GCE Mathematics June 009 6
Q6 z w =, z = i z + i (a) wz ( + i) = z wz+ iw= z iw= z wz i w i w= z( w) z = ( w) Complete method of rearranging to make z the subject. i w z = ( w) aef z i w = = w Putting in terms of their z w = d iw = w w = w w = 9 w u + iv = 9 u + iv u + v = 9 ( u ) + v u + v = 9u 8u + 9 + 9v 0= 8u 8u + 8v + 9 Applies w = u + iv, and uses Pthagoras correctl to get an equation in terms of u and v without an i's. Correct equation. dd 0 = u u + v + 9 9 8 Simplifies down to u + v ± αu ± βv ± δ = 0. ddd ( ) u v + + = 9 8 9 8 6 8 0 ( ) u v + = 9 9 8 6 (b) 9 {Circle} centre ( ) v, 0, radius One of centre or radius correct. 8 8 Both centre and radius correct. Circle indicated on the Argand diagram in the correct position in follow through quadrants. Ignore plotted coordinates. Bft (8) O u Region outside a circle indicated onl. B () [0] 6668/0 GCE Mathematics June 009 7
Q7 = a, a > (a) a Correct Shape. Ignore cusps. Correct coordinates. B B a O a (b) a a a =, > () { > a}, a = a a = a aef + a = 0 ()( a ) ± = Applies the quadratic formula or completes the square in order to find the roots. ± + 8a = Both correct simplified down solutions. { < a}, + a = a + a = a or a = a aef { = 0 ( ) = 0} = 0, = 0 = B (6) (c) a a a >, > + 8a < {or} + + 8a > is less than their least value is greater than their maimum value B ft B ft {or} 0< < For{ < a}, Lowest < < Highest 0< < () [] 6668/0 GCE Mathematics June 009 8
Q8 d t + 5 + 6 = e, = 0, = at t = 0. (a) AE, m + 5m + 6 = 0 ( m + )( m + ) = 0 m =,. So, mt mt Ae + Be, where m m. t t CF = Ae + Be t t Ae + Be d = ke = ke = ke t t t ke + 5( ke ) + 6ke = e ke = e k = t t t t t t { So, PI = e t } Substitutes k e t into the differential equation given in the question. Finds k =. So, A B their + their * t t t = e + e + e CF PI t t t Ae Be e == Finds d b differentiating their and their CF PI d* t = 0, = 0 0 = A + B + t = 0, = = A B A + B = A B = A =, B = 0 Applies t = 0, = 0 to and t = 0, = to d to form simultaneous equations. dd* So, t = e + e t t t = e + e cao (8) 6668/0 GCE Mathematics June 009 9
t = e + e t (b) t t e e 0 = = Differentiates their to give d and puts d equal to 0. = t e 0 t = ln A credible attempt to solve. d* t = ln or t = ln or awrt 0.55 ln ln ln ln So, = e + e = e + e = + Substitutes their t back into and an attempt to eliminate out the ln s. dd = + = = 9 uses eact values to give 9 AG d t = 9e + e t At t = d ln, = 9e + e ln ln d Finds d and substitutes their t into d* 9 = 9() + = + = + As = + = < 0 then is maimum. d 9 9 + < 0 and maimum conclusion. (7) [5] 6668/0 GCE Mathematics June 009 0