4754 Mark Scheme June 2006 Mark Scheme 4754 June 2006

Transcription

1 Mark Scheme 75 June 6

2 1 sin x cos x R sin(x α) R(sin x cos α cos x sin α) R cos α 1, R sin α R 1 + ( ), R tan α /1 α π/ sin x cos x sin(x π/) x coordinate of P is when x π/ π/ x 5π/6 y So coordinates are (5π/6, ) ft ft [6] R tan α or sinα /their R or cosα1/their R α π/, 6 or 1.5 (or better) radians www Using x-their απ/or 9 α exact radians only (not π/) their R (exact only) (i) + x A B C + + (1 + x) (1 x) 1 + x (1 + x) 1 x + x A(1 + x)(1 x) + B(1 x) + C(1 + x) x 1 5 5B B 1 x ¼ 1 5 C C 8 16 coeff t of x : A + C A. Clearing fractions (or any correct equations) B 1 www C www A needs justification (ii) (1 + x) 1 + ( )x + ( )( )x /! + 1 x + x + (1 x) ( 1)( x)+( 1)( )( x) /!+ 1 + x + 16x + + x 1 (1 + x) + (1 x) (1 + x) (1 x) 1 x + x + (1 + x + 16x ) + 6x + 5x ft Binomial series (coefficients unsimplified - for either) or (+x²)(1+x) (1-x) 1 expanded theira,b,c and their expansions sin(θ + α) sin θ sin θ cos α + cos θ sin α sin θ tan θ cos α + sin α tan θ sin α tan θ tan θ cos α tan θ ( cos α) tan θ sinα cosα * sin(θ + ) sin θ tan θ sin cos.59 θ 7.5, 7.5 [7] Using correct Compound angle formula in a valid equation dividing by cos θ collecting terms in tan θ or sin θ or dividing by tan θ oe www (can be all achieved for the method in reverse) tan θ sin cos -1 if given in radians -1 extra solutions in the range

3 (a) dx k dt x [] dx... dt k x (b) dy 1 dt y ydy 1dt 1 y t+ c When t, y 9 18 c y [ (1t+ 18)] (15(1t+18)) When t 1, y 15 [6] separating variables condone omission of c evaluating constant for their integral any correct expression for y for method allow substituting t1 in their expression cao 5 (i) x xe dx let u x, dv/dx e x v ½ e x 1 x 1 x xe e dx + 1 x 1 x xe e + c 1 x e (1+ x) + c * d [ xe e + c] e + xe + e dx xe x or x x x x x [] Integration by parts with u x, dv/dx e x x x xe e dx condone omission of c product rule (ii) V π y dx π ( ) π 1/ x x e dx xe x dx 1 x π e (1+ x) π( ¼ e.5 + ¼ ) 1 5 π (1 ) * e D Using formula condone omission of limits y²xe x condone omission of limits and π condone omission of π (need limits)

4 Section B 6 (i) At E, θ π x a(π sin π) aπ So OE aπ. Max height is when θ π y a( 1 cos π) a θπ, 18,cosθ-1 (ii) dy dy / dθ dx dx / dθ a sinθ a(1 cos θ ) sinθ (1 cos θ ) [] dy dy / dθ for theirs dx dx / dθ d d (sin θ) cos θ, (cos θ) sin θ both dθ dθ or equivalent www condone uncancelled a (iii) tan 1/ sinθ 1 (1 cos θ ) 1 sin θ (1 cos θ ) * When θ π/, sin θ / (1 cos θ)/ (1 + ½)/ BF a(1 + ½) a/* OF a( π/ /) [6] Or gradient1/ sin θ /, cos θ ½ or equiv. (iv) BC aπ a(π/ /) a(π/ + ) AF a/ a/ AD BC + AF a( π/ + + ) a(π/ + ) a. m ft [5] their OE -their OF

5 7 (i) AE ( ) 5 [] (ii) 15 uuur AE 5 1 Equation of BD is r 7 + λ 11 BD 15 λ D is (8, 19, 11) cao Any correct form 1 15 or r 7 + λ 11 λ or /5 as appropriate (iii) At A: At B: ( 1) + ( 7) At C: ( 8) + ( 6) Normal is 5 A,1, One verification (OR Normal, scalar product with 1 vector in the plane, two correct, verification with a point OR vector form of equation of plane eg ri+j+6k+μ(i+7j-5k)+ν(8i+6j+k) elimination of both parameters equation of plane Normal * ) 15 (iv).. AE AB is normal to plane 5 Equation is x + y + 5z. (v) Angle between planes is angle between normals and 5 5 ( ) cosθ 5 5 θ 6 scalar product with one vector in plane scalar product with another vector in plane x + y + 5z OR as * above OR for subst 1 point in x+y+5z, for subst further points correct equation, Normal Correct method for any vectors their normals only ( rearranged) or 1 cao

6 Comprehension Paper Qu Answer Mark Comment minutes hour minutes 5.5 seconds Accept all equivalent forms, with units. Allow. R ( T 19) ( T ) T 56.9 R will become negative in 56.5 and 5 seconds. R and attempting to solve. T56,56,56.9 any correct cao. The value of L is 1.5 and this is over hours or (1 minutes).(i) Substituting t in R L+ ( U L)e kt gives R L+ ( U L) 1 U.(ii) kt As t, e and so R L 5.(i) or R>1.5minutes or showing there is no solution for e... e 1 Increasing curve Asymptote A and B marked correctly 5.(ii) Any field event: long jump, high jump, triple jump, pole vault, javelin, shot, discus, hammer, etc. 6.(i) t t R e 6.(ii) ( ) R e 1.89 R e R hours minutes seconds Substituting their t 1, 1.5, etc.