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Edexcel_WS_C4_PaperI - Worked Solutions Edexcel C4 Paper I...

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Worked Solutions Edexcel C4 Paper I 1. ( a ) d y d θ = 1 ( 1 + cos θ) ( sin θ), d x d θ = 2 cos 2 θ d y d x = sin θ ( 1 + cos θ) 2 cos 2 θ where θ = π 6 , gradient = 1 2 1 + 3 2 · 2 · 1 2 = − 1 2 1 + 3 2 = − 1 2 + 3 = − ( 2 3 ) ( 2 + 3 )( 2 3 ) = 3 2 (5) ( b ) gradient = 0 where sin θ = 0 i.e. where θ = 0 at θ = 0, x = 0, y = ln 2 gradient is zero at (0 , ln 2) (3) 2. A d A = e 1 10 t d t A 2 2 = 10 e 1 10 t + c A = 20, t = 0: 400 2 = 10 + c , c = 190 A 2 2 = 10 e 1 10 t + 190 when t = 20 , A 2 2 = 10 e 2 + 190 A = 23 (2 sig. fig.) (7) 3. ( a ) 1 y d y d x + 3 x 2 2 = 0 d y d x = y( 2 3 x 2 ) (3) ( b ) (i) e x d y d x + y e x + 2 y d y d x = 0 d y d x ( e x + 2 y ) = − y e x d y d x = y e x e x + 2 y at (0 , 3) d y d x = 3 1 + 6 = − 3 7 (3) (ii) equation of tangent at (0 , 3) is y 3 = − 3 7 x 3 x + 7 y = 21 (2) 4. ( a ) (i) cos 2 x = 1 2 sin 2 x (1) hence sin 2 x = 1 2 ( 1 cos 2 x) (ii) sin 2 x = 1 2 ( 1 cos 2 x) d x = 1 2 x 1 4 sin 2 x + c (2) ( b ) Integrating by parts, π 8 0 x d d x 1 2 cos 2 x d x = x 2 cos 2 x π 8 0 + π 8 0 1 2 cos 2 x d x = x 2 cos 2 x + 1 4 sin 2 x π 8 0 = − π 16 · 1 2 + 1 4 · 1 2
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