A Level 2015 P1 Solutions for 9780_CWY_2018

A Level 2015 P1 Solutions (edited for New H2 Math Syllabus 9758) Qn Suggested Solutions 1(i) 2: aC y bx cx   3 2 2 1 1.6 1 ( 0.7) 2.4 1.6 (1) 3.6 ( 0.7) (2) d2 2 2 (3)d a b c a b c ya b a bxx                      1(ii) When C crosses the x -axis, 2 0ay bx cx    By GC, 0.589x (3 d.p.) 1(iii) 2 3.593 5.187 7.303 As , 5.187 7.303 yx x x yx       The other asymptote is the line 5.187 7.303yx  . 2(i) (ii) By GC, 1.73Ax  , 0.414Bx  , 1.73Cx  (3 s.f.) From the graph, 1 21 x xx   when 1.73 0.414x   or 1.73x . Solving simultaneously, 3.59345 3.593a  5.18691 5.187b  7.30274 7.303c

3(i) 1 1 2f f ... f n n n n n                   is the sum of areas of the n rectangles in the sketch. As n , the space between the top of the rectangles and the curve  fyx decreases to 0, so that this sum approaches the area under the curve.   1 0 1 1 2lim f f ... f f d n n xxn n n n                      3(ii)   33 3 3 3 3 3 1 3 1 1 2 ...lim 1 1 2lim ... 1 1 2lim f f ... f where f x x x n n n n n n n n n xxn n n n                                      1 0 11 1 0 0 3 4 33 4 fd 3d 4 xx x x x        4 Total perimeter of shapes,      2 22 4 2d x y x xx y       4 (1)2 dxy   Total area of shapes, 21 2 (2)xA xy    Sub (1) into (2): x y x

  2 2 2 4 1 22 22 xdx xA d xx    Let d 40d2 Ad xx    . 8 dx 2 2 d 40d A x   when 8 dx .  The area is maximum when 8 dx . Maximum area 2 21 2 8 8 3 2 2 d d d d             (shown!)

5(i)     1 242 2 3 0 Scaling by factor parallel t T o the -axi ranslate s 133 4yy x y x y x            The graph of 2yx undergoes a translation of 3 units in the positive direction parallel to the x-axis, followed by a stretch of ¼ units parallel to the y-axis. 5(ii) 5(iii) 6(i) Using standard series from MF26,   23 23 8 3 (2 ) (2 )ln 1 2 (2 ) 23 22 xxxx x x x          6(ii)                  2 23 2 3 4 31 1 21 1 ... 2! 3! 1 1 2 ...26 c c c c c cax bx ax c bx bx bx ab c c ab c c cax abcx x x                Comparing coefficients of x : 2a 2x : 122 bbc c  3x :   2 8 31b c c 

    2 8 3 8 31 bc bc b b   or   8 3 8 3 8 3 1 1 bc bc b b        53 53 ,bc   Coefficient of 4x in  1 c ax bx is        35 3 8 13 5 5 532 104 6 27   7(i) 3 5OC a , 5 11OD b 7(ii) 3 5BC OC OB    ab Line BC :  3 5 ,OB BC      abrb  3 5 1 (1)   r a b (shown!) 5 11AD OD OA    ba Line AD :  5 11 ra ba   5 111 , (2)      r a b 7(iii) When the lines BC an