1 http://www.manabino-academy.com

1 1 1.1............................................... 1 1.2............................................. 4 1.3............................................. 6 2 8 2.1............................................. 8 2.2 2......................................... 9 2.3....................................... 10 2.4 2......................................... 12 3 14 3.1.................................. 14 3.2.................................... 15 3.3...................................... 17 3.4............................................. 18 3.5............................................ 19 3.6............................................... 21 4 23 4.1........................................ 23 4.2........................................ 25 4.3......................................... 27 4.4......................................... 28 4.5......................................... 29

http://www.manabino-academy.com 1 1 1.1 1 v e f v e + f = 2 (() ( ) + ( ) = 2) 5 1 2 3 4 5

http://www.manabino-academy.com 2 6 1 1 1 1 4 1 1 ( ) = (1 ) ( ) 2 1 1

http://www.manabino-academy.com 3 () = (1 ) ( ) (1 ) 1 3 4 6 4 3 6 12 8 4 8 12 6 3 12 30 20 5 20 30 12

http://www.manabino-academy.com 4 1.2 2 2 1 2 3 1 2 (1) (2)

http://www.manabino-academy.com 5 (1) (2)

http://www.manabino-academy.com 6 1.3 1 1 2 3 1 2 1 3 (1) (2)

http://www.manabino-academy.com 7 (1) (2)

http://www.manabino-academy.com 8 2 2.1 3 3 1 1 3 2 1 1 1 2 2 2 3 1 1 4 1 3 5 1 4 1 2 1 2 2 2 1 2 1 2 2 ( ) 3 1 2 2 1 2 1 2 3 1 ( ) 3 1 2 1 3 1 1 1 4 1 3 1 5 3 1 4 1 1 1 1 1, 2, 3, 4

http://www.manabino-academy.com 9 2.2 2 2 3 ( i ) 1 ( ii ) (iii) (i), (ii) 2 (ii) 2 (iii) 2 2 2 2 D (1) (2) (3) D F E D,,, E, F, DE, EF, FD 8 (1) 8 D, E, F, EF 4 E, F 2 DE, DF D 2 (2) 8 D 4,, DE, FD 4 D (3) 8 D (1), E, F, EF 4 D (1), EF D

http://www.manabino-academy.com 10 2.3 3 ( i ) 1 ( ii ) (iii) P 2 1 Q 2 P 2 Q 2 P, Q PQ P 2 3 (1) (2) (3) (4) D (5) D F E G D H 1 1 1 (1) 2 D, FE (2) D,, E, F 4 2 2 D E EHD F FG (3) 2

http://www.manabino-academy.com 11 GHD, EFGH (4) D D 2 4 D E F D G D D DH (5) D D 4 EF, FG, GH, HE

http://www.manabino-academy.com 12 2.4 2 2 2 1 2 2 P, Q 2 l P l m Q l n m Q n n P Q m n P Q 2 P Q P l m 2 4 D (1) D (2) D E H F G (1) D FE, F F D FE F D FE D 4 FE, FG, GHD, EHD

http://www.manabino-academy.com 13 (2) D D EFGH

http://www.manabino-academy.com 14 3 3.1 1 3 1 (1) (2) (3) (1) (2) (3) (1) (2) 1 (3)

http://www.manabino-academy.com 15 3.2 1 1 3 2 (1) (2) (3) l 1 (1) (2) (3) D l l l l (1) l 1 D l (2) l 1 l

http://www.manabino-academy.com 16 (3) l 1 l

http://www.manabino-academy.com 17 3.3 1 (i) 1 (ii) 1 (iii) 1 3 3 1 (1) (2) (3) (1) (2) (3)

http://www.manabino-academy.com 18 3.4 1 ( i ) 2 ( ii ) 2 (iii) (i), (ii) 3 4 P G 3 D 3 3 D, P, F P H G E F D, P, F 3 D P DHG D P F FG D F 2 H 2 DHG EF DHG DP EF E F F DP EF D FG EHD FG PF EHD H D PF E F 3 4 3 P, Q, R (1) 2 (2) 3 P R P P G P G Q Q R

http://www.manabino-academy.com 19 3.5 ( i ) ( ii ) (iii) (iv) (ii), (iii) 3 5 (1) (3) (1) D (2) H G (3) E F D D D H G (1) (2) (3) E F D D FG E (1)E F H G (2)H E FG F, G, GF D (3) D

http://www.manabino-academy.com 20 (1) (3) (1) E (2) H (3) 3 5 P, Q, R D, E, 3 P, Q, R Q P D H R G F G E F D

http://www.manabino-academy.com 21 3.6 D-EFGH D H X G Y E F XY XY 1 X Y 3 6 (1) (3) (1) (2) (3) X Y X Y X Y

http://www.manabino-academy.com 22 (1) (2) (3)

http://www.manabino-academy.com 23 4 1 4.1 2 ( ) = ( ) 2 + ( ) 4 1 (1) (2) 5cm 4cm 4cm (1) (i) 2 4cm (ii) 4cm 4cm 4cm 5cm 5cm ( ) = 4 3 = 12 cm 2 ( ) = (3 + 4 + 3 + 4) 5 = 14 5 = 70 cm 2

http://www.manabino-academy.com 24 ( ) = ( ) 2 + ( ) = 12 2 + 70 = 94 cm 2 (2) (i) 2 (ii) 4cm 4cm ( ) = π 3 2 = 9π cm 2 ( ) = ( ) = 2 π 3 = 6π cm ( ) = 4 6π = 24π cm 2 () = ( ) 2 + ( ) = 9π 2 + 24π = 18π + 24π = 42π cm 2

http://www.manabino-academy.com 25 4.2 1 ( ) = ( ) + ( ) r cm R cm R cm x ( ) = (2 π R) x 360 cm r cm () = 2 π r cm r cm R cm x r cm R cm ( ) = () (2 π 1 R) x 360 = 2 π 1 r x = 360 r R () = 360 () = 1 2 = 1 2 ( ) ( ) () ( ) = 1 2 2πr R = πrr π ( ) ( ) 4 2 (1) (2) 6cm 9cm 5cm

http://www.manabino-academy.com 26 (1) (i) (ii) 6cm 5cm 5cm 6cm 5cm 5cm ( ) = 5 5 = 25 cm 2 ( ) = ( ) 1 2 1 5 6 3 4 = 60 cm 2 () = ( ) + ( ) = 25 + 60 = 85 cm 2 (2) (i) (ii) 9cm 9cm ( ) = π 3 2 = 9π cm 2 ( ) = π 9 3 = 27π cm 2 () = ( ) + ( ) = 9π + 27π = 36π cm 2

http://www.manabino-academy.com 27 4.3 () = ( ) () 4 3 (1) (2) 5cm 4cm 4cm (1) () = ( ) () = (4 3) 5 = 12 5 = 60 cm 3 () = () ( ) () () = ( ) () (2) ( ) = ( ) () = (π 3 2 ) 4 = 9π 4 = 36π cm 3

http://www.manabino-academy.com 28 4.4 1 3 () = () 1 3 = ( ) () 1 3 4 4 (1) (2) 4cm 5cm 4cm 5cm 6cm (1) 1 6 cm 4 cm ( ) = ( ) () 1 3 = (6 6 2 ) 4 1 3 1 = 48 cm3 (2) 3 cm 4 cm ( ) = ( ) () 1 3 = (π 3 3 1 ) 4 1 3 1 = 12π cm3

http://www.manabino-academy.com 29 4.5 O r O r 4 S S = πr 2 4 = 4πr 2 O r r 1 1 ( ) = 1 3 ( ) () r () = ( ) V V = ( ) = 1 3 ( ) () = 1 3 ( ) r = 1 3 4πr2 r = 4 3 πr3 4 5 3 cm () = ( ) 1 2 + () = 4π 3 2 1 2 + π 32 = 27π (cm 2 ) ( ) = () 1 2 = ( 4 3 π 33) 1 2 = 18π (cm3 )