熊本県数学問題正解

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2 ( ) (00 ) ( ) oboetene/plan/. ( ) (009 ) ( ).. oboetene/plan/eng.html 8 i

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4 i ( )0... ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) S ( ) A 7 ( ) A 8 ( ) A 9 ( ) A 9 ( ) A 0 ( ) A ( ) B ( ) A A A iii

5 .5.4 A A ( ) ( ) ( A ) ( B ) ( ) ( ) I II ( ) ( ) ( ) ( ) ( ) iv

6 (009) 009 ( ) I II A B /5 ( ) I II III A B C /5 ( ) I II III A B C /5 ( ) I II III A B C / ( ) I II III A B C /5 ( ) I II /7,8 ( ) I /7,8 ( ) I II A B /0, /5 (S ) I II A B( ) / (A B ) I II A( ) /7,8,9 I II A B( ) /9,0, ( A ) I II A /7,8,9,0, ( ) I A / I II( ) /4 ( ) I A( ) /4 ( ) I A /,, ( ) I /7 /7 ( ) I 9/ /0 ( ) I II /8 (4 ) oboetene/plan/

7 ... ( )0 f(x) f(x) = x x ( 7 t ) t dt () f(x) = 0 x () x 6 f(x) p q y = px y = qx + 4 x A B () AB x () A x a b a > b f(x) = a cos x + (a b) cos x sin x + b sin x () f(x) 6 a b () () a b f(x) 0 x π f(x) > 5 x 4 OABC OA BC OAB OAC () OB = OC () ABC G OG BC

8 () f(x) = x x = (x) x ( 7 t ) [ ] t x t dt = t t x = x x x (x) x f(x) = 0 x x x = 0 x(x x 9) = 0 x = 0, ± 5 () () f (x) = x x = (x + )(x ) f (x) = 0 x =, (x x).. x 6 f(x) x 6 f (x) f(x) f(x) x = ± 9

9 4 () y = px y = qx + 4 y px qx 4 = 0 D ( D = ( q) 4 p ) = q + p 4 p > 0 D > 0 α β (α < β) α + β = q p, αβ = 4p p > 0 αβ < 0 A B x α β AB x α + β q p > q > p (p, q) 6 (p, q) = (, ), (, 4), (, 5), (, 6), (, 5), (, 6) 6 6 = 6 () A x α q q + p p q + p q + p p q (p, q) (p, q) = (, ), (5, ) 6 = 8

10 .. 5 () f(x) = a cos x + (a b) cos x sin x + b sin x + cos x = a + x (a b) sin = a b ( sin x + cos x) + a + b = (a b) sin ( x + π 6 ( a > b a b > 0 sin x + π 6 (a b) + a + b + b cos x ) + a + b ) ( (a b) sin x + π ) + a + b 6 a + b f(x) a b (a b) + a + b a b a + b = 6 = a = 5, b = ( () () f(x) = sin x + π ) ( f(x) > 5 sin x + π ) > 6 0 x π π 6 x + π 6 π 6 + π π 6 < x + π 6 < 5 6 π 0 < x < π

11 6 4 () OA = a OB = b OC = c OA BC a ( c b) = 0 a b = a c a b θ a c θ a b cos θ = a c cos θ b cos θ = c cos θ OAB OAC a b sin θ = a c sin θ b sin θ = c sin θ ( b cos θ ) + ( b sin θ ) = ( c cos θ ) + ( c sin θ ) b (sin θ + cos θ ) = c (sin θ + cos θ ) b = c b = c OB = OC OAB = a b ( a b) OAC = a c ( a c) OAB = OAC a b = a c a b = a c b = c () OG = ( a + b + c) OG BC = { a + ( b + c)} ( c b) = a ( c b) + ( b + c) ( c b) = ( a c a b) + ( c b ) () a b = a c b = c OG BC = 0 OG BC

12 ( )0 p f(x) f(x) = p p x (t 6 + t )dt () f (x) x = p + () f(p + ) p > 0 0 < a < {a n } { a = a a n+ = log( + a n ) (n =,,, ) lim a n n () 0 < x < 0 < log( + x) < x 6 x 0.69 < log < 0.70 () 0 < a n < 6 n + (n =,,, ) lim a n n p q y = px y = qx + x A B () AB y () A x () OAB 90 O 4 () π x π cos x sin x > 0 x π 6 () 4 sin x dx cos x sin x π

13 8 () g(t) = t 6 + t G(t) p [ ] x f(x) = g(t)dt = G(t) p x p x = G(p) G(p x) x f (x) = 0 G (p x) (p x) = g(p x) ( ) = g(p x) = (p x) 6 + (p x) = {(p x) + } 4 f (x) (p x) + = 0 x = p + () x = p + f(p + ) = G(p) G( ) f(p + ) G(p) G (p) = g(p) = p 6 + p = (p )(p + ) = (p )(p + p + )(p + ) G(p) p > 0 p 0 G (p) 0 + G(p) p = G() G( ) = = 0 g(t) dt = (t 6 ) dt [ t 7 = 7 t ] 0 (t 6 + t ) dt = 40 7

14 .. 9 () x > 0 + x > log( + x) > 0 f(x) = (x 6 ) x log( + x) f (x) = x + x = x( x) ( + x) f(x) 0 x x 0 f (x) + 0 f(x) 0 0 log < log 4 log > 0 0 < x f(x) > 0 log( + x) < x 6 x 0 < x 0 < log( + x) < x 6 x 0 < x < 0 < log( + x) < x 6 x () 0 < a n < 6 (n =,,, ) (A) n + n = 0 < a = a < 0 < a < 6 + (A) n = k (A) 0 < a k < 6 k + < + a k < + 6 k + log < log( + a k ) < log ( 0 < a k+ < log + 6 ) k + ( + 6 k + 4 )

15 0 0 < 6 k + log 4 5 ( + 6 ) < 6 k + k + ( ) 6 = 6 k + 0 < a k+ < < 6k k + k = 6 (k + ) + 6 (k + ) + 6k (k + ) 5 n = k + (A), n (A) lim n 6 n + = 0 (A) lim a n = 0 n

16 .. () y = px y = qx + y px qx = 0 D D = ( q) 4 p ( ) = q + 4p p > 0 D > 0 α β (α < β) α + β = q p, αβ = p p > 0 αβ < 0 A B x α β AB y q α + β + = q q p + = q p + q p + < p q > 0 p q p q (p, q) (p, q) =(, ), (, ), (, ), (, ), (4, ), (4, ), (5, ), (5, ), (5, ), (6, ), (6, ), (6, ) 6 =

17 () A x α α = q q + 4p p q + 4p q + 4p p q (p, q) 6 (p, q) = (, ), (, ), (4, ), (5, 4), (6, ), (6, 5) 6 6 = 6

18 () A(α, qα + ) AO.. qα + α = q + α = q + p q q + 4p = q q + 4p A B AB q AO AB ( q ) q u =, + 4p v = (, q) OAB u v OAB 90 u v < 0 + q q + 4p q < 0 + q < q q + 4p p > + q q = p 6 q 6 p = () sin x cos x < 0 ( sin x π ) < 0 ( sin x π ) < 0 π x π 4 π x π π < x π < 0 π < x < π

19 4 () () π x π 6 cos x sin x > 0 π 6 4 sin x dx cos x sin x = π 0 π π 4 sin x 6 dx + cos x sin x 0 4 sin x cos x sin x dx 4 sin x = a( cos x sin x) + b( cos x sin x) a b 4 sin x = a( cos x sin x) + b( sin x cos x) = ( a b) sin x + ( a b) cos x 4 = a b 0 = a b a =, b = 4 sin x = ( cos x sin x) cos x sin x cos x sin x = = 0 π 0 π 4 sin x cos x sin x dx { ( cos x sin x) cos x sin x } dx [ x log cos x sin x ] 0 = π π = π 6 0 π sin x dx cos x sin x { } ( cos x sin x) dx cos x sin x [ = x log cos x sin x ] π 6 0 = π 6 + log 4 π sin x dx = π cos x sin x 6 + log π

20 ( )0 ( t ) (0, ) (, t) l l m l m P(x, y) () x y t () t P p > 0 {a n } {b n } a =, b = a n a n = b n b n + (n =,, 4, ) (a n + b n )(b n b n ) = pn + b n (n =,, 4, ) () a n b n () a n b n a n + b n () lim n a n b f(p) lim log f(p) n p 0 p p q y = px y = qx + x A B () AB y () A x () OAB 90 O 4 () π x π cos x sin x > 0 x π 6 () 4 sin x dx cos x sin x π

21 6 () (0, ) (, t) l y = (t )x + ( ) (0, ) (, t) ( )( 0 ) = ( ), ( )( t ) = ( + t + t (, ) ( + t, + t) m (t )x (t )y + 5 = 0 P l m (t t + )x = t + t t + = (t ) + 0 x = t + t t + y = t t t + () t = + tan θ ( π < θ < ) π () x = tan θ + tan θ = cos θ sin θ cos θ = cos θ sin θ + y = + tan θ + tan θ = cos θ + sin θ cos θ = sin θ + cos θ + x = cos θ sin θ y = sin θ + cos θ ( π < θ < π) P ( (x ) + y ) = 5, (x, y) (0, 0) 4 )

22 .. 7 () t P t ( ) t = 0, t = (, ) ( ) t =, x + y x y = 0 () x + y x y x + y x y = ( ) ( t + + t ) t t + t t + t + t t + t t t + = ( t + ) + (t ) ( t + ) + (t ) (t t + ) t t + = 5(t t + ) (t t + ) 5 t t + = 0 () a n a n = b n b n + a n b n = a n b n + a b = = {a n b n } a n b n = + (n ) = n +

23 8 () () a n = b n + n + (a n + b n )(b n b n ) = pn + b n (b n + n + b n )(b n b n ) = pn + b n {b n + (n + )}b n (b n + n)b n = pn + a n b n a n b n = pn + a n+ b n+ a n b n = p(n + ) + n = pn + (p + ) n a n b n = a b + {pk + (p + )} k= = + p n(n ) + (p + )(n ) = pn + (p + )n p = p + (p + ) p n = a n b n = pn + (p + )n p () b n + (n + )b n pn (p + )n + p = 0 p > 0 b n > 0 b n b n = (n + ) + (n + ) + 4p(n )(n + ) + n a n = (n + ) + (n + ) + 4p(n )(n + ) + n a n + b n = (n + ) + 4p(n )(n + ) + n

24 .. 9 a n + b n lim n a n b = lim n n = lim n a n b n lim n n (a n + b n ){(a n b n ) + a n b n } (a n b n ){(a n b n ) + a n b n } { (an ) } a n + b n b n + a nb n n n n a n b n n = lim n a n b n n a n + b n f(p) = lim n a n b n lim log f(p) = lim p 0 p p 0 p = lim p 0 log = lim p 0 log { (an b n n ) + a nb n n = p lim n a n + b n n ( + p) + 4p = + p ( + p) + 4p log + p ( + p) p ( + 4p) p ( + p) p ( + p) p {( + 4p) 4p } {( + p) p = log e e e = log = 0 } } = + 4p

25 0 () y = px y = qx + y px qx = 0 D D = ( q) 4 p ( ) = q + 4p p > 0 D > 0 α β (α < β) α + β = q p, αβ = p p > 0 αβ < 0 A B x α β AB y q α + β + = q q p + = q p + q p + < p q > 0 p q p q (p, q) (p, q) =(, ), (, ), (, ), (, ), (4, ), (4, ), (5, ), (5, ), (5, ), (6, ), (6, ), (6, ) 6 =

26 .. () A x α α = q q + 4p p q + 4p q + 4p p q (p, q) 6 (p, q) = (, ), (, ), (4, ), (5, 4), (6, ), (6, 5) 6 6 = 6

27 () A(α, qα + ) AO qα + α = q + α = q + p q q + 4p = q q + 4p A B AB q AO AB ( q ) q u =, + 4p v = (, q) OAB u v OAB 90 u v < 0 + q q + 4p q < 0 + q < q q + 4p p > + q q = p 6 q 6 p = () sin x cos x < 0 ( sin x π ) < 0 ( sin x π ) < 0 π x π 4 π x π π < x π < 0 π < x < π

28 .. () () π x π 6 cos x sin x > 0 π 6 4 sin x dx cos x sin x = π 0 π π 4 sin x 6 dx + cos x sin x 0 4 sin x cos x sin x dx 4 sin x = a( cos x sin x) + b( cos x sin x) a b 4 sin x = a( cos x sin x) + b( sin x cos x) = ( a b) sin x + ( a b) cos x 4 = a b 0 = a b a =, b = 4 sin x = ( cos x sin x) cos x sin x cos x sin x = = 0 π 0 π 4 sin x cos x sin x dx { ( cos x sin x) cos x sin x } dx [ x log cos x sin x ] 0 = π π = π 6 0 π sin x dx cos x sin x { } ( cos x sin x) dx cos x sin x [ = x log cos x sin x ] π 6 0 = π 6 + log 4 π sin x dx = π cos x sin x 6 + log π

29 4..4 ( ) C C C C : (x + 4) + (y + ) = C : x + (y 4) = 9 C : (x 4) + (y + ) = ( ) (0, 0) C ( ) (0, 0) C C C sin x cos x y = sin x ( ) t = tan x y t ( ) π x π y x {a n } {b n } a n = nπ 0 e x cos x dx, b n = nπ 0 e x sin x dx (n =,,,...) ( ) {a n + b n } {b n a n } ( ) a n b n ( ) lim n a n lim n b n

30 .. 5 ( ) C y y = kx C ( 4, ) kx y = 0 C k ( 4) ( ) = 5k 8k = 0 k + ( ) k = 0, 8 5 y = 0, y = 8 5 x ( ) C y y = kx kx y = 0 k ( 4) ( ) C > k + ( ) k 0 4 C k + ( ) > k 4 ( ) C k + ( ) > 5k 8k > 0 9k 7 < 0 5k + 4k + 8 > 0 8 k < 0, 5 < k 7 7 < k < k < 6 + 6, < k 5 5 ( ) y = kx < k < 0, 5 5 < k <

31 6 ( ) t = tan x sin x = t t cos x = + t + t sin x cos x sin x = t + t t + t t = + t + t + t t + t y = + t + t t + t ( ) π x π t y = + t + t t + t + t + t t + t = = 4( t) + ( t) ( t) ( t) 4 t + u = t y = u 4u + = (u ) y u = + u = u + x = π + x = 0

32 .. 7 ( ) {a n + b n } a n + b n = = nπ 0 nπ 0 nπ e x cos x dx + nπ 0 e x sin x dx {(e x ) cos x + e x (cos x) }dx = (e x cos x) dx 0 [ ] nπ = e x cos x 0 = ( ) ne`nı + {b n a n } b n a n = = nπ 0 nπ 0 nπ e x sin x dx nπ 0 e x cos x dx {(e x ) sin x + e x (sin x) }dx = (e x sin x) dx 0 [ ] nπ = e x sin x = 0 ( ) ( ) 0 a n = ( )ne`nı, b n = 5 ( )n e`nı 5 ( ) ( ) lim n! a n = 5, lim n! b n = 5

33 8... ( ) I ABC AB = 4 BC = 5 CA = 6 ) ABC ) ABC ) ABC 5 II a = a n+ = a n + 4 (n =,,, ) {a n } ) a a a 4 ) {a n } {b n } {b n } ) {a n } 4) {a n } n S n III ) y = x e x ) y = x e x ) C : y = x e x l : y = kx k 0 k

34 .. 9 I ) a = 5 b = 6 c = 4 cos A = = = 9 6 sin A > 0 ( ) 9 sin A = = ABC S S = bc sin A = = ) ABC R R = a sin A R = = 8 7 ABC ( ) 8 πr = π 7 = 64 7 π ) r ABC 5 4 S r 5 S = + + = 4r 5r 6r r ) r = r = r r = 7 = 7

35 0 II ) a = a + 4 = + 4 = 8 a = a + 4 = = 0 a 4 = a + 4 = = 44 ) a n+ = a n + 4 a n+ = a n+ + 4 a n+ a n+ = (a n+ a n ) a n+ a n+ = b n+ a n+ a n = b n b n+ = b n {b n } b = a a = 8 = 6 {b n } b n = 6 n` ) ) n n a n + k= 6 k = + 6(n ) = 6 n 4 a = a n n = {a n } a n = 6 n` 4 4) ) S n = n k= (6 k 4) = 6(n ) = 6 n 4n 6 4n

36 .. III ) y = xe x + x ( e x ) = (x x )e`x ) y = ( x)e x + (x x ) ( e x ) = (x 4x + )e x x 0 + y y y 4 0 e y 4 e O + x ) (t, t e t ) (t 0) y = kx y kt = t e t, k = (t t )e t k (t t )e t = 0 t 0 t = k = ( e, ) e

37 ... ( )60 () x x + ax + a + > 0 a () ABC AB = 4 BC = cos B = 7 8 CA sin C () x + ax + bx + 5 = 0 + i (i ) a b xy A(, ) B(, ) C(5, ) ABC () ABC ( ) () ABC f(x) = ax + bx + c f ( ) = 6 f () = f( ) = 6 () a b c () y = f(x) y = x +

38 .. () D/4 = a (a + ) = a a x D < 0 a a < 0 5 () CA = AB + BC AB BC cos B = = 4 CA > 0 CA = 4 = < a < + 5 cos C = BC + CA AB = + 4 = BC CA 4 ( sin C > 0 sin C = 5 = 4) 4 () + i ( + i) + a( + i) + b( + i) + 5 = 0 (a + b + 7) + (4a + b + )i = 0 a b a + b + 7 = 0 4a + b + = 0 a = b = 7

39 4 () AB y = (x ) y = x BC y = (x ) 5 y O B A C 5 x y = 4 x + 4 CA y = (x ) 5 y = x + ( ) AB BC CA y < x y > 4 x + 4 y < x + () ABC D(p, q) AD = BD AD = BD (p ) +(q ) = (p ) +(q ) BD = CD BD = CD p+4q = (p ) +(q ) = (p 5) +(q ) 8p+q = 7, p = 4 7 q = 4 4 ( 4 ABC 4, 7 ) 4

40 .. 5 () f(x) = ax + bx + c f (x) = ax + b f( ) = 6 a b + c = 6 f ( ) = 6 a + b = 6 f () = a + b = a = b = 4 c = () y = f(x) y = x + x x 4x + = x + x = 0, 0 x x + f(x) S S = {( x + ) (x 4x + )}dx 0 = x(x )dx 0 ( = 6 ) ( 0) = 4

41 6.. ( ) ( ) ( )60 () x a b c 5x + (x + )(x + ) = a x + + bx + c x + () 0 θ < π cos θ cos θ 0 θ () log (x ) + log x = log ABC AB = a BC = a + CA = a + a + () B () ABC sin A a f(x) = x +ax +bx (, f()) y = x 7 () a b () f(x) x

42 .. 7 () (x + )(x + ) 5x + = a(x + ) + (bx + c)(x + ) 5x + = (a + b)x + (b + c)x + a + c 0 = a + b, 5 = b + c, = a + c a = b = c = () (cos θ )( cos θ + ) 0 0 θ < π cos θ cos θ π θ 4 π () x > 0 x > 0 x > log (x )x = log (x )x = x x 8 = 0 x = +

43 8 () cos B = AB + BC CA AB BC = a + (a + ) (a + a + ) a(a + ) a(a + ) = a(a + ) = B = 60 () ABC = AB BC sin B = a(a + ) sin 60 = ) a(a + = a(a + ) 4 ABC = CA AB sin A 4 a(a + ) = a + a + a sin A (a + ) sin A = a + a +

44 .. 9 () f(x) = x + ax + bx f (x) = 6x + ax + b (, f()) y = x 7 f() = 7 = 9 y = f(x) f () = f() = 9 + a + b = 9 f () = 6 + a + b = a + b = 5 6a + b = 4 a = 9 b = () () f(x) = x 9x + x f (x) = 6x 8x + = 6(x )(x ) x f(x) x f (x) 0 + f(x) x = 9 x = 4

45 40.. ( )60 () y = 4 x x + ( x 4) { x + x 0 () x + x > 0 () ABC AB = BC = 6 CA = A x x + (m + )x + m + = 0 () () m ABC A = 45 B = 60 BC = a () ABC a () CA AB a

46 .. 4 () y = 4 x x + y y = (x ) 4 x 4 y x = 4 x = 0 4 O 4 x () (x + ) 0 (x + + )(x + ) 0 x + (x + )(x ) > 0 x <, < x x <, < x + + x () cos A = CA + AB BC CA AB = ( ) + ( 6) ( ) = 4( ) 8( ) = A = 60

47 4 () x + (m + )x + ( m + ) ( x + ) ( x + m + ) = 0 = 0 x =, m () m,, m = 0,, m = m = ± () R = BC sin A R = a sin 45 = a BC sin A = CA () sin B a sin 45 = CA CA = sin 60 a sin 60 sin 45 = AB = CA cos A + BC cos B 6 AB = a cos 45 + a cos 60 = a + a = 6 a + a

48 ( )60 () = + a + b 6 a b () x + 8x + k(k + ) = 0 k () ABC AB = 4 CA = 0 A = 60 BC ABC y = x 4ax + a + a + a ABCD AB = BC = cos B = () sin B cos D ()

49 44 () = + {( ) }{( ) + } = + ( ) ( ) = + = a b a = 4, b = 4 () x + 8x + k(k + ) = 0 D/4 = 4 k(k + ) = (k + 4)(k ) D > 0 (k + 4)(k ) > 0 (k + 4)(k ) < 0 4 < k < () BC = CA + AB CA AB cos A = cos 60 = = 56 BC > 0 BC = 56 = 9 ABC ABC = CA AB sin A = 0 4 sin 60 = 0 4 = 5

50 .. 45 y = x 4ax + a + a + y = (x ax) + a + a + = {(x a) a } + a + a + = (x a) a + a + a + = (x a) a + a + (a, a + a + ) { a > 0 a + a + > 0 a a < 0 (a + )(a ) < 0 < a < 0 < a < () cos B = ( ) sin B = = = ABCD D = 80 B cos D = cos(80 B) = cos B = () ABC CA = AB + BC AB BC cos B = + ( ) = = CA > 0 CA = CA sin B = R R = = 4

51 ( ) ( ) ( )

52 .. 47 () x + y = 0 y = x + 5 () f(θ) = cos θ sin θ (0 θ π) () x + ax + b = 0 + 7i a b (i ) y = x + ax + b () y = x + 4 y = x 4 a b () () {a n } a = a + a + 5a + + (n )a n = a n+ (n =,,, ) n () a n+ n a n () a n n 4 O A B C A O BC D ABD = 5 ADB = 40 () ACB () O AD a CD a 5 () y = x (x + ) () x (x + ) kx = 0 (k )

53 48 () (0, 0) y = x + 5 d = 5 + ( ) = 5 = 5 5 (x y 5 = 0) d r 0 r d = 0 5 = 5 () cos θ sin θ = ( sin θ) sin θ = sin θ sin θ + sin θ = t 0 θ π f(θ) = t t + t = t = f(θ) = t ( t + ) + θ = 7 6 π, 6 π θ = π () x + ax + b = 0 + 7i 7i α = + 7i α + β = αβ = 4 β = 7i γ α + β + γ = 0 αβ + βγ + γα = a αβγ = b (α + β) + γ = 0 αβ + (α + β)γ = a αβ γ = b + γ = 0 4 γ = a 4γ = b γ = a = 5 b = 7i a = 5 b =

54 .. 49 () x + ax + b = x + 4 x + ax + b = x 4 x + (a + )x + b 4 = 0 x + (a )x + b + 4 = 0 D = 0 (a + ) 4 (b 4) = 0 (a ) 4 (b + 4) = 0 4b = a + a + 7 4b = a 6a 7 a = b = 5 () y = x + 4 y = x 4 x x = a + x = a a = 4 y 5 x = x = y = x + 4 y = x 4 O x x x + 4 = x 4 x = S S = {(x x + 5) ( x + 4)}dx + {(x x + 5) (x 4)}dx = (x ) dx + (x ) dx [ ] (x ) [ ] (x ) = + = + =

55 50 () a + a + 5a + + (n )a n = a n+ a + a + 5a + + (n )a n = a n n = a = a n (n )a n = a n+ a n n a n = a n+ a = a n+ a n = n (n ) () () a = a 4 = a 5 = 4 a n+ = n a a a 4 a n a n+ = n n! a a = a n+ = n n! (n ) n = a n+ = n n! (n ) { (n = ) a n = n` (n )! (n ) 4 () DAC = ABC DAC = 5 ACB ACD C O ACB = CDA + DAC = = 65 () ABC () D C 40 A 5 B BAC = 80 ( ABC + ACB) = 80 ( ) = 90 BC ODA OD = AD + OA = a + 4 OC CD = OD OC = a + 4

56 () x y = (x )(x + ) = x 4 y 4 x < y = ( x + )(x + ) O 4 x = x () x (x + ) kx = 0 x (x + ) = kx y = x (x + ) y = kx k < k =, 0 < k < 0, 0 < k

57

58 .. 5 () y = f(x) (, 4) (, 4) y = 5 f(x) () { y 4x + 0 x + y 6x + 8y 0 () y = (log x) + log x k + 8 x = a (0 < a < ) 0 k a y = 4 x C y = x l () l C l () C l l y SOJODAIGAKU () 4 () 4 4,,, 4 4, 4, 4, 4 4 5, () 0 () 00 5 f(x) = x ax + a ( x ) 6 a

59 54 () f(x) = ax + bx + c (, 4) 9a b + c = 4 (, 4) a + b + c = 4 b = a, c = a f(x) = ax + (a )x a y = f(x) y 5 ax + (a )x a = 5 ax + (a )x a 7 = 0 D/4 = 0 (a ) a ( a 7) = 0 4a + 5a + = 0 (a + )(4a + ) = 0 a =, 4 a = b = 4 c = a = 4 b = 5 c = 5 4 f(x) = x 4x + f(x) = 4 x 5 x 5 4 () y 4x (x ) + (y + 4) 5 y y = 4x (x ) + (y + 4) = 5 O x (, 4) 5

60 .. 55 () log x = t y = y = t + kt + 8 ( t k ) + 4 k + 8 t = k x = k 4 k k + 8 = 0 k = ±4 x = a (0 < a < ) k = 4, a = 9 () y = 4 x y = x l m l l m = m = C l x = x = 4 y = 4 4 = 4 l (4, 4) y 4 = (x 4) y = x 4 () l l x x = x 4 x = S y { ( S = 0 4 x x )} 4 dx 4 { } + 4 x (x 4) dx ( = 4 x + x + ) O 4 x dx (, ) (x 4) dx 4 [ x = + x 4 + x ] + [ (x 4) 0 4 = 49 ] 4 4

61 56 () 7 O A 4 4 O A ( ) O 8C = 8 ( ) A 8C = 8 ( ) 4 4 9C 4 = 6 ( ) = 8 ( ) () () () 4 4!!! = 6 ( ) 8 4!! 8 4!! = 6 ( ) = 6 ( ) 4 6 4! = 04 ( ) = 70 ( ) 4 () 0 0, 0, 0,, ( ) = = 0 0 k () k k k(k + )

62 .. 57 ( 5 f(x) = x ax + a = x a ) a 4 + a y = f(x) x = a x x = ( ) M(a) a a < a M(a) = f( ) = ( ) a( ) + a = a + a < M(a) = f() = a + a = a + 9 a (a ) a < (a < ) x = a x = a x x

63 58 m(a) < a 6 < a m(a) = f() = a + a = a + 9 a a 6 ( a m(a) = f = ) a 4 + a a < a < m(a) = f( ) = ( ) a ( ) + a = a + < a a a < (6 < a) ( a 6) (a < ) x = a x = a x = a x x x a < M(a) m(a) = ( a + 9) (a + ) = 4a + 8 a < M(a) m(a) = ( a + 9) = a ) ( a 4 + a 4 a + 9 ) a 6 M(a) m(a) = (a + ) ( a 4 + a = a 4 + a + 6 < a M(a) m(a) = (a + ) ( a + 9) = 4a 8 M(a) (a) = 6 a =, 6

64 ( ) ( ) ( )

65 60 () y = f(x) y = x 8x + x p y q (, ) ( 4, ) p q f(x) ( ) 0 () 6 0 log 0 = 0.00 log 0 = () P (x) = x 7x + x + 8 Q(x) x 5 4x 7 Q(x) = 0 α P (α) y = x x + () P(, ) P Q Q () R P Q PQR R OABC OA M BC N OA = a OB = b OC = c () MN a b c MN () MN AB {a n } () a n () b = b n+ = a n + b n (n =,,, ) {b n } b n 5 f(x) = sin x cos x sin x cos x (0 x π) () sin x cos x = X f(x) X () f(x)

66 .. 6 () y = f(x) y = x 8x + f(x) = x + bx + c (, ) ( ) + b ( ) + c = ( 4, ) ( 4) + b ( 4) + c = b + c = 4b + c = 5 b = c = f(x) = x + x + y = x 8x + y = (x ) 7 y = x + x + y = (x + ) 5 (, 7) (, 5) p = = 5 q = 5 ( 7) = () log 0 6 = log 0 + log 0 = = log 0 = log 0 + log 0 = =.079 ( ) } 0 log 0 {6 = log log 0 = = 0.09 ( ) } 0 < log 0 {6 < 0 0 < 6 ( ) 0 < 0 0 ( ) 0 6 0

67 6 () x 7x + x + 8 = Q(x) (x 5) + 4x 7 x 7x x + 5 = Q(x) (x 5) x 7x x + 5 x 5 Q(x) Q(x) = x x Q(x) = 0 α α = ( ) ± ( ) 4 ( ) = ± P (x) x = α Q(α) = 0 P (α) = 4α 7 ( ± ) P = 4 ± 7 = 5 ± ( )

68 .. 6 () y = x x + y = x x = y = ( ) = P(, ) y = {x ( )} y = x + Q y = x x + y = x + y x x + = x + x x = 0 (x + ) (x ) = 0 x x = y = 7 Q(, 7) () < x < y PQ R 7 Q PQR y = x = < x < x = R(, ) PR PQR 6 PQR = 6 = 6 P O R x

69 64 () M OA OM = a N BC ON = b + c MN = ON OM = b + c a = a + b + c M a A O b B N c C OABC a b b c c a 60 a b = b c = c a = cos 60 = MN = MN ( = a + b + ) ( c a + b + ) c = 4 ( a + b + c a b + b c c a) = 4 ( ) = () MN = ( MN AB = a + b + ) c ( b a) = ( a + b a b + b c c a) = ( + + ) = MN AB θ cos θ = MN AB MN AB = = 0 θ 80 θ = 45

70 () {a n } a d 0 8 a + 9d = d) = 45 0(a a = d = a + 9d = 9 a n a n = + (n ) = n () {b n } {a n } n n n b n = b + a k = + (k ) k= k= = + (n )n (n ) = (n 7n + 6) b = b n n = b n b n = (n 7n + 6) 5 () sin x cos x = X sin x sin x cos x + cos x = X sin x cos x = X f(x) = sin x cos x + ( sin x cos x) = X + (X ) = X + X () sin x cos x = ( sin x π ) 4 0 x π X ( () f(x) = X + ) 5 4 X = + X = 5 4

71 66..8 ( )80 () x x+ = () 5 4 (a) (b) a y = x 4 x a () () () ABCD AB AP : PB = : P CD CQ : QD = : Q () PQ PC PD PQ AD BC () PQ AC R AR = RC PR = RQ AB AD BC

72 .. 67 () 6 4 x+ = 4 x+4 4 x + 4 x 0 x x+5 + (6 x+4 ) = 6 x x + 7 = 0 (4 x )(4 x 7) = 0 x x = x = 4 4 < x + 4 x > 0 x > x+5 (6 x+4 ) = 6 x + x 9 = 0 6( x + ) = 55 x > 6( x + ) > 64 x x = () (a) 5C 4 4 4! 5C 4! = 0 4 = 40 ( ) (b) C = 0 ( ) 4 4P 0 4 P = 0 4 = 40 ( ) 5 5 5C C = 5 ( )! 4 4P 5 4 P = 5 4 = 60 ( ) = 600 ( )

73 68 () x a y = x 4x + 4a x < a y = x + 4x 4a y = px + q y x (p + 4)x (q 4a) = 0 4 y x (p 4)x (q + 4a) = (p + 4) + 4(q 4a) = 0 (p 4) + 4(q + 4a) = 0 (p + 4q + 6) + 8(p a) = 0 (p + 4q + 6) 8(p a) = 0 p = a q = a 4 (p + 4) 4 x = = a + (p 4) 5 x = = a x a x < a y = ax a 4 () () S S = = a a a a {(x + 4x 4a) (ax a 4)}dx + a+ a {x (a )} dx + [ {x (a )} = = = 6 {(x 4x + 4a) (ax a 4)}dx ] a a a+ a {x (a + )} dx [ {x (a + )} + ] a+ a

74 .. 69 () Q CD : PQ = PC + PD + = PC + PD PC = PB + BC = AB + BC PD = PA + AD = AB + AD PQ = ( AB + ) BC + ( AB + ) AD = BC + AD () AR = x RQ = y AP = AR + RP = x + ( y) = x y CQ = CR + RQ = ( x) + y = x + y D A Q R C P B AB = AP = ( x y) = x y AD = CD CA = CQ ( x) = ( x + y) + x = x + y BC = AC AB = x ( x y ) = x + y x = AD + BC y = AD + BC 9 9 AB = ( AD + ) ( BC AD + ) BC 9 9 = 4 AD + BC

75 70..9 ( )80 () x x x kx + 4k + 5 > 0 k () ABCDEF (a) E B D (b) 6 C F A {a n },, 5, 0, 7, 6, () a n () 500 < a n < 500 a n f(x) = x x + x g(x) g(x) = () y = f(x) () g(x) x x x f(t) dt

76 .. 7 () f(x) = x kx+4k +5 x 0 k f(x) = (x k) k +4k+5 < k f() = k + 4k + 5 = k + 4 k k + 4 > 0 < k < 7 k f(k) = k + 4k + 5 k + 4k + 5 > 0 (k + )(k 5) < 0 k < k k < f( ) = ( ) k( ) + 4k + 5 = 0k + 4 0k + 4 > 0 k k < k < 7 < k k k < x = k x = k x = k x x x () (a) {AB,CD,EF}, {AB,DE,CF}, {AC,BE,DF}, {AC,DE,BF}, {AD,BC,EF}, {AD,BE,CF}, {AE,BC,DF}, {AE,CD,BF} 8 (b) 6 6C = = 0 ( ) 8 0 = 5 5 = 5

77 7 (),, 5, 7, 9, b n b n = n n n a n = + (k ) = + (n ) k= a = a n n = a n a n = + (n ) () () a = + ( ) = 485 a 4 = + (4 ) = 50 a 9 = + (9 ) = 445 a 40 = + (40 ) = 5 a n 9 k=4 { + (k ) } = 8 k= 8 ( + k ) = ( + k ) ( + k ) k= k= = )( 8 + ) 6 8(8 { + 6 ( } + )( + ) = ( + 795) = 540

78 () f(x) = x(x ) + x.. 7 x < 0 f(x) = x(x ) + ( x) 4 = x x 0 x < f(x) = x(x ) + x = x + x x f(x) = x(x ) + x = x O x y = f(x) () () 4 x 0 g(x) = x x (t t) dx = x x + 4 ( = x ) g(0) = 4 0 x g(x) = 0 x (t t) dt + x = x + x x ( t + t) dt g(x) g (x) = x + 6x x g (x) = 0 α α = g(x) x 0 α g (x) 0 + g(x) 4 ( g(x) = g (x) x g(α) ) + x 6 g(α) = α 6 = 6 = 8 6 y

79 74 x g(x) = x ( t + t) dt + = x x + x x t dt g(x) g (x) = x 4x + = (x ) + > 0 g(x) g() = 4 x g(x) = x x t dt = x x + ( = x ) + g() = 7 g(x) x = 8 6

80 ( )80 a + b () a + b + c = 0 (a + c)(b + c) + b + c (b + a)(c + a) + c + a (c + b)(a + b) () y = x C (a, a ) (a > 0) a l l C S l C x S S : S y x x + y 6x 0 D () D () (x, y) D (x 0) + y OABC OA OB OC L M N C LMN G OAB H () CG OA OB OC () OH OA OB

81 76 () a + b + c = 0 ( ) = a + b ( b)( a) + b + c ( c)( b) + c + a ( a)( c) = c(a + b ) + a(b + c ) + b(c + a ) abc = (b + c)a + (c + a)b + (a + b)c abc = ( a)a + ( b)b + ( c)c abc = a + b + c abc = a + b + c abc + abc abc = (a + b + c)(a + b + c ab bc ca) + abc abc = abc abc = () l (a, a ) a y a = a(x a) y = ax + a C l x y x = ax + a x = a, a S = a = = a a {( ax + a ) x } dx a (x + a)(x a) dx ( ) {a ( a)} = 9 6 a a S O a a (a, a ) S a x l x x x = a S = a 0 x dx + (a a)a = 5 6 a S : S = 9 a : 5 6 a = 7 : 5

82 () y { y x 6 (x ) + y O D y = x (x ) + y = 6 () (x 0) +y (0, 0) D (x, y) (0, 0) (0 0) + 0 = 00 (0, 0) y = x 6 y 0 = (x 0) y = x + 5 ( ) 5, 4 5 (, 0) ( ) ( ) = 5 > D y = x 6 (x ) + y = ( ±, ± 6 ) ( ) 5 5 ( D (0, 0) + ) 6, 5 5 ( + ( ) 6 0) + 5 = x

83 78 () G LMN OG = ( OL + OM + ON) OL = OA OM = OB A L O H G M N C ON = OC OG = 6 ( OA + OB + OC) B CG = OG OC = 6 ( OA + OB + OC) OC = 6 ( OA + OB 5 OC) () C CG = 6 {( CA CO) + ( CB CO) + 5 CO} = 6 ( CO + CA + CB) H CG k CH = k CG CH = k CO + k CA + k CB 6 6 H OAB k + k 6 + k 6 = k = 6 5 () CH = ( OA + OB 5 OC) = OA + OB OC 5 5 OH = OC + CH = ( OC + OA + OB ) OC 5 5 = OA + OB 5 5

84 S ( ) () = () = () 8x + 0x = ( 5 x 6 )( 7 x + 8 ) ( ) (4) x + 4x + 5 = x (5) (6) (7) x + x + x + x + = 8 x x + x + x + (8) x + x 6x 8 = (x )(x + x + 4 ) (9) x + y 6x + y 80 = (0) x + 4y = 0 x + y = 7 () 6 π θ π 4 cos θ 8 4 cos θ 9 () cos x ( ) sin x = 0 sin x + π () 9 log 8 = 4 (4) 6 x 6 4 x 6 = 0 x = 5 6

85 80 (5) y = x + 6x + x + 8 y = 7 8 x y = 9 x (6) y = x 40 y = 4x 4 x 4 (7) (8) 0 (k + 7) = k= k k= = k k= (9) a = (6, ) b = (, ) a b = (0) a = (6, ) b = (, ) θ = 50 5 () = = 5 () = ( 5 ) 5 ( 5 + )( 5 ) = ( 5 ) 5 = ( 5 ) = () 8x + 0x = (4x )(x + ) (4) x + 4x + 5 = (x + 4 ) x + 5 { ( = x + ) ( ) } + 5 ( = x + ) ( ) + 5 ( = x + ) +

86 .4. 8 (5) 4 4 4! = 4 = 48 ( ) (6) ! = 4 = 4 ( ) (7) x + x + x + x + x x + x + ) x +x + x +6x +4x x 4x + x 9x 6 5x +7 x + x + = (x + x + )(x ) + 5x + 7 x + x + x + x + = (x + x + )(x ) + 5x + 7 x + x + = x + 5x + 7 x + x + (8) P (x) = x + x 6x 8 P () = = 0 P (x) x x +x 6x 8 = (x )(x + 4x + 6) x + 4x + 6 x ) x + x 6x 8 x x (9) x + y 6x + y 80 = 0 (x 6x + 9) + (y + y + 6) = (x ) + (y + 6) = 5 4x 6x 4x x 6x 8 6x 8 0 (, 6) 5 5

87 8 (0) x + y = r x + 4y 0 = 0 d d = = 0 5 = d = r r = 4 x + y = 4 () 6 π θ π cos θ 4 cos θ 4 () ( ) = sin x + cos x P(, ) P OP = x OP 5 6 π ( ) = sin (x + 56 ) π y 5 π 6 O x () 9 log 8 = log 8 = log 64 = 64 M = a p log a M = p a log a M = M (4) 4 x = t t > 0 6 x = (4 x ) = t t 6t 6 = 0 (t + )(t 8) = 0 t = 8 4 x = 8 x =

88 (5) y = x + 6x + x + 8 y = x + x (a, a + 6a + a + 8) a + a + y (a + 6a + a + 8) = (a + a + )(x a) y = (a + a + )x a 6a = a 6a + 8 a + a 4 = 0 (a )(a + ) = 0 a =, a = y = 8x a = y = 9x (6) y = 4x 4 y = x (, 4) y x (, 0) 4 x A ABC y = x x x = 0 B C O x x = ABC S S = 0 x dx 4 = [ x ] 0 = (7) 0 k= 0 0 (k + 7) = k + 7 k= k= = + ) (0 = = 770

89 84 (8) k= k = ( ) 6 k= k = (6 ) k k= = 6 k k= ( ) = (6 + )( 6 ) 6 (6 ) = 6 = 6 + = 70 (9) a b = 6 + ( ) = 5 (0) cos θ = a b a b = ( ) = = 0 θ 80 θ =

90 A 7 ( )70 () a + b = a b = a b = a + b = () f(x) = x + 5x x 4 () (4) f(x) = ax 4x + a (i) x f(x) 0 a (ii) x f(x) 0 a () x = cos θ x sin θ sin θ = cos θ cos θ = cos θ = () y = cos θ 5 cos θ 0 cos θ y cos θ = y = y cos θ = y =

91 86 r > 0 x + y r A { y x y + x B x + y y C () B A r r () A B r 0 < r () C (4) B (5) B C (6) B C

92 () a b = (a + b)(a b) = = 4 6 (a + b) + (a b) = (a + b ) (a + b ) = ( ) + ( ) = + 8 = 0 a + b = 0 ( () x + 5x = x 5 ) ( x 4 f(x) = x 5 ) x = x = 8 () 4! = 4 = 4 ( ) 4! = 4 = 4 ( ) = 49 ( ) (4) f(x) = ax 4x + a D/4 = ( ) a(a ) = a + a + 4 = (a + )(a 4) (i) x a > 0 (a + )(a 4) 0 a, 4 a a 4 (ii) x a < 0 (a + )(a 4) 0 a, 4 a 4 4 a

93 88 () sin θ sin θ = sin θ cos θ sin θ = cos θ sin θ = cos θ( cos θ) = x( x ) = x + x cos θ cos θ = ( cos θ ) cos θ = (x )x = x x cos θ = cos(θ + θ) = cos θ cos θ sin θ sin θ = (x x) ( x + x) = 4x x () cos θ = x x cos θ = cos θ = x () y = cos θ 5 cos θ 0 cos θ = (4x x) 5(x ) 0x = 8x 0x 6x y = 4x 0x 6 = 4(6x 5x 4) = 4(x + )(x 4) y x y + 0 y 8 9 cos θ = 9 cos θ = 8. x + x. x x. 4x x

94 A r B 4 (, 0) (0, ) (, 0) (0, ) C x + (y ) 4 (0, ) () B A B (0, ±) A 0 + (±) r r () B x y y B d x + y = 0 d O x d = ( = ) + A B r d 0 < r () C π = 4π (4) B 6 = 6 (5) B C y PQRS S RQ RS P QS S S = = Q R O S x QRS = 0 S = π 0 60 = 4 π S + S = + 4 π

95 90 (6) (),(4),(5) S(C) S(B) S(B C) S(B C) S(B C) = S(B) + S(C) S(B C) = 6 + 4π ( + 4 ) π = π.. 4π π π

96 A 8 ( )70 () C : y = x 6x (, ) C x () log 45 log 60 + log = () AB = 0 BC = 6 C = 90 ABC B AC D AC = CD = (4) 4n n 4n 00 n m A(, ) m l x y P Q O () OP m OP = () OQ m OQ = () OPQ S S = m + m + (4) l OA S = (5) S m = S =

97 9 () ( ) ( ) A B B (i) A B (ii) C A B C D (iii) CD A B A () ( ) ( ) ( ) A B B (i) A B (ii) A B A (iii) A B (iv) A B

98 .4. 9 () y = x 6x = (x ) C (, ) C x x x 6x = 0 x = 0, 0 x y 0 S S = 0 { (x 6x)}dx = [ x + x ] 0 = 4 45 () log 45 log 60 + log = log 60 () C = 90 AC = 0 6 = 8 = log = A BD B 0 CD : DA = BC : BA CD : DA = 6 : 0 = : 5 CD = CA + 5 = 8 8 = (4) 4n = 6n n n = 6 4n n n = 6k (k ) 4n = 6n = 6 6k = k 4n 00 k 00 k k = 9 n = 6 9 = B 6 C D

99 94 l y = m(x ) ( ) y () P x y = 0 ( ) Q x = m + A OP = m + P O x l () Q y x = 0 ( ) y = m + OQ = m + () S = OP OQ S = ( ) m + (m + ) = 9 m + m + 6 (4) OA l OA = ( m) m = () S = = 69 (5) 9 m > 0 m > 0 S = 9 m + 9 m + 6 m m + 6 = 9 m = m m = S m =. m +. m

100 () (i) A B 4 4 8! = 70 ( 4!4! (ii) A C C B!!! 5! = 0 (!! ) (iii) A C D B CD 4! = 8 (!! ) CD A B 70 8 = 5 ( ) () (i) A B 7! = 40 (!!! ) (ii) A B 6! = 90 (!!! ) (iii) A B 5! = 0 (!!! ) (iv) (i) (iii) = ( )

101 A 9 ( )70 () x 6x 9 0 () x = x + x = x + 4 x = () x + x x x + = x x + (4) ABC A = 60 AB = 8 BC = 7 AC = AC = A B C a b c () a b c 6 () a b c 6 () a b c (4) a < b < c (5) a = b b < c (6) a b c O(0, 0) A(6, 0) B(0, 8) y = 9 x x > 0 P(p, q) ABP S () AB () OP AB p = S = () S p = S = (4) BAP p = (5) AP BP p =

102 () x 6x 9 0 (x )(x + ) 0 x, 0,,, 5 () x = x + x = + = ( + ) + ( )( + ) = + ( + ) = x + x = ( x + x) = ( ) x x = x + 4 x = 8 () x + x x x + = (x + )(x x + ) (x ) x x + = x x + 4 x x + (4) a = b + c bc cos A a = 7 c = 8 A = 60 7 = b + 8 b 8 cos 60 b 8b + 5 = 0 (b )(b 5) = 0 b =, 5 b = > 8 b = + 7 < 8. x x x

103 98 () 6 {,, } {,, 4} {,, } {,, }! = 6 ( ) {,, 4}!!! = ( ) {,, } ( ) = = 5 08 () 6 {, 6, 6} {,, 9} {,, 6} {,, 4} {, 6, 6} ( ) {,, 9} ( ) {,, 6} 6 ( ) {,, 4} ( ) = = 5 7 () 6 6P = = 0 ( ) 0 = (4) 6 6C = = 0 ( ) a b c 0 6 = 5 54 (5) 6 6C = 6 5 = 5 ( ) b(= a) c 5 6 = 5 7 (6) a b c Q 6 P c Q ( ) a b c b 6 a a b c P (a, b, c) ( (a, b, c) = (, 4, 4) ) 8! a b c = 56 ( 5!! ) 56 6 =

104 () AB 0 6 = 4 () P y = ( ) 9 x P p, OP 9 p 0 p 0 = 9 p 9 p OP AB y 9 p = 4 p = P(, 6) 4 B P y = x 8 x O 6 C C(6, 4) AC=4 B P x 8 B P C A 6 x S = 4 = 4 () P AB y S y = 9 x y = 9 x P 9 p 4 O 9 p = 4 p = 6 8 P(6, 4) B P A 6 x S = = 4 6

105 00 (4) BAP P A P y 9 p = p(x p) 9 y = 9 px 9 p A(6, 0) 0 = 9 p 6 9 p p p = 0 p > 0 p = (5) p > 0 AB BP p 6 AP 9 p 0 p 6 = p 9(p 6) BP 9 p ( 8) = p + 7 p 0 9p p 9(p 6) p + 7 9p = p + 5p 486 = 0 (p )(p + p + 6) = 0 p =

106 A 9 ( )70 () a b a < 0 y = ax 6ax + b x 6 4 a = b = y x = () x x + 5 = 0 α β α + β = α + β = () 5 a b c d e a b 5 () a = (, ) b = (, ) a b θ (0 < θ < π) sin θ = a + t b t = () n () 4 n n log 0 = 0.00 () n ( + x) n = a 0 + a x + a x + + a n x n a r+ = a = 5a n 5 n = a r () a b x 4x + ax + b = 0 i a = b =

107 0 () a < 0 y = a(x ) 9a + b ( x 6) x = x = 4 9a + b = 7a + b = 4 a = 7 6 b = 5 6 () α + β = = αβ = 5 = 5 α + β = (α + β) αβ = 5 = α + β = (α + β) αβ(α + β) = 5 = 8 () 5 a b c d e 5! = 5 4 = 0 ( ) a b!! = = ( ) 5 (5 )! = 4! = 4 = 4 ( ) 5C = 5 4 = 0 ( )

108 () cos θ = a b a b = + ( ) + ( ) + = ( sin θ = ) = = 0 sin θ = sin θ cos θ = ( ) = a + t b = (, ) + t(, ) = (t +, t ) a + t b = (t + ) + (t ) = 5t 5t + 8 ( = 5 t ) a + t b t = 7 4 = () n n{ + (n ) ( )} = n( n + 7) () 4 n (4 n ) = 4n 4 4 n > 4 n > n > 0 5 n log 0 > 5 log 0 = 0.00 > 0 n > 5 log 0 5 log 0 = 5 = n n = 9 5 n( n + 7) 4n 9

109 04 () a k = n C k (k = 0,,, n) a r+ = n C r+ a r nc r n! = (r + )!(n r )! n! r!(n r)! = n r r + a = 5a n 5 n C = 5 n C n 5 (= 5 n C 5 ) n(n )(n ) = 5 n 7n = 0 n n = 7 n(n )(n )(n )(n 4) 5 4 () i ( i) 4( i) + a( i) + b = 0 (a + b ) + ( a + 6) = 0 a b a + b = 0 a + 6 = 0 a = 6 b = 4 x 4x + 6x 4 = 0 (x )(x x + ) = 0 x =, ± i, + i. n r r , + i

110 A 0 ( )70 () () a = b = a b 6 a b a b = a b θ () 0 α β 80 { sin α + sin β = cos α + cos β = α β cos(β α) = cos α = α = β =

111 {,,, } {, 5,, 4} {,,, 4} {, 5,, 6} 4 {, 5,, 4} 4 4! {,,, } 4 4 x + y = C A(, 0) C C P Q(P Q 4 ) AP y = P (, ) AP AQ C C (x a) + y = r ( a > 0 ) AP C R (a, 0) S O a r = AOP ASR a r : r = : a = r =

112 () ) = ( =4( ) =4 + 5) (5 = () ( a b) (6 a b) = 600 ( a b) (6 a b) = 0 6 a 7 a b + b = 0 a = b = 6( ) 7 a b + = 0 a b = cos θ = a b a b = = 0 θ 80 θ = 45

113 08 () sin α + sin α sin β + sin β = cos α + cos α cos β + cos β = (cos α cos β + sin α sin β) = 4 cos β cos α + sin β sin α = 0 cos(β α) = 0 0 β α 80 β α = 90 4 β = α { sin α + cos α = cos α sin α = sin α cos α = 0 α 80 α = 60 5 β = {,,, } {, 5,, 4} {,,, 4} {, 5,, 6} {, 5, 4, 4} {5, 5, 4, 6} = 96 ( ) ! = 4 ( ) 4! = (! ) = = {, 5, 4, 4}, {5, 5, 4, 6}

114 A x + y = y = m(x ) y (m + )x 6m x + 9m = 0 D/4 = ( m ) (m + )(9m ) = 8m + D = 0 8m + = 0 m = ± AP m = 9x 6x + = 0 x = AP y = (x ) ( P, ) y a r = 4 AOP ASR SA : SR = OA : OP ( a) : r = : O P Q R S r a A x a = r =. (x ).. ( a).

115 0.4.7 A ( )70 () m n (m ) = 5n 6 m = n = () 5 50 log 0 = 0.00 log 0 5 = y = 5 50 log 0 y = log 0 5 = 0 y y = 0 < y < () () () (4) (5) k k

116 .4. ABC P AP + BP + CP = 0 PA = a PB = b PC = c () AP = AD D PD BD b c PD = BD = () PAB S BD BC BD = BC PDC S () a a = I b b = I a b = I I I I CP CA = BA BC = () (m ) = 5n 6 (m + 54m) (9m + 54) = 5n 6 m + 54m 9m n m + 54m = 5n 9m + 54 = 6 m > 0 m = n = 0 () log 0 5 = log 0 = log 0 0 log 0 = 0.00 = y = 5 50 log 0 y = log = 50 log 0 5 = = y y = 0 4: < y <

117 () 0 0C = 0 9 = 45 ( ) ( ) 45 () 8 C 8 C = 8 = () () () = 7 45 (4) 8 8C = 8 7 = 8 ( ) 8 45 (4) () 7 45 = 8 45 (5) ( ) X X ( 50) = 60 9 ( ) ( ) X 00 0 k X ( k) k 45 > 0 k < 00 8 = 00 8k 45 k k ( )

118 () PA = a PB = b PC = c AP = a BP = b CP = c AP + BP + CP = 0 ( ) ( a) + ( b) + ( c) = 0 a = b c.4. AD = AP PD = AD AP = AP AP = AP = PA = a ( = ) b c = b + c BD = PD ( ) PB = b + c b = b + c () () BD = ( c b) = ( PC PB) = BC D BC : PDC = PDB AP = AD P AD PDB = PAB 4 PAB = S 4 PDC = S P A B D C

119 4 () ( ) CP = PA + PB = a + b PC = a b CA = PA PC = a ( a b) = 4 a + b BA = PA PB = a b BC = PC PB = ( a b) b = a b CP CA = ( a + b) (4 a + b) = a a + 4 b b + 4 a b = I + 4I + 4I BA BC = ( a b) ( a b) = a a + b b = I + I. b + c b + c.. S I + 4I + 4I. I + I

120 B ( ) () x > x 4x + = 0 x = x + ( ) x x = x = 8 () AB = 4 BC = a CA = 6 ABC a 9 < a < 0 BAC = θ (a )( 4 5 a ) sin θ = 6 7 ABC 5 a = 8, 9 0 () a = ( x, 9) b = (, x + x ) a b = + x 4 5 a b x = 6 a b a b (4) y = cos θ + 7 cos θ + 4 (0 θ π) t = cos θ y = 7 t + 8 t + y y θ π θ 4 π

121 a b c 5 () abc = () a + b + c = () a + b = c 4 4 (4) a + b + c n+ n f(x) = nx + f(t) dt x n+ n f(x) a n = f(t) dt () a n = 47 n 48 n n () a k = 49 n 50 5 n 5 n k= () f(x) = n y = f(x) y = n n xf(x) x = 5 n, 54 n xf(x) 57

122 .4. 7 () x 4x + = 0 x > x = + x = ( + ) = x 0 x x + x = 4 x + + x = 6 x + x = 4 ( ) x ( + x = x + ) + = 4 + = 6 x () ABC CA AB < BC < CA + AB 6 4 < a < cos θ = a 6 4 < a < 0 = 5 a 48 sin θ = cos θ = ( cos θ)( + cos θ) ) ) 5 a 5 a = ( ( = a 4 48 sin θ > 0 sin θ = ABC ABC = CA AB sin θ 00 a 48 (a 4)(00 a ) 48 = 6 4 (a 4)(00 a ) 48 = (a 4)(00 a ) 4 ABC = 5 (a 4)(00 a ) = 5 4 (a 4)(00 a ) = 60 a 4 04a = 0 (a 64)(a 40) = 0 a = 8, 0

123 8 () a = ( x, 9) b = (, 9 x ) a b = x + 9(9 x ) = x + 8 ` x 8 a b a b = 0 x = t t > 0 x + 8 x 8 = 0 t + 8t 8 = 0 t 8t + 8 = 0 (t 9) = 0 t = 9 x = 9 x = 4 (4) cos θ + 7 cos θ + 4 = ( cos θ ) + 7 cos θ + 4 = cos θ + 7 cos θ + y = cos θ + 7 cos θ + 4 (0 θ π) t = cos θ t y = y = t + 7t + ( t ) 5 8 t = t = 0 y 0 t + 7t + { t + 7t + 0 t + 7t + (t + )(t + ) 0 t, t t + 7t 0 t(t + 7) 0 7 t 0 t 0 π θ π

124 .4. 9 () 6 6P = = 0 ( ) abc = 8 {,, 4}! = = 6 ( ) 6 0 = 0 () a + b + c = 8 {,, 5} {,, 4}!! 0 = 0 () a + b = c c = (a, b) = (, ), (, ) c = 4 (a, b) = (, ), (, ) c = 5 (a, b) = (, 4), (, ), (, ), (4, ) c = 6 (a, b) = (, 5), (, 4), (4, ), (5, ) 0 = 0 (4) a b c p p = 4 0 = a + b + c + + = 6 p = 6 0 = a + b + c = = p = 0 = a + b + c = p = = = a + b + c a + b + c p = {a, b, c} {, x, y} ( < x < y) + x + y = xy x y = 0 (x )(y ) = 4 x y x =, y = 4 x =, y = 6 a + b + c {,, 6}!! 0 = 0

125 0 () f(x) = nx + a n a n = n+ n [ (nt + a n )dt = nt + a n t = 4n + 4n + a n a n = 4n 4n () () n a k = k= n ( 4k 4k) k= ] n+ = 4 n(n + )(n + ) 4 n(n + ) 6 = 4n n 8n () () f(x) = nx 4n 4n n xf(x) = x (n + )x f(x) = xf(x) n nx 4n 4n = x (n + )x x (4n + )x + n(n + ) = 0 (x n){x (n + )} = 0 x = n, n + n x n + f(x) xf(x) n y = f(x) y = f(x) n S S = n+ n n+ = n ( = 6 {f(x) xf(x)} dx (x n){x (n + )} dx ) {(n + ) n} = 4 n

126

127 .5.5. A 70 ( ) (A ) ( ) ( ) 7 (70 ) , 6. 7.

128 .5.. () IC 8 IC 64 () o 5 6 log 0 = a 4b = a b. y = x 4x + 7 () (, 4) l () l x m 4. ABC A = 60 AB = 6 AC = 4 BC D AB : E AC : F AD DE EF () BC () sin B () BDE S (4) CDEF S 5. () z z z + z + z z () x ax + b = 0 x + ax + a b 6 = 0 a = b = () x y x + y x + y 6. p : y = x 4x + x y 4 x p () p () p p l () p l 7. 9

129 4. () 64 = () 5 6 log = log 0 5 = log 0 0 = (log 0 0 log 0 ) = ( 0.00) =.68 < log < 0 < 5 6 < (a + b)(a b) = a b a + b a b a + b > 0 a + b > a b a + b = a b =. () y = x 4x + 7 y = x 4 x = y = 4 = a = 7, b = l (, 4) y 4 = (x ) () l x x x + 6 = 0 y = x + 6 x = (, 0) (a, a 4a + 7) a 4 y (a 4a + 7) = (a 4)(x a) y = (a 4)x a + 7 (, 0) 0 = (a 4) a + 7 a 6a + 5 = 0 (a )(a 5) = 0 () a a = 5 y = 6x 8

130 () ABC C BC = cos 60 = F D = 8 BC > 0 BC = 7 BC () ABC sin A = CA sin B 7 sin 60 = 4 sin B 4 sin 60 sin B = = 7 7 A 60 E 6 B () D BC () BD = BC = 7 E AB : BE = AB = 4 () S = BDE = BD BE sin B = = (4) ABC = AB AC sin A = 6 4 sin 60 = 6 E AB : AE = AB = F AC : AF = 4 AC = AEF = AE AF sin A = sin 60 = () S = ABC ( BDE + AEF) = 6 ( ) + = 5

131 6 5. () Z = z + z + z z = z z + z z + z z Z = z z + z z + z z = z z + z z + z z = z z + z z + z z = z z + z z + z z = Z Z = Z Z () a = b = x ax + b = x + ax + a b 6 = 0 x + x = 0 x x 6 = 0 x =, x =, () x + y P P 4 (, 0) (0, ) (, 0) (0, ) x + y Q Q P Q

132 () p x y 4 y + 4 = (x ) 4(x ) + y = x 8x + 9 x p y = x 8x + 9 y = x + 8x 9 () p : y = x 4x + p : y = x + 8x 9 (, ), (5, 6) y ( ) = 6 ( ) (x ) 5 y = x 4 p p x y = 4x 8 y = x 4 () p l x 5 p l S S = 5 {( x + 8x 9) (x 4)}dx 5 = (x )(x 5)dx ( = ) (5 ) = C = = 84 ( ) 4C = 4 ( ) 5C 4 C = = 40 ( ) = 84 5C = 5 4 = 0 ( ) 0 84 = = 7 4

133 8.5. A 70 ( ) (A ) ( ) ( ) 8 (70 ) , 6. 7.

134 () x x 8 x = 0 () log x + log (x + 4) =. a + bi a b i () 6 5i () ( + i) 4 + i. A(, 0) B(, ) C(m, n) ABC n 0 () C () ABC (x x 0 ) + (y y 0 ) = r x 0 y 0 r () D CD 4. R ABCD A = 60 B = 75 BC = CD = () R () DA 5. y = x + ax + b y = x a b () b () () () ( 7. 7 sin x + π ) ( π ) + sin x cos x = 0 x π

135 0 8. f(x) = x 6x + 9x + k () f(x) g(x) y = g(x) x S () f(x) k () f(x) = 0 k. () x+4 = 6 x x = ( x ) 8 x = ( x ) x = t t > 0 6t + 6t t = 0 t(t + )(t 8) = 0 t > 0 t = 8 x = x = () x > 0 x + 4 > 0 x > 0 log x + log (x + 4) = x + 4 log = x x + 4 = x x x 8 = 0 (x + )(x 4) = 0 x > 0 x = 4. () 6 5i 4 + i = (6 5i)( i) ( + i)( i) = 6i + 5i (4 i ) = 6i + 5 ( ) (4 + ) = 7 6i 0 () ( + i) = + i + (i) + (i) = + 6i + i + 8i = + 6i + ( ) + 8( i) = i

136 . () AB = ( ) + ( 0) = 4 ABC AB = AC = BC AC = 4 (m ) + n = 4 m + n 4m = 0 BC = 4 (m ) + (n ) = 4 m + n 6m n + 8 = 0.5. m + n 8 = 0 m = 4 n (4 n) + n 4(4 n) = 0 4n 4 n = 0 4n(n ) = 0 n 0 n = C(, ) m = () ABC D ( ) ( ), D, x 0 = y 0 = r r = DA = ( ) + ( ) = () E(x, y ) E D C x + = y + = x = y = ( ),

137 4. () ABCD C = 80 D A = = 0 BCD C BD = ( ) + ( ) cos 0 = 9 BD > 0 BD = A B BCD R = BD sin C R = sin 0 = () BCD BCD = 0 DBC = 0 ABD = 75 DBC = 75 0 = 45 DA ABD sin ABD = R DA = sin 45 = 6 ( 5. () y = x + ax + b y = x + a ) a 4 + b ) a (, a 4 + b y = x ( a 4 + b = a ) + b b = 0 () y = x + ax y = x x x + ax = x x(x + a) = 0 x = 0, a y = x x = 0 y = 0 x = a y = a ) 4 ) a a4 (0, 0) (, a (, a 4 8

138 .5. () () x = a 4, y = a 8 a = 4x y = ( 4x) = x 8 y = x ! 5 4! = cos x + ( cos x) cos x = 0 6 cos x cos x = 0 ( cos x + )( cos x ) = 0 π x π cos x 0 cos x = x = π

139 4 8. () g(x) = f (x) = x x + 9 = (x )(x ) y = g(x) x y 0 S ( S = (x )(x ) dx = ) ( ) = 4 6 () f(x) x f (x) f(x) k + 4 k x = k + 4 x = k () f(x) = 0 y = f(x) x k k 0 4 k 0

140 A 70 ( ) ( ) (A ) ( ) 9 (70 ) , 6. 7.

141 6. x = 4 5 y = () x + y + xy () x + y x + y () x + y + xy. l : y = x + 5 x = y = ax + bx + () a b () l P l m () m. x + y = x + y ABC AB = AD = DC BC D AB 0 BAD 0 B D C () BD () sin C AC () ACD 6. θ () sin(θ + π) () sin θ ( () cos θ + π ) A

142 () x+ 8 = 0 () log (x + ) = 0 8. (x + x + x + )dx 4. () x + y = = 4( + 5) + 4( 5) ( + 5)( = 6 5) 4 xy = = 4 x + xy + y = (x + y) xy = 6 4 = () () x + y x + y = x + y (x + y) xy = = 4 () () x + y + xy = (x + y) xy + xy = = 7. () y = ax + bx + y = ax + b l : y = x + 5 y = ax + bx + x = P (, ) 4a + b + = 4a + b = a = b = () l m P(, ) y = (x ) y = x

143 8 () y = x y = x + x + y x x = 0 (x )(x + ) = 0 x =, y = x x = y = = x = y = ( ) = 4 ( (, ), ), 4. x + y = k y = x + k x + y = x + ( x + k) = 9x 8kx + k = 0 x D/4 = ( 4k) 9 (k ) = (k 9) = (k + )(k ) D 0 (k + )(k ) 0 (k + )(k ) 0 k x + y ! 4 4! 5! 4! = = 880 ( ) BD 5. () ABD sin 0 = sin 0 sin 0 BD = = sin 0 =

144 .5. 9 () ACD = 5 sin C = sin 5 = sin(45 0 ) = sin 45 cos 0 cos 45 sin 0 = = = 6 4 AC ABC sin 0 = sin 5 sin 0 AC = = 6 sin 5 = () AD = DC = ADC = 50 ACD = sin 50 = 6. () π < θ < π π < θ + π < π sin(θ + π) < 0 () π < θ < π π < θ < π sin θ < 0 ( () cos θ + π ) ( = sin θ + π ) = cos θ π < θ < π < cos θ < 0 ( cos θ + π ) > 0 7. () x+ 8 = 0 x+ = 6 x + = 6 x = 5 () log (x + ) = 0 log (x + ) = 4 x + = 4 x = ± 8. [ (x + x + x + )dx = 4 x4 + x + x + x ( = ) + { 4 ( )4 + ( ) + } ( ) + ( ) = 0 ]

145 A 4 70 ( ) ( ) (A ) ( ) 0 (70 ) , 6. 7.

146 ( ) log 64 () () log log 8 () ) ( ABC ABC sin B = 4 BC = 6 () AC () AB () cos A () 8 8 () () 4. f(x) = x + (a + )x + a + a x f(x) 0 a 5. y = x 5 y = x x + 6. π θ π () cos θ + cos θ tan θ = 0 ( () sin θ + π ) ( cos θ π ) = 0 7. f(x) = ax + bx + c f(x) = y = f(x) x = y = 6x + 7 a b c

147 4 ( ) ( ) 6. () log () log () 64 log 8 = log = 6 log = = log ( ) = ) ( = = AC. () sin B = R = 0 = 0 AC = R sin B 6 + = = + 4 () ABC S = AB BC sin B + = 6 + AB 6 AB = 4 () () () cos A = b + c a bc. () 8 = ( + ) + ( 6) ( = + + ) 4( + ) = 8C = 8 7 = 8 ( ) 4 8 {, 7}, {, 5}, {, 7}, {5, 7} 4 8 = 7 () 4 4 = 6 ( ) 6 (, ) = (, ), (5, ), (5, 4), (7, ), (7, 4), (7, 6) 6 6 = 8 () 4C 4 C = 6 6 = 6 ( ) 6 {(, 5), (, 4)}, {(, 7), (, 6)}, {(, 5), (, 6)}, {(, 7), (, 8)}, {(, 7), (4, 6)}, {(5, 7), (4, 8)} 6 6 = 6

148 4. f(x) D = (a + ) 4 a + = (a + )(a ).5. 4 f(x) x D 0 (a + )(a ) 0 a 5. y = x 5 y y = x x + y = x x = x = O 5 x P P(, ) P P Q y ( ) = (x ) y = x + Q Q(, ) PQ PQ = ( ) + { ( )} = (, ) (, ) 6. () cos θ + cos θ tan θ = 0 sin θ + cos θ sin θ cos θ = 0 sin θ sin θ = 0 (sin θ )( sin θ + ) = 0 sin θ = 0 sin θ + = 0 π θ π sin θ = θ = π 5 sin θ = θ = π 6 tan θ θ ± π θ = π 6

149 44 ( () sin θ + π ) ( cos θ π ) = 0 ) sin θ cos π + cos θ sin π ( cos θ cos π + sin θ sin π sin θ + cos θ cos θ = 0 sin θ = 0 ( ) sin θ + ( ) cos θ = 0 sin θ cos θ = 0 ( sin θ π ) = 0 4 π θ π 4 π θ π 4 π 4 θ π 4 = 0 θ = π 4 7. (ax + bx + c) = [ ax + ] bx + cx = a + c = a + c = f(x) = ax + bx + c f (x) = ax + b y = f(x) x = y = 6x + 7 (, ) 6 f() = a + b + c = f () = 6 a + b = 6 a = b = c = 0

150 A 5 70 ( ) (A ) (70 ) , 6. 7.

151 46. () x + y + z = x y z = x + 4y + z = 0 () (x ) + x > (x + 4) () α = + i α α i. 90 < θ < 80 sin θ = 4 9 cos θ tan θ. 4 () 6 () 6 () log 0 = 0.00 log 0 = x y = 4 y 4 4x + y x 6. x y 4 x + y 0 x y x + y x y 7. a a + a 8. 4 (x 5) dx

152 x + y + z = () x y z = x + 4y + z = 0 + 5x + y = 4 x + y = x = y = z = x = y = z = () (x ) + x > (x + 4) 7x > x > 7 () α = + i α = i (α ) = ( i) 4α 4α + = α α = sin θ + cos θ = cos θ = sin θ = ( ) 4 = < θ < 80 cos θ < cos θ = 8 = 9 ( tan θ = sin θ cos θ = 4 ) 65 9 = () ( ) 4 5 =

153 48 () ( ) ( ) 4 5 4C = () () () ( ) = log 0 54 = log 0 = log 0 + log 0 = =.7 log 0 5 = log 0 0 = log 0 0 log 0 = 0.00 = log 0 ( ) = 0 log log 0 5 = =.88 < log 0 ( ) < 4 log 0 0 < log 0 ( ) < log < < x y = 4 x = y + 4 4x + y = (x) + y = (y + 4) + y = y + 8y + 6 = (y + ) + 8 4x + y y 4 y = 4 y = (x, y) = (4, 4) 80 (x, y) = (, ) 8

154 y A A (, 4) k 7 (, 4) (, ) 0 x + y = k y A (, 4) k 4 A k O k 0 7 x (, 4) k k = 8 (, 4) k k = 0 x + y x = y = 4 8 x = y = = + ( )( + ) = + < = 8 < 5 < + < a 5 + a = + a = a + a = + = ( ) + = + ( + ) ( )( + ) = (x 5) dx = (9x 0x + 5)dx [ ] 4 = x 5x + 5x = ( ) ( ) = 9

155 = + 4 4x < x + < 5x + 6 x < < x < x < 4 x <, < x x 6x = 0 x = ± x = ± x = ± 4 x = ± 4 y = x + 6x + 5 (, 4) (, 4) (, 4) 4 (, 4) 5 y = x x + (0 x ) 4 4 6

156 x x 6 < 0 < x < < x < x <, < x 4 x <, < x 7 0 θ 80 cos θ + = 0 θ θ = 0 θ = 60 θ = 0 4 θ = 50 8 ABC BC = A = 0 C = 45 AB = A A B B C

157 5 4 (0 9) ( ) ABC AB = 5 CA = 5 cos BAC = 8 () sin BAC = () BC = () ABC (4) ABC 4 () () ()

158 + = ( + ) + ( ) + ( + )( ) ( ) 4 4x < x + x < x + < 5x + 6 x > < x < ( ).6. 5 = = x + ( )x = 0 x = ( ) ± ( ) ( ) = ± = ± ( ) 4 4 y = x + 6x + 5 y = (x + ) 4 (, 4) ( ) 5 y = x x + y = (x ) + y 6 0 x y x = 6 ( ) 4 6 (x )(x + ) < 0 < x < ( ) 7 cos θ = 0 θ 80 θ = 50 ( ) 4 O x

159 54 BC 8 sin A = AB sin C AB sin 0 = sin 45 AB = AB = ( ) 9 5! = 0 (!!! ) ( ) 0 7 7C = 5 ( ) 4 C 4 C = 4 = ( ) 5 ( ) 4 () sin A > 0 ( ) sin A = = () a = b + c bc cos A BC = = 6 BC > 0 BC = 6 BC () = R sin A R = BC sin A R = = 8 7 = (4) ABC = bc sin A = = ( )

160 () 6 = 4 ( ) 4 = 8 () 6 = ( ) 4 4 = 6 6 = 5 6 () 4 4C ( ) ( ) = 8 ( )

161 56.6. ( ) ( 5) x + y = x + y = xy = x x (a )x + a a + = 0 a = x = a b a > 0 y = ax 6ax + b ( x 4) a = b = 4 tan θ = + sin θ + sin θ = 5 ABC AB = 6 CA = 4 BAC = 0 ABC D AD = ( 5) BAC BC a b x + ax + b = 0 x = + i (i ) a = b = 0 x π sin x cos x + = 0 x = π x y = 9 x x+ + x = x = log 4 x y+6 = 0 (5, ) (, ) 5 0 x x dx =

162 ABC AB = 5 CA = 7 ABC = 60 O AB BC O O BC = ABC O R ( ) 4 O r ( ) 4 a b f(x) = x + ax + bx + 4 f() = 7 f ( ) = 0 ( ) a b f(x) x f(x) = k k α

163 58 x + y = (x + y) xy(x + y) = xy xy = 4 x (a )x + a a + = 0 D = 0 { (a )} 4 (a a + ) = 0 a = 7 4 (a ) x = = a = 7 4 = 5 4 y = ax 6ax + b y = a(x ) 9a + b 4 a > 0 x 4 x = x = 5a + b = 9a + b = a > 0 a = b = 7 + sin θ + ( sin θ) + ( + sin θ) = sin θ ( + sin θ)( sin θ) = sin θ = cos θ = ( + tan θ) = { + ( ) } = 6 5 ABC = 4 6 sin 0 = 4 6 = 6 AD = x ABD + ACD = 6 x sin 60 + sin 60 4 x = 6 x + 4 x = 5 x + x = x ABD + ACD = ABC 5 x = 6 x = 5 ( )

164 i ( + i) + a( + i) + b = 0 (a + b ) + (a + )i = 0 a + b a + a + b = 0 a + = 0 a = b = 4 ( sin x π ) + = 0 ( sin x π ) = 0 x π π x π π x π = π 6 x = π 6 x = t x t y y = t t + = y t = t = ( t ) + 4 x = x = log 4

165 60 4 (5, ) A x y + 6 = 0 l l A B(s, t) l AB t s 5 l AB t s 5 = ( s + 5 AB s + 5 t + s + t = 7 ), t + s = t = 7 l + 6 = 0 s t = 9 (, 7) 5 0 x x = x + x x = x 0 x x dx = = = 0 0 x( x + ) dx + ( x + x) dx + [ x + x ] 0 [ x + x x(x ) dx (x x) dx ] = = 8 ( )

166 .6. 6 b = c + a ca cos B 7 = 5 + a 5 a cos 60 a 5a 4 = 0 (a + )(a 8) = 0 a > 0 a = 8 ABC S S = ca sin B = 5 8 sin 60 = 5 8 = 0 s = a + b + c = s = 0 ABC R S = Rs 0 = R 0 R = 4 O O B BO = R BO = r O O = R + r BO = BO + O O R = r + (R + r) r = R r = A O B O r R C ( ) 8 0

167 6 4 f(x) = x + ax + bx + 4 f (x) = x + ax + b f() = 7 + a + b + 4 = 7 a + b = f ( ) = 0 a + b = 0 a b = a = b = 9 f(x) = x x 9x + 4 f (x) = x 6x 9 = (x + )(x ) f (x) = 0 x =, f(x) x f (x) f(x) 9 x = 9 x = f(x) = k y = f(x) y = k x 9 4 k O < k <4 y y = f(x) x f(x) = 4 y = k x x 9x + 4 = 4 ± 5 x = 0, α < α < + 5

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( ) 18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................

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