(1) a x ax + 2 = x + a 2 (2) x 1 = 3 x 1 x = x < 1 x = (3) x x 3 = 6 (4) x 1 2 = 3 x =, 71 (1) ax = b ( i ) a \= 0 x = b a (ii)

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Download "(1) a x ax + 2 = x + a 2 (2) x 1 = 3 x 1 x = x < 1 x = (3) x x 3 = 6 (4) x 1 2 = 3 x =, 71 (1) ax = b ( i ) a \= 0 x = b a (ii)"

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1 (1) a a + = + a () 1 = 3 1 = < 1 = (3) = 6 (4) 1 = 3 =, 71 (1) a = b ( i ) a \= 0 = b a (ii) a = 0 0 = b { b = 0 b \= 0 () (3) < < (4) 1 = ±3

2 71 (1) a + = + a (a 1) = a 1 ( i ) a 1 = 0 a = = 1 1 (ii) a 1 \= 0 a \= 1 1 a 1 = a a 1 ( i ) (ii) a = 1, a \= 1 = a a 1 () ( i ) 1 1 = 1 1 = 3 = 4 (ii) < 1 1 = ( 1) ( 1) = 3 = =, 4 (3) ( i ) 1 ( + 1) ( 3) = 6 = 4 3 (ii) 1 < < 3 ( + 1) ( 3) = 6 = 1 < < 3 y 3 y = 1 O 1 4 y = y 6 5 (iii) 3 ( + 1) + ( 3) = 6 = = 4 3, 8 3 (4) 1 = 3 1 = ±3 1 = 3 ( i ) 1 ( 1) = 3 = 6 1 (ii) < 1 ( 1) = 3 = 4 < O = 4, 6

3 (1) () (3) { { { + y = 4 + y = 4 + y = 4 (1) () (3) 4y = 0 + 4y = 0 + 4y = 8 73 a y { (a ) + 4ay = 1 a = (3a + 1)y = a a = 74 { ( + y 1)( + y + 3) = 0 y = 1 (, y) { a + by = p c + dy = q 1 d b (ad bc) = pd qb 3 1 c a (ad bc)y = pc qa 4 ( i ) ad bc \= (ii) ad bc = 0 a : b = c : d { a : b : p = c : d : q 1 a : b : p \= c : d : q 1 74 y = 1 y = ±1 y = 1 1

4 7 (1) { + y = 4 1 4y = 0 1 8y = 8 y = 1 = ( 1 ) () { + y = y = y = 8 4 (3) { + y = y = y = 8 6 = 4 t, y = t t 73 { (a ) + 4ay = 1 1 (3a + 1)y = a (a ) : 4a = 1 : (3a + 1) 4a = (3a + 1)(a ) 3a a = 0 a = 3, 1 ( i ) a = y = y = 3 (ii) a = y = 1 4y = 1 a = 3 a = 1 74 y = 1 y = ±1 y = 1 ( i ) y = 1 1 { + ( 1) 1}{ + ( 1) + 3} = 0 = 1, 1 3 (ii) y = { + ( + 1) 1}{ + ( + 1) + 3} = 0 = 0, 5 3 = 5 3

5 (1) 4 5 = 0 =, () 4 = 0 =, (3) + 3 = 1 + =, (4) 4 + = 0 =, 76 (1) 4 5 = 0 () 8 = (3) 6 = 4 =, 75 + (a + b) + ab = ( + a)( + b) ac + (ad + bc) + bd = (a + b)(c + d) a + b + c = 0 (a \= 0) = b ± b 4ac a a + b + c = 0 (a \= 0) = b ± b ac a 76 (1) 0 < 0 = 4 5 = 0 () 0 < 0 (3) < 0 A = B B > 0 A = ±B

6 75 (1) 4 5 = 0 ( + 1)( 5) = 0 = 1, 5 () 4 = 0 ( + )( ) = 0 =, (3) + 3 = = 0 ( + 1)( 1) = 0 = 1, 1 (4) = ± ( ) = ± 76 (1) = 4 5 = 0 ( + 1)( 5) = 0 0 = 5 = ±5 () ( i ) 0 8 = ( + )( 4) = 0 0 = 4 (ii) < 0 8 = = 8 < 0 = =, 4 (3) 6 = 4 6 = ± = = = 0 D D = < = = 0 ( + 1)(5 6) = 0 = 1,

7 a + b = 0 1, + b + a = 0 1 α α = 78 a + b = 0 = 1, α + a + 3b = 0 1 = α a b 77 f() = 0 g() = 0 α f(α) = 0 g(α) = 0 a b α a b 78 = 1 a + b = 0 a b 1 b a α a α a a 1

8 77 α 1 α + aα + b = 0 1, α + bα + a = 0 1 (a b)α = a b (a b)(α 1) = 0 3 a = b 1 1 α = 1 1 ( ) b = (a + 1) 1 : + a (a + 1) = 0 ( 1)( + a + 1) = 0 : (a + 1) + a = 0 ( 1)( a) = 0 ( a 1) + a = 1 78 a + b = a + 3b = 0 1 = a + b = 0 b = a 1 1 a a 1 = 0 = 1, a + 1 α = a a 3(a + 1) = 0 α (a + 1) + a(a + 1) 3(a + 1) = 0 a = ±1 ( i ) a = 1 b = 1 : = 0 ( + 1)( ) = 0 : + 6 = 0 ( + 3)( ) = = (ii) a = 1 b = 0 1 : + = 0 ( + 1) = 0 : = 0 ( 1) = = 0 (a, b) = (1, ), ( 1, 0)

9 (1) 3 5 < + 1 < + 3 { 4 5 () > < 5 < > 1 81 a a 79 (1) A < B < C { A < B B < C () { f() 0 g() > 0 f() 0 g() > a b { b 0 ( i ) a = 0 b < 0 (ii) a > 0 b a (iii) a < 0 b a

10 79 (1) 3 5 < + 1 < 6 < < + 3 < > 1 < < 3 () > 3 5 > < < 5 < 8 < 4 < + 1 > > 1 > < 1 ( < 4 > 3) < 1 < 4 3 < < a 1 4 (a 4) 1 5 a 4 < 0 a < 1 a 4 1 a 4 = 5 1 = 5(a 4) a = a < a = 19 10

11 (1) 1 < < 3 () + 4 < 6 83 (1) y = () a (1) a < () a < a (3) a < + 1 a (4) a (3) a < () (1) y = 6 y = 10

12 84 (1) a < p (p > 0) p < < p () p < < q q > 0 (3), (4) y = a y = + 1 y = a y = a y = a a 1 y = y = a y = + 1 a < 1 a = 1 a > 1

13 8 (1) 1 < < 3 3 < < 1 1 < < 3 1 < < 1, 3 < < 5 y y = 3 1 O () ( i ) < 0 ( 4) < 6 > 3 < 0 3 < < 0 (ii) 0 < 4 ( 4) < 6 < 0 < 4 0 < (iii) 4 + ( 4) < 6 < < < 83 (1) ( i ) < 3 y = ( + 3) ( 1) = (ii) 3 < 1 y = ( + 3) ( 1) = 4 (iii) 1 y = ( + 3) + ( 1) = + y = y = + 4 y O 4 y = y () (1) y = 6 y = , O

14 84 (1) a < < a < a < < a + () a + > 0 a > (3) y = + a y = + 1 = a 1 y = a a 1 < a a 1 < a a > 1 a 0 a + (4) (3) > a 1 1 y = + 1 a 1 a

15 (1) ( 3) 5 < 0 { () 3 < > 0 (3) (a + 1) a > 0 a (4) a (a ) + (4 a) 0 85 (1) () (3) ( ) = 0 = a + 1 (4) a = 0 1

16 85 (1) ( 3) 5 < < 0 ( 5)( + 1) < 0 1 < < 5 () 3 < 0 ( + 1)( 3) < 0 1 < < > 0 < < < 3 < (3) (a + 1) a > 0 { (a + )}( + 1) > 0 a + 1 a + = 1 1 a + ( i ) a + < 1 a < 3 < a +, > 1 (ii) a + = 1 a = 3 \= 1 (iii) a + > 1 a > 3 < 1, > a + (4) (a ) + (4 a) 0 {(a ) + }( 1) 0 1 ( i ) a = ( 1) 0 1 ( (ii) a > 1 : + ) ( 1) 0 a a < 0 < 1 (iii) a < 1 : a a, 1 ( + ) ( 1) 0 a 1 a 0 < a <, 1 a a = 0, = 1 a < 0, a 1 > 1 a > 0

17 (1) 5 < 1 () (3) < 1 (4) (1) () (3) < a (a > 0) a < < a a 0 a (4) < ( + 3)

18 86 (1) ( i ) < < 0 ( + 1)( 4) < < 4 (ii) 1 < 0 < 1 5 < ( 1) 6 < 0 ( + )( 3) < 0 < 1 < < 1 ( i ) (ii) < < 4 () ( i ) ( 1) ( + 1)( 4) (ii) 1 < 0 < 1 + 3( 1) ( + 5)( ) 0 < 1 5 < 1 ( i ) (ii) 5 4 (3) < 1 1 < < 1 1 < > 0 ( )( 1) > 0 < 1 > < 1 3 < 0 ( 3) < 0 0 < < < < 1, < < 3 (4) ( + 3) ( + 3) ( 4)( + 3) ( + 3)( 6) = 3,

19 (1) 3 + k 0 5 k () a + b > 0 < < 1 a = b = (3) + < 4 < < 87 α < β α < < β ( α)( β) < 0 α < < β ( α)( β) > 0

20 87 (1) 5 ( + )( 5) k k = 10 5 () < < 1 ( + )( 1) > 0 + > 0 a b a = 1, b = (3) + < a + a < 0 4 < < b = 4 + a = a = 0 a = 8 + < < 0 ( + 4)( ) < 0 4 < < + 1

21 a + a 3 0 a 89 8 < 0 1 < < ( 8)( + k) > 0 k 90 k f() = k + k + 3 f() > 0 k f() < 0 3 k 88 y = f() α β f() 0 f(α) 0 f(β) 0 α β 89 a b f() < 0 a b f()g() > 0 a b g() < 0 90 f() > 0 f() > 0 y = f() y > 0 < 0 +

22 88 f() = a + a 3 y = f() 0 f() 0 f(0) 0 f() 0 { a a + a a 3 0 y = f() 89 8 < 0 ( 4)( + ) < 0 < < 4 1 < < 8 < 0 1 < < ( 8)( + k) > 0 1 < < + k < 0 f() = + k y = f() y = f() f( 1) 0 f() 0 { f( 1) = 3 + k 0 1 f() = k 0 k 3

23 90 f() = k + k + 3 ( = k ) k 4 + k + 3 f() > 0 f() > 0 ( ) f k = k 4 + k + 3 > 0 k 4k 1 < 0 (k 6)(k + ) < 0 < k < 6 k + f() < 0 3 f() 0 f(3) < 0 f(4) 0 f() = 7 k 0 f(3) = 1 k < 0 f(4) = 19 3k 0 6 < k k

24 (a + ) + a 3 < y < (a 1) ( ) y a y ( ) a < a < y ( ) a < a < 9 a 1 4 f() = a, g() = a (1) f() g() a a () f() g() a a (3) 1 f( 1 ) g( ) a a (4) 1 f( 1 ) g( ) a a

25 91 f() < y < g() ( ) y ( ) ( ) y y y f() g() ( ) y f() < g() f() < g() y ( ) y ( ) y f() < y y< g() y f() < g() 9 (1) f() g() f() g() 0 y = g() y = f() a a () f() g() f() g() 0 (1 4) g() f() 0 (1 4) (3) 1 f( 1 ) g( ) f( 1) f( 1) g( ) f( 1 ) g( ) 1 f( 1 ) g( ) (4) (3) f( 1 ) g( ) 1 f( 1 ) g( )

26 91 f() = + (a + ) + a 3 g() = (a 1) y f() < y < g() f() < g() h() = g() f() = (a + 1) a + 1 h() > 0 ( h() = a + 1 ) 4a + 1a a + 1a 7 > 0 8 4a + 1a 7 < 0 y = g() y = h() y = f() (a + 7)(a 1) < 0 7 < a < 1 y f() < y < g() f() < g() ( f() = a + ) + a + 8a 8 4 ( g() = a 1 ) a a a + 8a 8 4 a + 6a + 1 < < a a < a < y = g() y = y = f()

27 9 g() = = ( ) + 5 (1) y = f() = a (1, 6) a = 6 a a 6 () y = f() = a y = g() 1 4 f() = g() (a + 4) + 9 = a D D = (a + 4) 4 9 = a + 8a 0 = (a + 10)(a ) D = 0 a = 10, 1 4 = a + 4 a = a a y 9 6 O y 9 6 O (3) 1 4 f() m f g() M g m f M g m f = f(1) = a, M g = g(4) = 9 y 9 a 9 6 (4) 1 4 f() M f g() m g M f m g M f = f(4) = 4a, m g = g() = 5 4a 5 a 5 4 O y O 1 4

28 (1) +( 4k)+k +1 = 0 k k = = () 4 + k + 3 = 0 k (3) a + a + a 1 = 0 (a + 1) (a + 1) + 1 = 0 a < a < 94 + (t + k + 1) + (kt + 6) = 0 1 t 1 t k k + k 0 k k 1 t 1 1 t k k + k 0 k k

29 3 93 a + b + c = 0 (a \= 0) = b ± b 4ac a b 4ac D Discriminant D > 0 D = 0 1 D < 0 a + b + c = 0 D = (b ) 4ac = 4(b ac) D 4 (1) a + b + c = 0 D = 0 = b a D = 0 b a > 0 (3) D 1 D D 1 > 0 D > 0 94 D 0 D t (D = f(t)) p.90

30 3 93 (1) + ( 4k) + k + 1 = D D 4 = (1 k) (k + 1) = 4k 5k = k(4k 5) D = 0 k = 0, = (1 k) = k 1 k > 1 k k = 5 4 = 3 () 4 + k + 3 = 0 D D = k = k 48 0 k 4 3, k 4 3 (3) { a + a + a 1 = 0 (a + 1) (a + 1) + 1 = 0 D 1 D D 1 > 0 D > 0 D 1 = a 4(a + a 1) = (3a )(a + ) > 0 < a < 3 D 4 = (a + 1) (a + 1) = a > 0 a > 0 0 < a < 3

31 (t + k + 1) + (kt + 6) = 0 D D = (t + k + 1) 4(kt + 6) 0 4t + 4(k + 1)t + (k + 1) 4kt 4 0 4t + 4t + k + k 3 0 f(t) = 4t + 4t + k + k 3 ( f(t) = 4 t + 1 ) + k + k 4 f(t) 0 1 t 1 t 1 t 1 f(t) 0 f(t) t = 1 k +k 4 k k + k 4 0 y = f(t) (k + 6)(k 4) 0 k 6 4 k 1 t 1 1 t f(t) 0 1 t 1 f(t) 0 y = f(t) t = 1 f(t) t = 1 f(1) = k + k 15 k k + k t = 1 y 1 t y = f(t) + 1 O 1 t (k + 5)(k 3) 0 k 5 3 k

32 3. 95 (a 1) + (a ) = 0 (1) () (1) a () α β 0 < α < 1 < β < a 96 a + a + 6 = 0 a (1) () (3) 1 97 m = 0 m (1) 1 () (3) 1

33 3 95 (1) 0 () f() = (a 1) + (a ) = 0, 1, f() 96 f() = a + a + 6 y = f() y (1) 0 f(0) () (3) y 97 m \= 0 m f() = 1 m m = 0 B > 0 A \= 0 A A ( 0 B A > 0 AB > 0

34 3 95 (1) D D 4 = (a 1) (a ) = a 3 0 a 3 () f() = (a 1) + (a ) y = f() f(0) > 0 f(1) < 0 f() > 0 + f(0) = (a ) > 0 a \= 1 β + f(1) = a 6a + 7 < 0 0 α 3 < a < 3 + f() = a 8a + 1 = (a )(a 6) > 0 a <, a > 6 3 < a < 96 f() = a + a + 6 y = f() f() = ( a) a + a + 6 y = a y a + a + 6 (1) f(0) = a + 6 < 0 a < 6 () a + a + 6 < 0 a <, 3 < a a < 0 a < 0 f(0) = a + 6 > 0 a > 6 6 < a < (3) 1 a + a a, 3 a a > 1 a > 1 f(1) = a + 7 > 0 a < 7 3 a < 7 f(1) O a f(0) y f(0) a + a + 6 y O + 1 a a + a O

35 3 97 m \= 0 D = 1 + 8m 0 m m( \= 0) 1 m m = 0 f() = 1 m m (1) 1 f( 1) > 0 : f( 1) = 1 1 m = m 1 m > 0 m(m 1) > 0 m < 0, m > 1 y = f() = 1 m 1 m 1 m > m > 0 m(m + 1) > 0 m m < 1, m > m > 1 () f(1) < 0 f(1) = 1 3 m = m 3 m < 0 m(m 3) < 0 0 < m < 3 > m y = f() y = f() - 1 (3) ( i ) 1 < < 1 (ii) f( 1) > 0 f(1) > 0 ( i ) 1 < 1 m < m m > 0 1 m m < 0 m(m + 1) > 0 m(m 1) > y = f() 1 m + 1 m < 1 m > 0m < 0 m > 1 m < 1, m > 1 4 (ii) f( 1) > 0 f(1) > 0 m < 0, m > 1 m < 0, m > 3 m < 0, m > m > 3

48 * *2

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