Answers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l =
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1 C ALGEBRA Answers - Worksheet A a 7 b c d e 0. f 0. g h 0 i j k 6 8 or 0. l or 8 a 7 b 0 c 7 d 6 e f g 6 h 8 8 i 6 j k 6 l a 9 b c d 9 7 e 00 0 f 8 9 a b 7 7 c 6 d 9 e 6 6 f 6 8 g 9 h 0 0 i j k 9 l 9 7 m n 00 0 o 6 6 p q r 6 a + 8 b c d e + 6 f a + b + c d e f a b 8 + c d + e f ( )( + )
2 C ALGEBRA Answers - Worksheet A page 8 a b c d 7 7 e 6 f 7 7 g 7 7 h i j k 8 0 l b 0 c a + d e 0 + f ( ) a x + x x + b x x x x 6 x ± x x ± c x d x + (x ) 6x x + 0(x ) 6x + x 0 + x ( + ) x + a ( ) b + + ( + ) + a + b c + ( + ) ( + ) ( 6 + ) or 6 + d ( ) + ( ) e +
3 C ALGEBRA Answers - Worksheet A page f g h i + ( + ) ( 7 ) ( 7) + + ( ) or j ( ) ( ) or 6 k 7 + (7 + ) (7 + ) l 6 + 6( + ) + ( + ) x x + x( ) x x (+ ) 9 l l ( + ) + 9 6( + ) 8 9 a b 6 ( 6) ( + )( ) + ( ) ( ) + c ( + 0)( 0 + ) d ( )( ) ( 8) or 9 e + ( )(+ ) f + ( )( + ) (7 + ) g + + ( + )(+ ) ( + ) h ( 7 )( 7 + ) ( + 7)
4 C ALGEBRA Answers - Worksheet B a 6 b 6 c d 6 e f 6 g 8 h i 7 6 ( ) 6 or j ( ) 8 or 6 6 k l a 8 b 7 c 0 d e 9 f a 8p 7 b x 0 c 6n d y e b f p q g x y h 6r s i x y 7 j a b 8 k r s 6 00r s l p q rs a b c 6 9 ( ) 6 d g 6 6 e 6 h 8 7 ( ) ( ) f 9 7 i 9 7 j k l m 8 n o 8 p 9 q or r ( ) a c ( ) 8 b ( 6) 8 d ( 7) 9 ( ) ( ) e ( 9) f ( 8) g ( 6) 6 6 h 8 6 ( ) ( ) i k ( ) ( ) j ( ) ( ) or l ( 6) ( ) ( ) 6 6 m or 0. n ( ) ( ) ( ) o q ( ) ( ) or 0.6 p ( ) ( ) ( ) or r ( ) ( ) ( ) ( ) ( ) ( )
5 C ALGEBRA Answers - Worksheet B page 6 a 7 b 6 + c 6 d 6 ( 9) e 8 f 9 ( ) i 9 ( ) j g ( 8) 9 h 6 6 or k 7 ( ) 9 or l ( ) 6 00 ( ) ( 8) or 8 7 a x b y 6 c p d 8x e y f + 8b 8b g x x h 6 a a i ( ) 9 0 p p j 9x k + 6 y y l t m + 7 b b n + 6 y y o + ( ) 6 x x p 9 + ( ) a a 8 a x 6 6 b x c x d x x ( ) x 8 e x 8 f x ± 6 ± g x ± 8 ± h x 7 x x (± ) ± 6 x (± ) ± 7 x 7 x ( ) 9 9 a e x b x ( x ) x f 6 x x x g c x x x d ( x ) x h x x x 6 x x x 0 a x b x c x d 9 x e x f x a ( ) 6 b ( ) c ( ) d 6 ( ) e 6 ( ) f ( ) a ( ) x x b ( ) x + x + c x x ( ) d ( ) x x e ( ) x x f ( ) x + x 6 a x y b x y c (x ) y d ( ) x x ( x ) y e x 8y f ( ) x x 8 y
6 C ALGEBRA Answers - Worksheet B page a x 6 b x c x + 7 d ( ) x x x 6 x x + x x x x e x f 6 x g ( ) x x h ( ) x + 6x + x x x 6x + x x x 7 x 6 i (7 ) x+ 7 x+ 7 j x ( ) k (6 ) x+ 6 x 6 l ( ) x x x + x x x x x 8 9 x x a x + ( ) x x b x ( ) x+ x+ c ( ) x x x d ( ) x x x x + x x x + x x x x x x x x e ( ) x + ( ) x f ( ) x ( ) x g 6 x (6 ) x + h ( ) x ( ) x x + x 6x 6 x 6 x 6 x + x 8x x + x 6x 6 x x x + x 8x x x x x i ( ) x x j ( ) x x k ( ) x ( ) x l ( ) x + ( ) x x x x x x 6x x x x x x x x 6x x x x x x x 7 6 a x b x + 6x c x d x + 8x e x + x f x g x + x h x x i x 6 + x x j x 9 + 6x + x + x k x + x l x x x + x x 9 + 7x + x x + x x x + x 7 a x + b t t c x x y d 6y y y y e p + p f w w g x ( x+ ) x + h x t ( t t ) t t
7 C ALGEBRA Answers - Worksheet C a 9 + x x x 0 + x b 8 8 x ± ± 9 6 x > 0 x x ( ) x x + a 8 x x + b 8 7 x x x 7 n ( + 7) a 7 b [ k ] 7 a i xy t t t 8 x x + 6 ii y ( t ) 6t 6t + x( ) 6 + b 6t + t 0 x t + t ( ) 6t + t (0 + ) t + (6 + )( + ) a 6 y + 6 x (6 ) x 0 a + 6 y + 6 x + y + x b + ( + ) + y x ( + ) b x y x (x ) x x + + x y ( )
8 C ALGEBRA Answers - Worksheet C page ( ) x + ( ) x + a a + b 0 x + 6x + a and b integers a 0 x + 6x + a x also a + b b 7 a ( ) t a ( ) 6 t t b + 0 t 7 b ( ) y ( ) y + c + + (+ )( + ) + y y + ( + )( + ) y y y 7 + a a ( b ) 9 9 b 6 a area ( )( + ) 9 8 a ( b ) b (6 + ) b b ( a ) a ( + ) or + b ( a ) a b AC ( ) + ( + ) AC 0 c tan ( ACB) ( )( ) + ( )( ) (6 + ) (8 ) 8 7 a i x + x y 8 ( + ) + p( + ) + q ii x ( ) x x ( x ) y p + p + q b y y 0 p, q rational + p y(y ) 0 p y 0 or and p + q x 0 (no solutions) or x q 9 x
9 C ALGEBRA Answers - Worksheet D a x + 8x + b x + x x + c x + 6x x + 7 d x x + 8x x x 8 e x 7x + x x x + 6 f x + x x x x + x x 9x x + 8 x + x x x + x g x 7 9x + x + x x 6 + x x h x + 8x 6 + x + x 8 x 7 x 6 9x + x x + x + 8x i + x x 6x 0 x + x j 6x + 0x 6x + x + x x x x + x + x k 8x + 6x x x x l 7x 6 + x + 7x 8 8x 6 x + x + 8 8x x + 6x 6x 8 x 6 x + x + 7x + x a y y + y + 6 b t t + t + 6 t t + t t + 8 t 6t t + t + c x 6x x x + x x x + 7 x x x + 6 d 6 + m + m 6m + 6 6m + m m m + m 0 + m + m m + m m e u + u + u + u u + u 6 u u + u + u a x x + x + + 8x x b x + 7x x + 9x x + 8x x + x x 6 + 9x x + x x + x c 0x + 8x x + x + 7x d 8x + x + x x x 0x x + x x x + x e x + 9x x x + x 0x f 6x x + x + x 7x + x x 0x + 9x + x x + x + x x x a LHS (x + )(x x + ) x(x x + ) + (x x + ) x 6x + x + x x + x x + 0x + RHS b LHS ( + x x )( x + x ) ( x + x ) + x( x + x ) x ( x + x ) x + x + x x + x x + x x x + x x RHS c LHS ( x) ( x)(9 6x + x ) (9 6x + x ) x(9 6x + x ) 7 8x + x 9x + 6x x 7 7x + 9x x RHS
10 C ALGEBRA Answers - Worksheet D page a x(x + x 6) + (x + x 6) b x(x x + 7) (x x + 7) x + x 6x + x + x 6 x 6x + x x + x x + 6x x 6 x x + 9x c ( + x x ) 7x( + x x ) d (x )(x ) (x )(9x x + ) 8 + 0x x x x + 7x x(9x x + ) (9x x + ) 7x 9x + 6x + 8 7x 6x + x 8x + x 8 7x x + 6x 8 e x (x x + 9) + (x x + 9) f x(x x + x + ) (x x + x + ) x x + 9x + 6x x + 7 x x + 0x + 8x x + x x x x + x x + 7 x x x + x + x g x (x + x + ) + x(x + x + ) + (x + x + ) x + x + x + x + 6x + x + x + x + x + x + x + 7x + h x (x x + ) + x(x x + ) (x x + ) x x + x + x x + x 6x + x x + x x + 7x i x (x x 8) x(x x 8) + (x x 8) 6x x x 0x + 0x + 0x + x 8x 6 6x x + x 6 j x (x + x 6) + x(x + x 6) 6(x + x 6) x + x 6x + x + x x 6x x + 6 x + x 8x x + 6 k x (x + x + ) + x (x + x + ) + (x + x + ) x 7 + x + x + 8x 6 + x + x + x + x + x 7 + 8x 6 + x + 6x + x + x + l 6( + x x + x ) x( + x x + x ) + x ( + x x + x ) 8 + 6x 6x + x 6x x + x x + x + x x 6 + x 7 x 7 x 6 x + x x + 6x 6x a (p )(p + p + ) p (p + p + ) (p + p + ) p + p + p p p p + p + 0p p b t(t + t + ) + (t + t + ) + t(t + t + 7) + (t + t + 7) t + t + t + t + 6t t + t + 7t + t + t + 8 t + 0t + t + 8 c x (x + x ) 6(x + x ) + x(x + x x + 6) (x + x x + 6) x + x 8x 6x 6x + + x + 6x x + 8x x x + x 6 x + x 9x + x + 8 d u(u u ) + (u u ) u (u + u ) u(u + u ) + (u + u ) u u u + u 8u 6 u 0u + 6u u u + u + u + u u 9u + u u + u 9
11 C ALGEBRA Answers - Worksheet E a (x + )(x + ) b (x + )(x + ) c (y )(y ) d (x ) e (y + )(y ) f (a + )(a ) g (x + )(x ) h (p + )(p + 7) i (x + )(x ) j (m )(m 8) k (t + 6)(t ) l (y )(y 8) m (r + )(r ) n (y + 7)(y 9) o (a + ) p (x + )(x 6) q (x )(x ) r (s + 8)(s + ) s (p + 7)(p ) t (m 0)(m + 9) a (x + )(x + ) b (p + )(p + ) c (y )(y ) d ( + m)( m) e (r + )(r ) f ( + y)( y) g (a )(a ) h (x + )(x ) i (x + )(x + ) j (s ) k (m + )(m ) l ( + y)( y) m (u + )(u + ) n (p + )(p ) o (8x + )(x + ) p (6r )(r + ) a (x )(x ) 0 b (x + )(x + ) 0 c (x + )(x ) 0 d x 7x 8 0 x or x or x or (x + )(x 8) 0 x or 8 e (x + )(x ) 0 f x x 0 g x x 0 h (x + 9)(x + ) 0 x or (x + 6)(x 7) 0 x(x ) 0 x 9 or x 6 or 7 x 0 or i x + x 60 0 j x x 0 k (x )(x ) 0 l x x 6x (x + 0)(x 6) 0 (x + )(x 7) 0 x or x 7x + 0 x 0 or 6 x or 7 (x )(x ) 0 x or m x + x 0 n x x 0 o x x 6 0 p 6x 9x (x )(x + ) 0 (x )(x + ) 0 (x + )(x 6) 0 (x )(x ) 0 x or x or x or 6 x or q (x + ) 0 r x x + 0 s x +0x+ x t 6x x x + 6 x (x )(x ) 0 x + x x x 6 0 x or (x + )(x + ) 0 (6x + )(x 6) 0 x or x or 6 6 a (y y + 6) b x(x + x ) c p(p ) d m(m + 7m + 6) (y )(y ) x(x )(x + ) p(p + )(p ) m(m + )(m + 6) e (a + )(a + ) f (t + )(t ) g ( + x x ) h (r r ) ( x)( + x) (r 7)(r + ) i x(x x + ) j y (y + y 8) k (m + )(m ) l p(p p + ) x(x )(x ) y (y + 6)(y ) (m +)(m +)(m ) p(p )
12 C ALGEBRA Answers - Worksheet E page a x x + 0 b x + x c x 9 0 (x )(x ) 0 (x + )(x + ) 0 (x + )(x ) 0 x or x or x or y y y (0, 6) (0, ) (, 0) (, 0) O x O (, 0) (, 0) x (, 0) (, 0) O x (0, 9) d x x 0 e x 0x + 0 f x x x(x ) 0 (x ) 0 (x )(x ) 0 x 0 or x x or y y y (0, ) (0, 0) (0, 0) O (, 0) x O (, 0) x O (, 0) x (, 0) g x + x 0 h + x x 0 i x x + 0 x x + 0 x x 0 (x )(x ) 0 (x )(x ) 0 (x + )(x ) 0 x or x or x or y y y (0, ) O (, 0) (, 0) x (0, ) (, 0) (, 0) (0, ) O x O (, 0) x (, 0) j x + x k 8x + x 0 l + 7x x 0 (x + )(x + 6) 0 (x )(x ) 0 x 7x 0 x 6 or x or (x + )(x ) 0 x or y y y (0, 6) (0, ) (0, ) ( 6, 0) O x O (, 0) x (, 0) O (, 0) x (, 0) (, 0)
13 C ALGEBRA Answers - Worksheet E page m x 7x n 6x + 7x 0 o 6x + x 0 (x )(x ) 0 6x 7x + 0 (6x )(x + ) 0 x or (x )(x ) 0 x or 6 x or y y y (0, 6) (, 0) O (, 0) x (, 0) O ( 6, 0) x O (, 0) x (0, ) (0, ) (, 0) 6 a x x + 0 b x 0 x c x(x x ) 0 d 0x x 9 (x )(x ) 0 x x 0 0 x(x )(x + ) 0 x 0x x or (x + )(x ) 0 x, 0 or (x )(x 9) 0 x or x or 9 x ± or ± e + x x 0 f x 6 x(x ) g (x + )(x + ) h x x x x 0 x 6 x x x + 8x + x x 0 (x + )(x ) 0 x x x + 8x + 0 (x + )(x ) 0 x or (x )(x ) 0 (x + 6)(x + ) 0 x (no sol s) or x or x 6 or x ± i x + 7x 0 j x(x + ) x k x(x + ) (x + ) l 7 x(x + ) (x + ) (x )(x + ) 0 x + x x x + x x x 6x x + x (no sol s) or x + x 0 x x 6 0 x + 8x 0 x ± (x )(x + ) 0 (x + )(x ) 0 (x )(x + ) 0 x or x or x or
14 C ALGEBRA Answers - Worksheet F a (x + ) + b (x ) + c (x ) + d (x + ) 9 (x + ) + (x ) + (x ) e (x + ) + 8 f (x ) 6 g (x + 6) h (x ) + (x + ) + (x ) (x + 6) 6 (x ) i (x + ) 9 9 j (x ) + 8 k (x + ) 9 + l (x + ) (x + ) 8 (x ) + (x + ) + (x + ) m (x 9) n (x ) o (x + 9 ) p (x 7 ) 9 (x 9) + 9 (x ) (x + 9 ) (x 7 ) 7 q (x ) 9 + r (x ) + 7 s (x + ) 9 + t (x ) 6 (x ) + (x ) + 7 (x + ) (x ) 6 a [x + x] + b [x x] 7 c [x x] + d [x + 6x] + [(x + ) ] + [(x ) ] 7 [(x ) ] + [(x + ) 9] + (x + ) + (x ) (x ) (x + ) e [x + x] f [x 0x] + g [x + x] h [x x] + [(x + ) ] [(x ) ] + [(x + ) ] [(x ) 9 ] + (x + ) (x ) + 6 (x + ) (x ) 7 i [x 8x] + 8 j [x x] k [x + 8x] + 70 l [x + x] + [(x ) 6] + 8 [(x ) [(x + ) 6] + 70 [(x + ) 6 (x ) (x ) 7 (x + ) 0 (x + ) 9 8 m [x + x] 7 n [x x] o [x + x] + p [x + x] [(x + ) 9 6 [(x ) ] [(x + ) ] + 9 ) 9 6 (x + ) 7 (x ) + (x + ) + (x + ) 7 6 a (y ) + 0 b (p + ) 0 c (x ) d (r + ) (y ) (p + ) (x ) (r + ) 8 y ± p + ± x ± r + ± 8 ± y ± p ± x ± r ± e (x ) f (a 6) g (m ) h (t 7 ) (x ) (a 6) (m ) (t 7 ) x ± ± a 6 ± ± 6 m ± t 7 ± x ± a 6 ± 6 m ( ± ) t (7 ± )
15 C ALGEBRA Answers - Worksheet F page i (u + 7 ) 9 j y y + 0 k p + 6p l x + 6x 9 (u + 7 ) ) + 0 (p + ) 9 (x + ) 9 9 u + 7 ± (y ) (p + ) (x + ) 7 u 7 ± y ± ± p + ± ± x + ± 7 ± 6 u or y ± p ± x ± 6 m m m n x 8x o t + t p a 7 a + 0 (m ) x 7x (t + 6 ) (a 7 6 ) (m ) (x 7 ) (t + 6 ) (a 7 6 ) 7 6 m ± (x 7 ) 0 t + ± a m ( ± ) x 7 t ( ± ) a (7 ± 7 ) 6 a y (x ) + 7 b y (x + ) c y (x ) 9 + y (x ) + 6 y (x + ) y (x ) 8 y 6 at x, minimum y at x, minimum y 8 at x, minimum d y (x + ) + e y [x x] + f y (x + ) 9 y (x + ) + 0 y [(x ) ] + y (x + ) 7 y 0 at x, minimum y (x ) + 8 y 7 at x, minimum y 8 at x, maximum g y [x + x] + h y [x x] i y [x + x] + 7 y [(x + ) ] + y [(x ) ] y [(x + ) y (x + ) y (x ) + y (x + ) + y at x, minimum y at x, maximum y at x, maximum j y [x x] + 9 k y [x + x] 8 l y [x + x] + 7 y [(x ) 9 ] + 9 y [(x + ) y [(x + ) ] + 7 y (x ) y (x + ) y (x + ) + y 0 at x, minimum y at x, minimum y at x, maximum a y (x ) + b y (x + ) c y (x ) + y (x ) y (x + ) y (x ) + minimum (, ) minimum (, ) minimum (, ) y y y (0, ) O x (0, ) (, ) O x (0, ) O x (, ) (, )
16 C ALGEBRA Answers - Worksheet F page d y (x + ) e y (x + ) + f y [x x] + 8 y (x + ) + y (x + ) y [(x ) ] + 8 minimum (, ) minimum (, 0) y (x ) + 9 maximum (, 9) y y y (0, 0) (0, 8) (, ) (0, ) O x (, 0) O x O x g y [x 8x] 7 h y [x + x] 7 i y (x ) y [(x ) 6] 7 y [(x + ) ] 7 y (x ) 9 y (x ) + 9 y (x + ) minimum (, 9 ) maximum (, 9) maximum (, ) y y y (, 9) (, ) O x (0, ) O x (0, 7) (0, 7) O x (, 9 ) j y (x + ) 9 + k y [x + x] + l y [x x] y (x + ) + y [(x + ) ] + y [(x ) ] minimum (, ) y (x + ) y (x ) 7 minimum (, ) maximum (, 7) y y y (0, ) (0, ) O (, 7) x (, ) (0, ) O x O x (, ) m y [x + x] + n y [x x] + o y [x x] + y [(x + ) 6 y [(x ) y [(x ) ] + 9 y (x + ) y (x ) y (x ) + maximum (, 9 ) 8 minimum (, 0) minimum (, ) (, 9 8 ) y y y (0, ) (0, ) (0, ) (, 9) (, ) O x O (, 0) x O x 6 a (x ) x + kx 0 (x ) (x + k) k 0 b x (x + k) k + x + k ± x k ± k + k +
17 C ALGEBRA Answers - Worksheet G ax + bx + c 0 x + b x + c 0 a a (x + ba ) (x + ba ) b a b a b x + ± b a x b + c 0 a c b a ac a ± b ac a a ± ac a b ac a b± b ac a a x x ± 6 ± b t t 8± ± c y 0 ± 00 6 y 0 ± 6 d r r ± + 8 ± x ± t ± y 7 or r ± e a a 8 ± 8 ± 0 f m m 0 g x m ± + 0 ± 08 h u 6± 6 x ( ± ) u 6± ( ± ) a 9 ± m ( ± ) u i y ± + 0 j x x 0 k p y ( ± ) x ± x ± p 6 7± 9 6 l t t 0 ( 7 ± 7 ) ± t t ± 6 7 x or t 7 ± 7 m r + r 9 0 n 6u + u 0 o y 8y 0 p x 8x 0 r r ± ± 8 u u ± 6+ ± 0 y y 8 ± ± 9 6 x 8± x 8 ± 8 r 7 ± 8 u 6 ( ± 0 ) y ± 9 x ± x 8x + 0 x 8± 6 8 ± 0 ± 0 ( 0, 0) and ( + 0, 0)
18 C ALGEBRA Answers - Worksheet G page a b ac > 0 b b ac 0 c b ac < 0 a b c d x x x x 6 a b ac b b ac c b ac d b ac real and distinct not real not real real and distinct e b ac 0 f b ac g b ac h b ac 7 real and equal real and distinct real and distinct not real i b ac j b ac k b ac 0 l b ac real and distinct not real real and equal not real m b ac 7 n b ac o 9 b ac p 6 b ac 7 not real real and distinct real and distinct not real 7 equal roots b ac 0 p 0 p 8 repeated root b ac 0 q + q 0 q(q + ) 0 q 0 q 9 x + rx x + 0 has equal roots b ac 0 (r ) 6 0 r r 0 (r + )(r 6) 0 r or 6
19 C ALGEBRA Answers - Worksheet H a x(0 x x ) a AB (6 + ) + (k ) 6 + k k + x( + x)( x) k k + 6 b x( + x)( x) 0 b k k x, 0 or k k 0 (k + )(k 7) 0 k or 7 a x x a y [x + x] + x x 0 [(x + ) 6 (x + )(x ) 0 (x + ) x or turning point is (, 9 8 b 9 ( x) x( x) b x x 0 x 9x + 0 x + x 0 (x )(x ) 0 (x )(x + ) 0, x or x or y y x x (0, ) (, 0) O (, 0) x x ± 0+ 8 ± 98 ± 7 6 a y [x x] + k [(x ) 9 ] + k (x ) 7 + k x-coordinate of P or b y-coord of P k 7 7 k curve is y x 9x + coordinates of Q are (0, ) 7 y 0y equal roots b ac 0 (y )(y 8) 0 k( k) 0 y x or 8 k k + 0 x or (k )(k ) 0 k or 9 a f(x) [x x] + 0 a x ± 6 [(x ) ] + ( ± ) 6 (x ) + 7 b x(x ) (x + ) b turning point is (, 7) x x 6 0 c (x ) + 7 x ± 6+ (x ) ± 0 x ± ± 0
20 C ALGEBRA Answers - Worksheet H page a (x k) k a x 6x 0 (x k) k 6± 6+ 6 x x k ± k 6 ± x k ± k 6 b y(y + y ) 0 b k y(y )(y + ) 0 x 6 ± 6 6 y, 0 or 6 ± 0 6 ± a x 0 y p a f(x) [x x] + y 0 x or p [(x ) 9 ] + y (x ) + y (x + )(x p) A, B, C b minimum value of f(x) (, 0) O (p, 0) x (0, p) b x 0 y q y 0 x q [ q > 0] y (0, q ) y (x + q) O ( q, 0) x a x ( x ) t 6 a (k ) b let t x t + t 6 0 (k ) + (t )(t + ) 0 b x kx + k 0 t or discriminant b ac but x t x 8 or 7 8 k (k ) k 8k + 0 using a (k ) + for all real k, (k ) 0 discriminant > 0 real and distinct roots for all real k 7 a (x + x )(x x ) x (x x ) + x(x x ) (x x ) x x x + x 6x 8x x + 9x + x x x + x + b (x + x )(x x ) 0 (x + )(x )(x + )(x ) 0 x,, or
21 C ALGEBRA Answers - Worksheet I a x x + b x 6 x c x + 6 x x x x x, y x, y x, y d subtracting e x + y + 0 f 6x + 6y y + 0 x y + 0 x 6y 0 y subtracting adding x 7, y 7y + 0 x 7 0 y x x, y x, y a x + x b x + x + x c x x + x 7 x x 6 0 x x 0 x + x 6 0 (x + )(x ) 0 (x + )(x ) 0 (x )(x + ) 0 x or x or x or (, 0) and (, ) (, ) and (, 7) (, ) and (, ) a subtracting b adding c y x x x 0 x 7x + 0 sub (x + )(x ) 0 (x )(x ) 0 x + (x ) x or x or x x 0 x, y x, y 7 x(x ) 0 or x, y 7 or x, y 6 x 0 or x 0, y or x, y d y x + 0 e y x f y x sub. sub. sub. x + x(x + 0) + 0 x x( x) ( x) 7 x x ( x) 0 x + x + 0 x x + x 0 (x + )(x + ) 0 x ± (x )(x + ) 0 x or x, y x or x, y or x, y x, y or x, y 8 or x, y g y x h x y i y x sub. sub. sub. x + x( x) + ( x) (y) y y 0 x + x( x) x x 0 y y 0 x 6x (x + )(x ) 0 y(y ) 0 (x )(x ) 0 x or y 0 or x or x, y x 0, y 0 x, y 0 or x, y or x 8, y or x, y
22 C ALGEBRA Answers - Worksheet I page j y x k y x 7 l y x sub. sub. sub. x + (x ) (x ) 8 x x(x 7) + (x 7) x x + ( x) 0 x 7x x 7x + 0 x 7x + 0 (x )(x ) 0 (x )(x ) 0 (x )(x ) 0 x or x or x or x, y x, y x, y or x, y 7 or x, y or x, y m x y + 0 n y x o x y 7 sub. sub. sub. (y+0) y(y+0)+y 6 x + x ( x ) 6 (y 7) +(y 7)+y 0 y + 0y + 0 x x + 0 y 9y (y + 6)(y + ) 0 (x )(x ) 0 (y )(y ) 0 y 6 or x or y or x, y 6 x, y x, y or x, y or x, y or x, y a subtracting b y x c y 7 x + y + 0 sub. sub. y + y + y 0 x(x ) 6 x (7 x) + 0 y + y 0 x x 6 0 x(7 x) + x 0 (y )(y + ) 0 (x + )(x 6) 0 8x 0x + 0 y or x or 6 (x )(x ) 0 x, y x, y 6 x or or x, y or x 6, y x, y or x, y x x x + 6 x ( ) y x y x x 0 ( ) x ( ) + y x 6 + y (x + )(x ) 0 6x 6 y x or 6x 6 x P and Q are the points (, 6) and (, ) x PQ ( + ) + ( 6) x, y PQ 7 AB A + B 6 9 A and B integers AB 6 () and A + B 9 () () A B 9 sub. () B(B 9) 6 B 9B 0 (B + )(B ) 0 B or B integer B A, B
23 C ALGEBRA Answers - Worksheet J a x < 6 b x c x > 8 d x 6 x < x 7 x > x e x f x < g 9x h x < x x < x 6 x < i x < j x 0 k < x l x x x > x 0 a y > 7 b p c 6 < x p x > d a e < u f b 9 a u > b 9 g x < 8 h y i 0 p x < 6 p j r > 6 k 6t t l 6 + x x r > 8 7 7t 6x 8 t x m 7y + 6y + < 0 n 0 8x > 6x o u u + < 9 y < > x 7u < x < u < 7 a (x )(x ) < 0 b (x + )(x ) 0 c (x + )(x + ) < 0 d x + x 8 0 (x + )(x ) 0 < x < x < x < x e (x )(x ) > 0 f x + x > 0 g (x + 7)(x + ) 0 h x 9x < 0 (x + 6)(x ) > 0 (x + )(x ) < x < or x > x < 6 or x > x 7 or x < x < i x + x 6 0 j (x + 6)(x + ) > 0 k x 7x 0 < 0 l x 0x (x + 9)(x 7) 0 (x + )(x 0) < 0 (x 7)(x ) x 9 or x 7 x < 6 or x > < x < 0 x 7 or x
24 C ALGEBRA Answers - Worksheet J page a (x )(x ) 0 b (r + )(r ) < 0 c p + p 0 (p )(p + ) 0 x < r < p d (y )(y + ) > 0 e (m + )(m + ) < 0 f x 9x (x )(x ) 0 y < or y > < m < x or x g a 8a + < 0 h x + x 7 x i y + 9y > y 0 (a )(a ) < 0 x + 6x 7 0 y + 7y + 0 > 0 (x + 7)(x ) 0 (y + )(y + ) > 0 7 < a < 7 x y < or y > j x + x > x + 6 k u 6u < u l t + t 6t x + x 6 > 0 u u + > 0 t 8t 0 (x + )(x ) < 0 (u )(u ) > 0 (t + )(t ) 0 < x < u < or u > t m y y + y n p + p o 6 + x < 6 x x y 6y + 0 p + p 0 x + 9x + 0 < 0 (y )(y ) 0 (p + 7)(p ) 0 (x + )(x + ) < 0 7 y p 7 or p < x <
25 C ALGEBRA Answers - Worksheet J page a for critical values b for critical values c for critical values d for critical values ± + 6± 6 6 6± 6+ ± 6 x x x x ± x x 6 ± x 6 ± ± x x ± x ± x ± x ± < x < + x < or x > + < x < + x or x + 6 a equal roots b real and distinct roots b ac 0 b ac > 0 6 k 0 k > 0 k 9 > k k < c no real roots d real roots b ac < 0 b ac 0 9 k < 0 k < k (k + )(k ) 0 k > 9 k or k e equal roots f no real roots b ac 0 b ac < 0 + k 0 k + k < 0 k k(k + ) < 0 0 < k < 0 g real and distinct roots h equal roots b ac > 0 b ac 0 (k ) > 0 k 8k 0 > k k(k 8) 0 k < k 0 or 8 i no real roots j real roots b ac < 0 b ac 0 k (k ) < 0 (k ) 6 0 k 8k + < 0 k k 0 (k )(k 6) < 0 6 (k + )(k 7) 0 7 < k < 6 k or k 7
26 C ALGEBRA Answers - Worksheet K a > y n n < 0 y < 8 (n + )(n ) < 0 b (x )(x 6) 0 6 < n < x or x 6 n integer n, 0,,, a (x + 8). x 9x 6x + < x 8 0.x 9x x + < 0 x 6 (9x )(x ) < 0 b x(x + 8) 80 9 x + 8x (x + 8)(x 0) x 8 or x 0 but x > 0 (width > 0) and x 6 0 x 6 x y t t t t sub. y(y + 8) 0 t + t 0 y + 8y 0 0 (t + )(t ) 0 (y + 0)(y ) 0 0 t or t 0 y x + y y y y + 8 max value of (x + y) () a x + x kx let height be h h (r ) r real and distinct roots but h b ac > 0 h ( k) 6 > 0 (r ) r 76 k + k 6 > 0 r r 70 0 k k 60 > 0 (r + 7)(r 0) 0 b (k + 6)(k 0) > r 0 k < 6 or k > 0 maximum value of r 0
27 C ALGEBRA Answers - Worksheet L a x a (x ) 9 + x (x ) + x b y b y 0 7 y 7 y (, ) O x 9 a ( ) 9 7 x + 9 x b x 6 9 x 9 x x 7 6 a x 0 ± 00 0 ± 8 0 ± 6 a x 9 9x 9x x (x 7) 0 ± x 7 b b + (y + ) y(y + ) y + y 0 + (y + )(y ) 0 x < or x > + y or 7 y x + 8 a x 6 sub. x 6 x x(x + ) + (x + ) 7 0 b ( + )( + ) 9 x ( ) x ± + x, y or x, y a, b 9 a let A be (, ) 0 a x 7 AP (k ) + (k ) x 7 AP < (k ) + (k ) < 9 b x + x < k 6k + < 0 x + x < 0 b (k )(k ) < 0 (x )(x + ) < 0 < k < < x <
28 C ALGEBRA Answers - Worksheet L page a f(x) [x x] + a [(x ) ] + (x ) b (, ) b 6 x 6x c (x ) 0 x x ± x ± x ± 6 x no real roots a AM AC + b ac < 0 BM AB AM k (k + 6) < 0 ( + ) ( + ) k k 6 < ( ) (k + )(k ) < 0 BM < k < b AC BM ( + ) ( + ) 6 + ( ) y + 7 ( ) y + 6 LHS x (x x 9) x(x x 9) y + y (x x 9) y x x 9x x + x + 9x + 6x 9x 7 x x 7 A, B and C 7 7 a x + x + k 0 (x + ) + k 0 x + ± k x ± k b real roots only if k 0 k c k x ± 8 x ±
29 C ALGEBRA Answers - Worksheet M x + x + a 8 6 x + x 0 0 b x + x x x + 9 (x + )(x ) 0 x + x a (, 0) 0 8 p + q () x 98 x (, 0) p + q () x + 98 () () p p x sub. q x + b x-coord x y (, ) a real and distinct roots 6 ( ) x y b ac > 0 x y () 6k k > 0 k k > 0 ( ) x y + k(k ) > 0 8x y + () b () and () y x + 8x x 0 x, y k < 0 or k > 7 a LHS (x 7 ) a (y + )(y ) < 0 (x 7 ) 7 a, b b x 7 < y < b x( x) (x ) x + x 0 x ± + 8 x ± ± 9 x + ( ) x 0 a t t t x + x t + t 0 x x + 0 t ± + 0 ± (x )(x ) 0 t ( ± ) x or b (x + )(x ) 0 x [no solutions] or x ± ±
30 C ALGEBRA Answers - Worksheet M page x + x 0 a x + ( x ) 9y (x + 7)(x ) 0 b 9y 0y (9y )(y ) 0 x 7 or x y x, 9 x, 0 a 7 y x + 7 b 6 < 7 < 8 sub. x ( x + 7 ) < < 9 8x (x + 7) 6 0 n 8 x x 8 0 (x + )(x ) 0 x or x, y or x, y 8 a a x + ( ) y 0 x + y 6 y + b ( ) 6 8 x + 7 b ( ) x z ( 6) x z x y 7 z (y 7) z y 6 7 a (x k) k k 0 8 a x k ± k y y + k y y x k ± k + k y + y 0 b real roots k + k 0 b (y )(y + ) 0 k(k + ) 0 y, x y, 0 k or k 0
31 C ALGEBRA Answers - Worksheet N a ( ) 9 or x 6 y b x 7 sub. (6 y) y(6 y) + y 6 x 7 y 8y + 0 x 9 (y )(y ) 0 y or x, y or x 7, y a 8 + a f(x) (x ) (x ) + b a, b and c + b turning point is (, ) y (0, ) O y f(x) (, ) x a S.A πr + πrh πr + πr 6 8x x x S.A 8π πr + πr 8π x( x ) 0 r + r 6 x 0 x b (r + 6)(r ) 0 r + r 6 0 x ( ) 6 r 6 maximum value of r 7 a x 8 a t t b ( ) y + ( ) y b t t y + y (t )(t ) 0 y t, x t, 9 9 x + kx + + k 0 0 a ( ) x (x + k) k + + k 0 6x x + k ± k x x k ± k b ( ) y real k k 0 y + k < 0 y no real roots
32 C ALGEBRA Answers - Worksheet N page x 7x < x x + x x < 0 + ( + ) ( )( ) (x + )(x ) < 0 ( )( ) + < x < + ( ) + 9 a (y )(y + 9) 0 a i x ( ) x x ( x ) y 9 y or ii x x y b equal roots b let y x y 9( y) + 0 b ac 0 y 9y + 0 k 6 0 (y )(y ) 0 k ± 8 y x or x or x y + 6 a (x + a) a + b 0 sub. (x + a) a b (y + ) + y(y + ) + y 9 x + a ± a b a b a± a b y + y 0 x a ± (y )(y + ) 0 x ± a b a b y or b for repeated root, a b 0 (, ) and (, ) b a 7 a f(x) [x 6x] a + [(x ) 9] (x ) + b y (6 ) ( ) real x (x ) 0 y ± ( ) (x ) + y ± ( ) f(x) y ± ( 0 ) b x x + 9 < 9 y 0 or + 0 x 6x + < 0 (x )(x ) < 0 < x <
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