Chapter 13 Exercise 13.1

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Chapter 1 Exercise 1.1 Q. 1. Q.. Q.. Q. 4. Q.. x + 1 + x 1 (x + 1) + 4x + (x 1) + 9x 4x + + 9x 1x 1 p p + (p ) p 1 (p + ) + p 4 p 1 p 4 p 19 y 4 4 y (y 4) 4(y ) 1 y 1 8y + 1 y + 8 1 y 1 + y 1 + 1 1 1y + y + 1 1 18y + 7 1 x + 1 7 x + 1 7 + x + 14 1 + x + 14 (x + 1) 14() + 1(x + ) 14 4x + 8 + x + _ 14 x 4 14 Q.. Q. 7. 1 1 + 4q 1 1 7q + 4(4q ) 1 (7q ) 1 1 + 4(4q ) (7q ) 1 1 + 1q 8 14q + 4 1 + q 1 q 1 x 1 x 4 (x 1) 1 + 7 1 (x 4) 1 + 7 1 1x x + + 7 1 x + 4 1 Q. 8. r 4r + Q. 9. Q.. r r 1 1 4r + r 1r 1 1() 1 (4r + ) 1 1r 4 1r 9 r 1 r 4 1 x 9 x 9 x x ( x) x 1 x 9 ( x). x x 9 + x x x 19 1 x + x. 1 x +. x x + 1 x 17 x (r) 1 Active Maths (Strands 1 ): Ch 1 Solutions 1

Q. 11. Q. 1. Q. 1. Q. 14. Q. 1 Q. 1. Q. 17. 1 x x. 1 x x x x x x 8 x + 1 1 x x + 4 x _ 8. x 1. (x + 1) (x + 1). x 8x x 1 (x + 1)x 7x 1 (x + 1)x. x (x + 4) (x + 4). x x x 8 (x + 4)x x 8 (x + 4)x x 1 + x 1 _ (x 1) + (x 1) (x 1)(x 1) 19x 7 (x 1)(x 1) 1 x + 4x 1(4x ) (x + )(4x ) (x + ) (x + )(4x ) 4x x 1 (x + )(4x ) x 17 (x + )(4x ) 1 x + 1 + 7 x 1(7 x) + (x + 1) (x + 1)(7 x) _ 7 x + x + (x + 1)(7 x) 9x + 9 (x + 1)(7 x) x 1 4x _ (4x ) (x 1) (x 1)(4x ) 1x 9 x + (x 1)(4x ) x 4 (x 1)(4x ) Q. 18. Q. 19. Q.. Q. 1. Q.. x + x + (x + ) + ( x) ( x)(x + ) x + 1 + x ( x)(x + ) 4x + ( x)(x + ) x x (x ) (x ) (x )(x ) x 4x + (x )(x ) x (x )(x ) 11 x + x 11x + x 4 (x )( x) 18 9x (x )( x) 9( x) (x )( x) 9 x OR 11 x + x 11 x x 11 x 9 x x 4 1 4 x x 4 + 1 x 4 4 x 4 7 x + x 7 x x x 11( x) + (x ) (x )( x) Active Maths (Strands 1 ): Ch 1 Solutions

Q.. (i) 1 x x + 4 1 4 + 4 1 8 Q. 4. (i) 4 8 8 1 8 1 x x + 4 1(x + 4) (x) x. (x + 4) x + 4 x x(x + 4) x + 4 x(x + 4) (iii) Let x x + 4 x(x + 4) + 4 (() + 4) + 4 (4 + 4) 1 1 8 Let x (iv) The answers to parts (i) and (iii) are the same. x 1 + Let x x + 4 ( ) 1 + ( ) + 4 1 + + 4 11 ( ) (11) 18 4 x 1 + x + 4 _ (x + 4) + (x 1) (x 1)(x + 4) x + 1 + 4x (x 1)(x + 4) x + (x 1)(x + 4) (x + 1) (x 1)(x + 4) (iii) Let x (x + 1) (x 1)(x + 4) ( + 1) (. 1)(. + 4) 4 11 4 Q.. 4 (i) x + 8 x Let x 1 4 x + 8 x 4 ( 1_ ) + 8 1_ 4 1_ + 8 1_ 4 1_ _ 4_ 8 1 1 1 4 1 1 + 8 1 8 1 1 4 1 4 4 x + 8 x 4(x ) + 8(x ) (x )(x ) 4x 1 + 4x 1 (x )(x ) 44x 8 (x )(x ) Active Maths (Strands 1 ): Ch 1 Solutions

Q.. (i) (iii) Let x 1 44 ( 1_ ) 8 (( 1_ ) ) ( 1_ ) 8 ( 1_ )( 1_ ) + 1_ _ 1 _ 4 1 4 4 x + 1 + x + 1_ + 1 + 1_ + + + + 4 x + 1 + x + (x + ) + (x + 1) (x + 1)(x + ) x + + x + (x + 1)(x + ) 1x + 1 (x + 1)(x + ) (iii) Let x 1 1 ( 1_ ) + 1 ( 1_ + 1 )( 1_ + ) 1 + 1 _ 1 1 4 4 1 4 4 4 1 Exercise 1. Q. 1. Q.. Q.. Q. 4. Q.. Q.. Q. 7. Q. 8. Q. 9. Q.. Q. 11. Q. 1. Q. 1. Q. 14. Q. 1. Q. 1. (x + 1)(x ) (x ) (x + 1) (x + 4)(x + ) (x + 4)(x + ) x + (x + ) (x + 4) x + 4 x + x + (x + ) (x + ) 4x + (x + 1) (x + 1) x + 1 x 1 x 1 (x + 1) 18y 4 y x 1 x 4 x x x 4x 4 x 4 (x 1) (x 1) 1 (x + ) x + (9y ) (y 1) 9y y 1 (x 4)(x + 4) (x 4) (x + 4) x + 4 x (x ) x (x ) (x 8)(x + 8) (x 4) (x 4)(x + 8) (x 4) (x + 8) 4x + 1 or 4(x + 4) y 1 1 y 1(1 y) 1 (1 y) x + x + 8 (x + 4)(x + ) x + 4 (x + 4) x + 4 8p q pq 8. p. p. p. q. q. p. q. q a bc ac 4p a a b c a c c ab c x x (x + )(x ) x x + x + 18p q r 18. p. p. p. q. q. r. r p 4 q r p. p. p. p. q. q. r 18. r p 4 Active Maths (Strands 1 ): Ch 1 Solutions

Q. 17. Q. 18. Q. 19. a + 4a (a + a ) a + a 1x + 18x x x _ x(x + x ) x x + x 4x + 7x (x + )(4x 1) x + (x + ) 4x 1 Q.. (i) x x + 1 (x 7)(x ) (x )(x x + 1) x x + 1x x + x x 1x + 71x (iii) x 1x + 71x x (x )(x 7)(x ) (x ) (x )(x 7) Q.. 9x x (x + 1)(x ) x + 1 (x + 1) x Q. 1. (i) 1x 1 1(x 1) 1(x 1)(x + 1) (x + 1)(x + 1) (iii) 1x 1 x + x + 1 1(x 1)(x + 1) (x + 1)(x + 1) 1(x 1) x + 1 Q.. (i) x + 9x + (x + )(x + 4) (x + 1)(x + 9x + ) x + 9x + x + x + 9x + x + x + 9x + _ (iii) x + x + 9x + (x + 4) _ (x + 1)(x + 9x + ) (x + 4) (x + 1). (x + ). (x + 4) (x + 4) (x + 1)(x + ) (iv) Let x Solution ( + 1)( + ) 7 1 _ x + x + 9x + x + 4 + ( ) + 9() + + 4 8 + 4 + 8 + 1 1 (iv) Let x 4: (4 )(4 7) ( 1)( ) x 4: 4 1(4 ) + 71(4) 4 4 4 + 84 Q. 4. (i) x 17x (x + 1)(x ) 1(x 17x ) x(x 17x ) x 17x x + 17x + x Q.. Q.. Q. 7. Q. 8. x + x 11x _ (1 x)(x + 1)(x ) (iii) x + 1 (1 x)(x ) (iv) ( 1) + ( 1) 11( 1) ( 1) + 1 8 14 (1 + 1)( 1 ) 14 (x + 1)(x + ) (x + )(x 4) x + 1 x 4 x + 9x + 4 x + 11x + (x + 1)(x + 4) (x + 1)(x + ) x + 4 x + x 8x 1 x x 4(x x ) (x x ) 4 ax + ay cx cy ax + ay + cx + cy a(x + y) c(x + y) a(x + y) + c(x + y) (x + y)(a c) (x + y)(a + c) a c a + c Active Maths (Strands 1 ): Ch 1 Solutions

Q. 9. Q.. (x x 1) 4(x + 1x + 1) (x + 4)(x ) (x + 4)(x + ) x (x + ) a b b a (a b) (b a)(b + a) 1. (b a) (b a)(b + a) 1 b + a Exercise 1. Q. 1. (i) x + 7x + 1 (x + 4)(x + ) x + 7x + 1 (x + 4)(x + ) x + (x + ) x + 4 (iii) x + 4 x + x + 7x + 1 x + x 4x + 1 Answer: x + 4 4x + 1 Q.. (i) x + 11x + (x + 1)(x + ) x + 11x + (x + 1)(x + ) x + (x + ) x + 1 (iii) x + 1 x + x + 11x + x + x x + x + (iii) x + 7 x 9 x x x 9x 7x x Answer: x + 7 7x Q. 4. (i) x 7x (x + 4)(x ) Q.. Q.. x 7x (x + 4)(x ) x (x ) x + 4 (iii) x + 4 x x 7x x 1x 8x x Answer: x + 4 8x 4 x 8x 8x + 1 8x 4x 4x x + 1 Answer: 8x 4 4x + 1 x 7 x + x 11x x + x Answer: x 7 1x 1 1x 1 8x Answer: x + 1 Q.. (i) x x (x 9)(x + 7) x x (x 9)(x + 7) x 9 (x 9) x + 7 Active Maths (Strands 1 ): Ch 1 Solutions

Q. 7. x + x + 1 x + x + x + 7x + x + x x + 7x x + x Answer: x + x + 1 x + x + Q. 8. x + 4x + 1 x x + x 7x x x 4x 7x 4x 8x Answer: x + 4x + 1 Q. 9. x + x + 1 x + 8x + x + x x + Answer: x + x + x x Q.. 8x x + 1x + 14x 1 1x + 4x Answer: 8x x 1 x 11 Q. 11. x + x 8 x + 9 x + 1x + x 7 x + 9x 4x + x 4x + 18x 1x 7 1x 7 Answer: x + x 8 Q. 1. x 8x x x x x + 1 Q. 1. Q. 14. x + x Answer: x 8x 4x x + 1 + 4x +1x 18x + 1 + 18x + 1 x + x + x + 1 4x + x + 8x + 4x + x 4x + 8x Active Maths (Strands 1 ): Ch 1 Solutions 4x + x x + Answer: x + x + x + 7x + 4x x + 14x + 4x + 1x 14x + x 8xx + 1x 8x + x 4x x +4x + Answer: 7x + 4x Q. 1. x x + 8 7x x 1x + 74x 48 x x 1x + 74x Answer: x x + 8 1x + 18x x 48 x 48 7

Q. 1. x 8x 8 x 1 x 9x + 8 x x 8x + x 8x + 8x 8x 8 + 8 8x 8 + 8 Answer: x 8x 8 Q. 17. x x + x 4 x + x x 4 x 4 Answer: x Q. 18. x + x 4x 4x x x x Answer: x + x Q. 19. x + x x 1 9x 19x + 9x + x x 19x + x + x 18x + + 18x + Answer: x + x Q.. (i) (x + )(x 7) x x 14 (x 1)(x x 14) x x 8x x + x + 14 x 11x x + 14 (iii) x 11x x + 14 x 7 (iv) x x 14 x 1x 11x x + 14 x x x x x + x Answer: x x 14 (x 1)(x x 14) (x 7) _ (x 1)(x + )(x 7) (x 7) (x 1)(x + ) 8x + 14 8x + 14 Q. 1. (i) (x + )(x 4) x x 1 (x 1)(x x 1) x 1x x x + x + 1 x 17x 1x + 1 (iii) x 17x 1x + 1 x + (x 1)(x x 1) (x + ) (x 1)(x + )(x 4) (x + ) (x 1)(x 4) 8 Active Maths (Strands 1 ): Ch 1 Solutions

(iv) x x 1 x 1 x 17x 1x + 1 x x 1x 1x 1x + x x + 1 x + 1 Answer: x x 1 Q.. x + x 1 x + 1 x + x x 1 x + x x x x + x 4x 1 4x 1 Answer: x + x 1 (x + 4)(x ) Q.. x x 1 x + 1 x 1x 1 x + x x 1x x x 1x x 1 1x 11 Q. 4. k + 7k k k + 9k k + k k 14k k 14k k k k + k + Answer: k + 7k Let k k + 7k + 7() k + 9k k + () + 9() () + 4 + 81 1 + 18 1 Answer: x x 1 (x 4)(x + ) Q.. (i) 4a + a 7 a + a + 9 a 4a + a + a 7 (4a + a ) 1a + a (1a + 18a) 18a 7 18a 7 Answer: a + a + 19 Active Maths (Strands 1 ): Ch 1 Solutions 9

Q.. (i) x 1 (x 1)(x + 1) 4x 7x + (4x )(x 1) (iii) (x + 1)(4x 7x + ) 4x 7x + x + 4x 7x + 4x x 4x + (iv) 4x x 4x + x 1 _ (x + 1)(4x 7x + ) (x + 1)(x 1) 4x 7x + (4x )(x 1) x 1 (x 1) 4x Q. 7. 8x 1 x 1 4x + x + 1 x 18x + x + x 1 8x 4x 4x + x 4x x x x 11 Answer: 4x + x + 1 x 1 Q. 8. 9x 1x + 1 x + 4 7x + x + x + 4 7x + x x + x x 48x 48x x + 4 Answer: 9x 1x + 1 Revision Exercises Q. 1. (a) (i) x + 1 x + x + 7 + x + 48x + 4 (x + 1) + (x + ) x + + x + 1 x + 17 x 1 + x + 7 7x 7 + x + 7 9x + 7 Active Maths (Strands 1 ): Ch 1 Solutions

(iii) x 1 (iv) 7x (b) (i) a a 1 a + b a + b x 1 x 1 (x 1) (x 1) 1x x + 1x 1 + x + (a + b) (a + b) (iii) x 4 x 4 (x ) (x )(x + ) x + (c) (i) x + x + x + 1x + x + x x x + x + Answer: x + 4a + a + 1a + 14a + 4 1a + 8a a a + 4 Answer: 4a + a + 4 (iii) c + 7 c 1c + 11c 14 1c c 1c c 14 Answer: c + 7 1c 14 (x )(x + 4) Q.. (a) (i) x (x + 4) 4(x + 1) 4 (x + 1) 7x (x 1) + (x + ) 7x 4x + + x + 1 8x + 1 Active Maths (Strands 1 ): Ch 1 Solutions 11

(iii) x + x 1 (x + )(x ) x x + (x + ) (iv) 4a b ab (v) x x x + 1 (b) (i) x 1 (iii) 4 a. a. b. b. b. a. b x 4 x + x 1 4 x 7 x ab (x )(x + 1) x (x + 1) + x + (x 1) 1(x 4) + (x + ) 1x 4x + + x + x + 4 (x 1) + (x ) (x )(x 1) 9x + x (x )(x 1) 11x 1 (x )(x 1) _ 4( x) 7(x ) (x )( x) x (x )( x) 11(. x) (x )( x) 11 x (c) (i) x + 1x + (x + 8x + 1) (x + 4)(x + 4) Q.. (a) (i) x + 1x + (x + 4)(x + 4) (x + 4) (x + 4) (iii) x + 1x + (x + 4) () + 1() + 81 x 4x 1 (x ) (x ) 1 x 9 x 4x + (x )(x + ) (x )(x 1) x + x 1 (iii) x 7x + (x )(x 1) x 1 (x ) x 1 (iv) x + x 8 (x 7)(x + 4) x + 8 (x + 4) x 7 1 Active Maths (Strands 1 ): Ch 1 Solutions

(b) (i) x + 7x 4 (x 1)(x + 4) x 1 x + 4 (x + 4) x + 7x x x + 9x x + x x 14x x 14x x x + x + Answer: x + 7x (iii) x x + 1 x + x 14x 9x + x + 4x 18x 9x 18x 1x x x + x + Answer: x x + 1 (iv) 4x + x + x 1 8x + x 8x 4x 4x + x Answer: 4x + x + 4x x x x (c) 8 x 1 x 8 x + 1 x x Let x 9 Solution 9 4 8 Using x 1 x 8 9 1 9 8 4 1 4 Same Active Maths (Strands 1 ): Ch 1 Solutions 1

Q. 4. (a) (i) x x 4x 1x + 9 4x x x + 9 x + 9 So 4x 1x + 9 x x x + 4x 4x 11x + 1x 1x + 1x x 1x 1x 1x 4x 1x x + 1x + So 1x + 1x 1x + x 4x 1 + 4x (iii) x + x + x x 19x x x x 19x x x x x So x 19x x x + x + (iv) x + x + 1 x 1 xx + 8x 1 x + x 9xx 1 Answer: x + x + 1 9x + x x x 1 x + 1 (b) (i) x 1 8 1 x x 1 + 8 x 1 11 x 1 x + x + x (x + ) (x + )(x 1) 1 x 1 14 Active Maths (Strands 1 ): Ch 1 Solutions

(iii) x 1 8 1 x + x + x + x 11 x 1 + (x + ) (x 1)(x + ) 11 x 1 + 1 x 1 1 x 1 Q.. (i) Divisor Quotient Dividend x 1 x x 1x + 4x 1 x + 1 8x + x 1 x 1 x x 4 x 4x x + 4 x x x + 4 1x 4x + 17x 1 x x 4 x 1 x 4x x + 4 x + x x x + 4 + x + x 4x + 4 + 4x + 4 (x )(x x + 4) x(x x + 4) (x x + 4) 1x x + 8x 18x + 9x 1 1x 4x + 17x 1 x + x 1 x + x 48x 8 x 48x x + 48x 48x 8 Division: x + p q x 8 Active Maths (Strands 1 ): Ch 1 Solutions 1

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