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Summer Mathematics C1 Transition Answers 1. Write in index notation a) 6 4 x 6 2 = 66 b) (53 ) 4 = 512 c) 8 7 ÷ 8 3 = 84 d) 5 2 ÷ 5 4 = 52 e) 4 3 x 4 3 = 40 f) 4 3 x 2 8 = 214 2. Find the value of a) 10 3. 1 1 1 = 0.1= 10 b) 2 5 = 32 1 4 c) 81 = 3 1 =5 d) 7 x 200 - 180 = √36 = 6 c) 75 = 5√3 5 72 + 18 = 10√2 − 6√2 + 3√2 = 7√2 7 =7 b) Rationalize and simplify as appropriate 5. Make “a” the subject of the formula in each case a) 4a + 12b = 32c b) 3d = a 10 a) 1 4. 3 = √3 3 c) 2f = 5a 2 - 11q 𝑎 + 3𝑏 = 8𝑐 9𝑑2 = 𝑎 − 10 2𝑓 + 11𝑞 = 5𝑎2 𝑎 = 8𝑐 − 3𝑏 𝑎 = 9𝑑2 + 10 √ 2𝑓+11𝑞 5 5 b) 8. 9. √5 = 5√3= 75 √5√3 √15 = 15 15 d) -2a +14 = 6ba + 14t 14 − 14𝑡 = 𝑎(6𝑏 + 2) 𝑎= =a 14−14𝑡 6𝑏+2 7−7𝑡 =3𝑏+1 Expand the following simplifying the answer where possible a) (x – 3)(x + 4) b) (2x – 7)(x + 8) c) (3x +2)(3x -2) =𝑥 2 + 𝑥 − 12 = 2𝑥 2 + 9𝑥 − 56 =9𝑥 2 − 4 e) Simplify ( 5 2 ) 2 leaving your answer in the form a + b c . = 5 + 4√5 + 4 = 9 + 4√5 7. 2 1 3 e) 49 = 343 f) = 4 8 3 2 Simplify these surds a) 6. d) 25 1 2 d) (5x + 13) 2 =25𝑥 2 + 130𝑥 + 169 Factorise each of these quadratic expressions a) x 2 + 3x + 2 b) x 2 + 20 - 9x c) x 2 – 49 = (𝑥 + 2)(𝑥 + 1) d) 49 – x 2 = (7 − 𝑥)(7 + 𝑥) = (𝑥 + 7)(𝑥 − 7) f) -5x – 6 + 6x 2 = (2𝑥 − 3)(3𝑥 − 2) = (𝑥 − 4)(𝑥 − 5) e) 2x 2 + 7x + 3 = (2𝑥 + 1)(𝑥 + 3) a) x 2 – x – 6 = 0 (𝑥 − 3)(𝑥 + 2) = 0 𝑥 = 3 𝑥 = −2 b) 3x 2 + 5x = 2 3𝑥 2 + 5𝑥 − 2 = 0 (3𝑥 − 1)(𝑥 + 2) = 0 1 𝑥 = 3 𝑥 = −2 Some quadratic expressions do not factorize. You may be asked to complete the square. Solve these quadratic equations a) Show that ( x + 3 ) 2 - 8 x 2 + 6x +1 (𝑥 + 3)2 − 8 = 𝑥 2 + 6𝑥 + 9 − 8 = 𝑥 2 + 6𝑥 + 1 b) Write x 2 + 4x – 3 in the form ( x + a ) 2 + b 𝑥 2 + 4𝑥 − 3 = (𝑥 + 2)2 − 4 − 3 = (𝑥 + 2)2 − 7 c) Hence, solve x 2 + 4x – 3 = 0 leaving your answer in surd form. (𝑥 + 2)2 − 7 = 0 𝑥 + 2 = ±√7 𝑥 = −2 ± √7 10. Solve these equations simultaneously (remember to find values for x and y) by elimination… a) 4x + y =10 2x – y = 8 by substitution… b) 4x – 5y = 4 2y = 25 – 6x 2𝑥 − (10 − 4𝑥) = 8 𝑥 = 3 𝑦 = −2 12𝑥 = 50 − 4𝑦 12 + 15𝑦 = 50 − 4𝑦 𝑦 = 2 𝑥 = 3.5 c) x + y = 11 xy = 30 𝑥(11 − 𝑥) = 30 0 = 𝑥 2 − 11𝑥 + 30 𝑥=5 𝑥=6 𝑦=6 𝑦=5 Substitution above! Elimination below. a) 4x + y =10 2x – y = 8 6𝑥 = 18 𝑥 = 3 𝑦 = −2 b) 4x – 5y = 4 2y = 25 – 6x 12𝑥 = 50 − 4𝑦 12𝑥 = 12 + 15𝑦 0 = 38 − 19𝑦 𝑦 = 2 𝑥 = 3.5 d) 3x + y = 14 x + y 2 = 28 𝑥 + (14 − 3𝑥)2 = 28 9𝑥 2 − 83𝑥 + 168 = 0 56 𝑥=3 𝑥= 9 𝑦=5 𝑦=− 14 3