. Algebra 1 Basic algebraic operations Evaluation... 1 Simplification, removing brackets, FOIL, squares... 1 Factorisation, common factor, difference of two squares, quadratic (trinomial)... 1 Solving linear equations... Simultaneous Equations... Functions, evaluating, finding values... Quadratic equations using factorisation, using the formula... Inequalities solving... 4 Changing the subject of the formula... 5 Algebraic Fractions simplifying, common denominator... 6 Algebraic fraction equations solving... 6 Indices... 7 Surds... 8 Evaluation 1. Evaluate 0 p q where p = 1 and q = 6 KU Simplification. Simplify 4( ) 5(4 + 1) Remove the brackets and collect like terms ( a b)( a 5b) KU 1 + + 4 4 KU 4. Remove the brackets and simplify your answer ( )( ) ( ) 5. Remove the brackets and simplify ( y 4) KU 6. Multiply out the brackets and simplify. ( )( 4 1) + Factorisation 7. Factorise 8. Factorise 6 9 KU 4a 9b KU 9. a) Factorise the epression 9 6 + y b) Hence simplify 9 y y 1 KU KU 10. a) Factorise a 9b 1 KU a 9b b) Hence simplify KU a + 6b 11. a) Factorise 9 1 KU 4(5 + ) b) Epress in its simplest form KU 5 9 15 0 1. Epress 9 16 in its simplest form - 1 -
1. i) Factorise completely 6 1 KU 6 ii) Epress in its simplest form. KU 9 14. Factorise 1 10 KU Solve Linear Equations 15. Solve the equation ( ) 5 1+ = 7 16. Solve the equation 5 + a = a 15 Simultaneous Equations 17. Solve algebraically, the system of equations a + 4b = 7 a 5b = 17 18. Solve the system of equations 5a + b = 9 7a b = 5 Functions 1.. f ( ) =, evaluate f (-) KU h( t) 15t t = Find h(-) KU. Given that f ( ) = + + 5 1 evaluate f ( ) 4. f ( ) = 9 6 (a) Evaluate f (-) 1 KU (b) Given that f (t) = 11, find t KU 5. The function f () is given by the formula f ( ) = 7, where is a real number. (a) Find the value of f ( ). KU (b) Find the values of a for which f ( a ) = 0. 4 6. f ( ) = find ( 1 ) f KU 7. ( ) f = a) Find f(4) 1 KU - -
b) Given that f ( ) = 7, find. 8. f ( ) = Find the eact value of f() Give your answer as a fraction with a rational denominator. KU 9. f ( ) = Find the eact value of f(1), giving your answer as a surd, in its simplest form. KU Quadratic Equations 1. Solve algebraically, the equation. Solve algebraically, the equation. Solve algebraically, the equation = 7 6y y 0 = KU 9 5 0 = 4. Solve for : 5. Solve the equation 6. Solve the equation 7 15 0 + = 5 1 0 + = p p 10 0 = where p is a real number. 7. Two functions are given below: f ( ) = + 1 g( ) = 5 + Find the values of for which f() = g() 8. Find the two roots of the equation 4 = 0 (Answer correct to 1 decimal place). 9. Solve the equation + 6 = 0 Give your answers correct to significant figures. 4 KU 5 KU - -
Inequalities 1. Solve the inequality 8 > ( + 5). Solve algebraically the inequality y < 4 ( y + ). Solve the inequality ( 6) < 4. Solve algebraically the inequality 6 < 5(1 ) 5. Solve algebraically, the inequality + 5 8 16 6. Solve the inequality 5( ) 4(1 ) where is a positive integer. 5 KU 7. An inequality, like 4 + 10 6 + + 6, can be solved by then ii) i) solving 4 + 10 6 + and solving 6 + + 6 looking carefully at the two sets of answers to decide on the correct solution to the original inequality. a) Solve + 1 5 + + 4 KU b) Write down the set of all possible solutions where is an INTEGER. 1 KU - 4 -
Changing the subject of the formula 1. Y ( v w) = Change the subject of the formula to v. 5. P 1 ( m s) = Change the subject of the formula to m KU 6. L = 8 + Change the subject of the formula to Y. Y k m 4. Change the subject of the formula to k. d = KU t 5. Q p T = + Change the subject of the formula to T. KU 6. M = R t Change the subject of the formula to R. 7. Change the subject of the formula to b. = 4 A b c 8. a) Change the subject of the formula Q = s + t, to s b) Find the value of s when Q =.5 and t =. KU 9. The frequency, F hertz of the sound you hear as you drive past a factory siren at a speed of v metres per second is given by the formula F f 1 v = s where f is the true frequency of the sound emitted by the siren and s is the speed of sound. Change the subject of the above formula to v. - 5 -
Algebraic Fractions 1. Epress as a single fraction in its simplest form 1 1, 0 KU. Epress as a single fraction in its simplest form +, 0. Epress as a single fraction in its simplest form 5, 0 or ( ) Fraction Equations 1. Solve the equation. Solve the equation + 1 = 4 + 4 + 1 = 1, where is a real number.. Solve algebraically the equation ( 5 + ) = 4 4. Solve the equation 1 + = 4 4 KU 5. Solve this equation for : 1 = 4 KU 4 6. Solve algebraically, the equation 7. Solve algebraically, the equation ( 1) + = 4 ( 1 ) m m = 5-6 -
Indices 1. Evaluate 7 KU 10 4. Epress in its simplest form ( ) y y KU. Simplify 7 a ( a + 5) KU 4. Epress y 4y 6y 5 1 in its simplest form. 5. Epress 4 y y y in its simplest form. KU 6. Epress b 1 b b in its simplest form. KU 7. Remove the brackets and simplify 8. Remove the brackets and simplify b b + b 1 1 1 1 1 a a + a KU - 7 -
Surds 1. Epress 50 as a surd in its simplest form. 1 KU. Simplify 7 KU. Simplify 48 KU 4. Epress as a surd in its simplest form. KU 5. Epress 7 + 50 as a surd in its simplest form 6. Epress + 8 as a surd in its simplest form. 7. Multiply out the brackets ( 6 ) Epress your answer as a surd in its simplest form. KU 8. f ( ) = Find the eact value of f(1), giving your answer as a surd, in its simplest form. KU 9. Epress 5 as a fraction with a rational denominator. KU 10. Simplify 4 Epress your answer as a fraction with a rational denominator 11. f ( ) = Find the eact value of f() Give your answer as a fraction with a rational denominator. KU 1. A function f is given by f ( ) = 4 Find the value of ( ) f KU - 8 -