PiXL Independence: Mathematics Answer Booklet KS4 HIGHER Topic - Factorising, Inequalities, Quadratics Contents: Answers 1
I. Basic Skills Check Answer the following questions. In order to improve your basic arithmetic you should attempt these without a calculator where possible. Skills Check 1 1. A coat is reduced by 1% to a price of 59.84. Calculate the original cost of the coat. 68. What is 84 of 16?. Write 58 as a product of its prime factors. x9 4. Factorise x² -6x -16. (x 8)(x + ) 5. Write 760,000,000 in standard form. 7.6 x 10 8 6. Solve: 4x + 7 1 x 5 7. Use prime factors to find the lowest common multiple of 70 and 80. 560 8. List the first 5 terms of the sequence -n-. -5, -8, -11, -14, -17 9. Find the total perimeter of the sector shown, correct to one decimal place. 15 6cm 6.1cm 15 9 10. Calculate (1.5 10 ) (4. 10 ), giving your answer in standard form correct to two significant figures..0 x10 5 Skills Check 1. The height of a student is measured to the nearest cm, if it is recorded as 1cm what is the maximum and minimum height of the student? UB= 1.5 LB=11.5 1. Calculate (6.1 10 ) (.4 10 ), giving your answer in standard form correct to two significant figures. 1.5 x10 16. Write 40 as a product of prime factors. What is the LCM of 40 and 5?
5 = 40 1 = 5 LCM = 50 4. Factorise x² -x - 80 (x-10) (x+8). 5. Write 0.000000 in standard form..0 x10-6 6. Solve: 17 4 x x 7 7. Calculate the total surface area of a cylinder with radius cm and length 11cm. Give your final answer to one decimal place. Circle area = 18π Curved surface = 66π cm Total = 6.9cm 11cm 8. Find the nth term of the sequence: -, 1, 6, 1.. n 9. Find the reciprocal of the number.6, giving your answer as a fraction. 5/18 10. The masses of a group of pupils are displayed in this table. Calculate an estimate of the mean mass. Mass (x kg) Frequency MP MPx f 40 x 50 4 45 180 50 x 60 8 55 440 60 x 70 5 65 5 70 x 80 75 5 1170/0 = 58.5 kg Skills Check 1. Find the lower bound for the perimeter of this parallelogram if the measurements shown are correct to the nearest centimetre.
16cm 10cm = 50cm LB = 9.5 and 15.5. Work out 5 19, simplifying your answer as far as possible. 1 6 4 = 1 7 1. Use prime factors to find the lowest common multiple of 11 and 84. LCM = 6 4. Vicki rolls a dice 0 times. Her scores are recorded in this table. Calculate the mode, median and mean of her scores. Score 1 4 5 6 Frequency 4 4 0 5 6 1 Mode = 5 Median = 4 Mean =.4 5. Calculate the total surface area of a cone with radius 6cm and vertical height 8cm. Give your answer as a single multiple of π. 8cm Curved surface = 60π Base = 6π 6cm Total = 96π 14 5 6. Work out ( 10 ) ( 410 ), giving your answer in standard form. 1. x10 0 7. Work out 95.05 7 4.15 8. Find the nth term of the sequence:, 1, 7, 48. n 4
9. Use prime factors to find the highest common factor of 150 and 900. 150 10. Write the recurring decimal 0.101010101. as a fraction in its simplest form x = 0.101010.. 100x = 10.101010 99x = 10 x = 10 99 II. Short Exam Questions Section 1 Factorising and Simplifying 1. Expand and simplify each of these expressions: a. (5x x 4) 15x x + 11 b. x(4x 5) 8x 10x c. 4(x 7) (x ) 4x + 8 x + 6 = x + 4 d. ( x 7)(x ) x + 9x + 14 e. ( 4x 1)(x 5) 8x + 0x x 5 = 8x + 18x 5 f. ( 5x ) 5x 0x + 9. Factorise each of these expressions by removing common factors: a. xy 5y y(x + 5) b. 1x 18x 6x (x ) c. xy 4y 8 4y(x + 1). In an exam, Robert factorises the expression 18y -6y to give the answer 9y(y -4y). Explain why he would not receive full marks. This has not FULLY been factorised. It should be 18y (y ) 4. Factorise each of these quadratic expressions using double brackets: a. x 9x 14 (x + 7)(x + ) b. x x 70 (x 7)(x + 10) c. x x 90 d. x 11x 0 (x 10)(x + 9) (x 6)(x 5) 5. Factorise each of these harder quadratic expressions: 5
a. x 11x 5 (x + 1)(x + 5) b. x 10x 7 (x 7)(x 1) c. 5x 1x 6 (5x )(x + ) 6. Factorise each of these using the difference of two squares: a. y 6 (y 6)(y + 6) b. 49y 81 (7y + 9)(7y 9) c. x 9y (x + y)(x y) 7. Factorise each of these expressions fully: a. 5y 0y 40 5(x + )(x + 4) b. y 16y y(y 16) c. 6y 6y 54y 6y(y ) 8. Use factorisation to work out Show every step in your working. (5+47) (5-47)= 100 x 6 = 600 5 47 without a calculator. 9. Anna s back garden consists of a rectangular lawn measuring 9 metres by 7 metres, surrounded by a gravel path of width x metres. Find, and simplify, an expression for the total area of the garden. x Gravel Path Path = (7 + x)(9 + x) = 6 + 14x + 18x + 4x x Lawn 7 = 4x + x + 6 9 6
Section - Solving Equations 1. Solve each of these equations. a) 4x 15 x = 4.4 y b) 4 9 y = 15 c) 5m 8 m 1 m = 7 d) 4 x x 16 x = e) (k 1) 5(k ) k = 7 f) (m ) (m 4) 10 m = 4. The perimeter of the rectangle shown opposite is 8cm. What is the value of x. (+x)+(4x-5) = 8 x 6+4x+8x-10 = 8 1x = 4 X =.5 4x 5. Solve each of these equations. a) d 1 4 d = 45 b) y 1 5y y = 1 c) m m 7 9 6 m = 18 7
x 4 x 11 4. Solve the equation 10. 11x + 4x + 1 = 10 44 15x 10 = 10 44 15x = 450 x = 0 5. Simplify each of these algebraic fractions as far as possible. a) 4 h 1 h b) 5 c 0c 1 4c c) 15y 1 5y 7 1 4 d) e) x x x 7x 10 6x 1 16 4x (x+5)(x+) 6(x+) (x+4)(x 4) x(x+4) = (x+5) 6 = (x 4) x 6. Simplify these fractional multiplications and divisions. a) y y 4 4 b) 4m 15p 5p 8m m c) x 1 x 5x 4 9 x+1 9 x +5x+4 = 9(x+1) = (x+1)(x+4) x+4 7. Write each of these expressions as a single algebraic fraction. Simplify your answers where appropriate. a) y y 4 6 7y 1 b) c) x 4 x 1 5 6 1 x x 4 6(x+4)+5(x 1) 0 x+4+x 6 (x )(x+4) = = 6x+4+5x 5 = 11x+9 0 0 x (x )(x+4) 8. You are given the equation 10 180 15. x x 4 8
Setting out each stage of your working clearly, show that this equation can be transformed into the form x 19x 8 0. 10x + 480 + 180x 540 = 15(x )(x + 4) 00x 60 = 15(x + x 1) 0x 4=x + x 1 Which gives x 19x 8 = 0 as required Section Inequalities and Simultaneous Equations 1. Solve these inequalities and represent the solutions on a number line. a) x < 4 x < 8 b) x 5 > 17 x > 11 c) (x + 5) 16 x d) 7x 5 x + x e) x + 1 < x + x < 1. Write down inequalities to describe each of these number lines. b) c) - - -1 0 1 4 x > - - -1 0 1 4 x <. Write down all the integer values of x included in the inequality 4 x. -4, -, -, -1, 0, 1, 4. Write down all the integer values of x included in the inequality 4 x. -, -1, 0, 1 5. Tickets to a fair cost 4.75 for adults and.50 for children. A coach party of 48 people arrives and their tickets cost a total of 147. Form a pair of simultaneous equations, and solve them to find the number of adults and children on this coach. 9
4.75A +.50C =147 A+ C = 48 4.75A+.5C =147.50A+.5C=10.5A = 7 Adults=1 Children=6 6. The length of a rectangle is 6cm more than its width. The area of the rectangle is 55cm. a. Form a quadratic equation to represent this information. w(w + 6) = 55 w + 6w = 55 w + 6w-55 = 0 (w + 11)(w 5) = 0 w = 11 or 5 Since length it can t be 11. W = 5 b. Solve your equation to find the dimensions of this rectangle. Width = 5, Length = 11 7. Solve these simultaneous equations algebraically: 7x 6y 9 8x 15y 54 Multiplying to get equal numbers of y gives: 5x 0y 45 16x 0y 108 51x so x Adding these gives 15 Substituting into the first equation gives 1 6y 9 6y 1 y 8. Bobby buys 8 tins of beans. Large tins cost 7p and small tins cost 49p. Altogether his beans cost 17.80. How many large tins and how many small tins did he buy? The question leads to these simultaneous equations: L S 8 7L 49S 1780 Multiplying the first equation by 49: Subtracting these gives: 49L 49S 17 7L 49S 1780 4L L 408 17 So Bobby buys 17 large and 11 small tins of beans 10
9. Solve the inequality 6x 4x 15 and show your answer on a number line. x 15 x 18 x 9 7 8 9 10 10. Solve these non-linear simultaneous equations: ANS x x 5x 1 19 x 8x 0 0 x 10x 0 x 10, x x y 19 y x 5x Solutions: x = -10 with y = 49 and x = with y = 1. 1 Section4 - Mixed Questions 1. Find and simplify expressions for the perimeter and area of this compound shape. x 6 4x 1 x 7 Perimeter = total of all six sides = x 4x 1 x 6 x 7 6 x 6 14x 14 or 14 x 1 x or 4 x x 14 Area = 4x 1 6x 7 1x 9x 4. The perimeter of the rectangle shown opposite is cm. Find the value of x and the area of the rectangle. x 11
x x 1 so x-6 4x so 6x- 4 so x 6. Length = 1cm, width = cm so area = 9cm. Ben cycles from Acton to Beeswell and back again, a distance of 60 miles in each direction. On the outward journey he averages x mph but on the return journey his average is mph less. The total journey takes 1 hours. Write this information as an equation, and then show that it can be rearranged to make 1x 15 0. x (Remember that time taken = distance ) speed 60 x 60 x Question leads to 1. Cross-multiplying gives 60x 180 60x1xx which leads to 0 1x 156x 180. Dividing by 1 gives 1x 15 0 x, as required. 4. Simplify as far as possible A 15A a) 8B 16B x x b) 6 9 7 x - c) x 5 4 ANS: A 8B 16B 9x 6 x x b. 15A 5AB 54 54 x 18 x 8 7x 5 9x 7 c. x 5x 4 x 5x 4 or 9x x 5x 4 1
5. Simplify as far as possible; a. 1k k d d 8 5 = 4d b. d h = 6. Expand and simplify the expression 5x x 4 x. 6 7d h ANS 6x 5x 15 x 4 x = x 15 x 4 x = x x 15x 8x 4x 60 = x 9x 11x 60 7. Natalie is a years old. Write down expressions in terms of a for the following people s ages: a) Joyce, who is 10 years older than Natalie a+10 b) John, who is half Natalie s age. a/ c) Gavin, who is twice Joyce s age. (a+10) a d) Steven, who is 4 years older than John. 8. The angles in a triangle are x, x and 5x. Write an equation to find the value of x. x + x + 5x = 180 9x = 180 x = 0 Write down the size of each angle in the triangle. 0 o, 60 o, 100 o 9. The four angles of a quadrilateral are 45, 105, (4x 15) and 5x. a) Form an equation, in terms of x, using this information. 45 + 105 + 4x 15 + 5x = 60 15 + 9x = 60 x = 5 b) Solve your equation and work out the size of the largest angle of the quadrilateral. Angles= 45, 105, 85, 100 so largest angle = 100 o 10. The length of a rectangle is 5cm more than its width. The area of the rectangle is 18cm. Form and solve a quadratic equation to find the width of this rectangle. Give your answer correct to two decimal places. w5 18 w w 5w18 0 5 5 41 18 5 5 7 w w (Watch minus signs here!) Width =.4cm (or -7.4cm) 1
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