Mathematics Higher Tier, June /1H (Paper 1, non calculator)

Transcription

1 Link to past paper on AQA website: The question paper associated with these worked answers is available to download for free from the AQA website. You can navigate around the website in a few ways but one way is to go to QUALIFICATIONS, GCSE, MATHS, MATHEMATICS, MATHEMATICS A (LINEAR), KEY MATERIALS. These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton Tuition for further details. Question 1 substitute the values in for A and B = = = -2 Question 2 a) probabilities must always add up to 1 prob (green) = 1 prob (all other colours) 1 ( ) = = 0.2 b) Yes, Jack is correct number of reds = 0.6 x 100 = Page 1

2 Question 3 we need to work out the area of the rectangle and the area of the trapezium and add the two together Area of rectangle 6 x 12 = 72 cm 2 Area of trapezium Given in formulae sheet ½(a + b)h a = 12, b = 8, h = 3 area = ½ x (12 + 8) x 3 = ½x 20 x 3 = 10 x 3 = 30 cm 2 total area = = 102 cm 2 Question 4 a) n n + 5 x 2 2(n + 5) b) = 4 4 x 2 = 8 2 x (-1 + 5) = 8 output is Page 2

3 Question 5 a) we know the answer will use the digits is unchanged but 1.37 is 100 times smaller than 137 so the answer will be 100 times smaller too 46 x 1.37 = also from estimating 46 x x 1 so is about right b) we know from the question that = 137 the numerator is unchanged but the denominator is 10 times bigger which means that the answer will be 10 times smaller = 13.7 also from estimating = = 12 so 13.7 is about right c) we know 46 x 137 = 6302 so replace the denominator with this = 10 Question 6 a) multiply both sides by w 28 = 7w 7w = 28 divide both sides by 7 w = 4 b) subtract 3x from both sides 4x 2 = 8 add 2 to both sides 4x = 10 divide both sides by 4 x = 2.5 c) multiply both sides by 4 3y + 11 = 8 subtract 11 from both sides 3y = -3 divide both sides by 3 y = Page 3

4 Question 7 a) angles on a straight line add up to 180⁰ x + 2x = x = 180 subtract 105 from both sides 3x = 75 divide both sides by 3 x = 25⁰ b) y = 2x since alternate angles are equal y = 2 x 25 = 50⁰ Question 8 a) when you divide fractions you change the to a x and flip the second fraction over x cancel down numbers on the top with numbers on the bottom x = = 1 b) when you subtract fractions you must have a common denominator (number on the bottom). There is no need to convert the mixed fractions to top heavy fractions. The common denominator here could be 15 4 = 4 1 = = Page 4

5 Question 9 Question: How many hours homework did you do in total last week (Monday to Sunday)? Give your answer by ticking the most appropriate box. Only tick one box. 0 hours More than 0 but less than 1 hour More than 1 but less than 2 hours More than 2 but less than 3 hours More than 3 but less than 4 hours More than 4 but less than 5 hours 5 hours or more You must make sure that there are no gaps and also no overlaps so that it is clear where the pupil should put their tick. Question 10 a) for all polygons the sum of the exterior angles will be 360⁰ for all regular polygons the sum of the exterior angles will be 360 and the exterior angles will all be equal for a regular octagon the exterior angle will be = 45⁰ b) the interior angle is 168⁰. angles on a straight line add up to 180⁰ so the exterior angle will be = 12⁰ 360 number of sides = 12 so number of sides = = Page 5

6 Question = is the discount, 140 was the original price. To find the percentage we put 28 out of 140 as a fraction and then to convert to a percentage we multiply by 100 x 100 = x cancel down as 10 goes into the top and the bottom x cancel down as 14 goes into the top and the bottom x cancel down as 2 goes into the top and the bottom x = = 20% Page 6

7 Question 12 a) x y 7 note note Note 1: (-1) 2 (4 x -1) + 2 = = = 7 Note 2: (2) 2 (4 x 2) + 2 = = -2 b) c) the equation x 2 4x + 2 = 0 is just the same as y= 0 so if we see where our curve crosses the line y = 0 we will have the solutions y = 0 is just the x axis x = 0.6 or x = Page 7

8 Question 13 a) to determine the smallest number the most important part is the power of x 10-3 is the smallest number b) to determine the largest number the most important part is the power of x 10 6 is the largest number c) the power of 10 is -1 so we need to move the decimal point one place to the left 0.18 d) 4.5 x 10 5 x 10-2 = 4.5 x = 4.5 x 10 3 Question 14 Similar shapes are exactly the same shape but they may not necessarily be the same size or the same way around a) the angles in both shapes will be exactly the same so angle EFD will be the same as BCA = 37⁰ angles in a triangle add up to 180⁰ = 65⁰ = 115⁰ angle DEF = 115⁰ b) we first find two sides that correspond, these are BC and EF, 8 and 20. We are going from the smaller triangle to the larger one so we want the bigger number on top. Our scale factor is = DF = x 15 = = 37.5 cm Page 8

9 Question 15 a) to calculate a four point moving average you add all the four points up and then divide by = = b) we need to work backwards ( March 09) 4 = multiply both sides by March 09 = March 09 = subtract from both sides March 09 = Page 9

10 Question 16 a) (x + 5)(x 2) + 3(x + 1) = expanding the brackets x 2 + 5x 2x x + 3 grouping terms x 2 + 6x 7 b) factorising means putting into brackets this is a quadratic so we need two brackets find two numbers that multiply to give -7 but combine to give +6 these two numbers are +7 and -1 (x + 7)(x 1) c) we know from part a) that the expression is the same as x 2 + 6x 7 so we can just set this = 0 x 2 + 6x 7 = 0 from part b) we now know that (x + 7)(x 1) = 0 If two brackets multiply to give 0 then one of the brackets must equal 0 If x + 7 = 0, then x = -7 If x -1 = 0, then x = 1 x = -7 or x = 1 Question 17 a) first look for a number that goes into both terms (2) then look for a letter that goes into both terms (x) 2x(3x 5y) b) 5 3 x a 4 x 3 x b 3 = 125a 12 b 3 alternatively 5a 4 b x 5a 4 b x 5a 4 b = 125a 12 b Page 10

11 Question 18 anything to the power of 0.5 means the square root of a negative power has the effect of flipping the number = 81 = = 1/6 2 = so we have 9 x = Question 19 this equation can be factorised to give (x -2)(x -7) we know this because it goes through the x axis at 2 and 7 If we now expand the brackets we have (x -2)(x -7) x 2 2x 7x + 14 grouping terms x 2 9x + 14 so comparing coefficients b = -9 c = Page 11

12 Question 20 The amount of money taken on Monday was (90 x 1.20) + (130 x 2.00) + (80 x 3.50) + (60 x 5) = = 948 this represents 20% of the whole week, we want 100% so we simply multiply by 5 ( = 5) 948 x 5 = x Question 21 we need to first find the volume of the ball and the volume of the cylinder then we calculate the volume of the empty space then work out the fraction that is empty volume ball this is a sphere from formulae sheet volume = πr3 volume cylinder from formulae sheet = area of cross section x height cross section is a circle, area circle = πr 2 height is 2r volume = πr2 x 2r = 2πr 3 volume of empty space = 2πr 3 - πr3 = πr3 - πr3 = πr3 to find the fraction we put volume of space over volume of cylinder ( πr3 )/( 2πr 3 ) the π and r 3 cancel out and we are left with / = Page 12

13 Question 22 multiply both sides by 3 x 8 = 2 x 3 x 3 x 8 = 2 x 3 = 6 divide both sides by 8 = square both sides w = = Page 13

14 Question 23 the answer should be in surd form so we know that this will not factorise we can use the quadratic formula or complete the square Quadratic formula from formulae sheet a = 1, b = -4, c = Completing the square (x 2) = 0 (x 2) 2 19 = 0 add 19 to both sides (x 2) 2 = 19 square root both sides x 2 = 19 add 2 to both sides Page 14

15 Question 24 angle DAE = 40⁰ by alternate segment theorem angle CAE = angle CAD + angle DAE = = 72⁰ angle ACE = 40⁰ because angles in the same segment are equal ACE is a triangle so the angles in the triangle will add up to 180⁰ x = 180 ( ) = = 68⁰ Question 25 a) the two events are independent so probability of caramel and caramel is each separate probability multiplied together x p = multiply both sides by 8 5p = 2 divide both sides by 5 p = b) prob neither pick a caramel = prob (Emilia doesn t pick a caramel) x prob (Sophie doesn t pick a caramel) (1 - ) x (1 - ) = x = If you found this paper helpful then visit where you will find plenty more. It should be noted that Chatterton Tuition is responsible for these solutions. The solutions have not been produced nor approved by AQA. In addition these solutions may not necessarily constitute the only possible solutions Page 15