maths past paper 1 answer

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expression, in terms of n, for the nth term of this sequence. 2 Show that 2 Answer ALL questions. Write your answers in the spaces provided. You must write down all the stages in your working. 3 n 3n 2 369 147 7 ام x 3 X 35 = 3 10 12 10 = 8 3 4 155 13 3n-2 (Total for Question 1 is 2 marks) 8. 15 13 35 4 (Total for Question 2 is 3 marks) P6 2 2 7 7 RA0220 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3 The diagram shows four graphs. YA Graph A YA y = -x³ Equation y = x³ X y = x² X Graph C Graph D Each of the equations in the table is the equation of one of the graphs. Complete the table. 1 y = - X Graph B у, Letter of graph B с D A YA P6 2 2 7 7 RA 0 3 2 0 X X (Total for Question 3 is 2 marks) 3 Turn over ► 4 The diagram shows four triangles. 55+45 = 100 180-100 = 80° 55° 24 × 80 10 cm Triangle A Profit 10 cm Triangle C Two of these triangles are congruent. Write down the letters of these two triangles. 5 Sean pays £10 for 24 chocolate bars. He sells all 24 chocolate bars for 50p each. Work out Sean's percentage profit. % profit 55° 0.5 = £12 = 45° = 8 cm 10 2 £12 - change original 8 cm x 100 Triangle B 80° 45° -£10 = £2 Triangle D (55° A D (Total for Question 4 is 1 mark) x 100 20% 10 cm 10 cm P6 2 2 7 7 RA04 20 and 20 (Total for Question 5 is 3 marks) % DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6 ADC is a triangle. D Angle EBC= 148° Angle ADC = 63° 63° AED and ABC are straight lines. EB is parallel to DC. E 63° 180- 148 = 32° EAB = 85° A 85 Work out the size of angle EAB. You must give a reason for each stage of your working. Angles on AEB = 63° Corresponding angles are to 180° Angles in 32 a 148° B a triangle 63 +32 = 95 180-95 = 85° straight line equal P6 2 2 7 7 RA05 20 add to add 180° (Total for Question 6 is 5 marks) 5 Turn over ► 7 The table shows information about the heights, in cm, of a group of Year 9 girls. least height median greatest height 15 899 16 45770 17 23442 18 22 6 150 cm 165 cm 170 cm are taller on This stem and leaf diagram shows information about the heights, in cm, of a group of 15 Year 9 boys. Range average = Key: 15 8 represents 158 cm Median = 168 Jeast height = 158 greatest height = 182 Compare the distribution of the heights of the girls with the distribution of the heights of the boys. The boys their Median height. is greater. The boys heights greater. 170-150 = 20 are More spread out, Range: = 182-158 = 24 P6 2 2 7 7 RA 06 20 their range. (Total for Question 7 is 3 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8 The diagram shows a prism placed on a horizontal floor. The prism has height 3 m The volume of the prism is 18 m³ The pressure on the floor due to the prism is 75 newtons/m² Work out the force exerted by the prism on the floor. area of base volume = 18 are a = 3 m area 18 3 3 0.000672 6.72 × 105 1/³=6m² 9 Write these numbers in order of size. Start with the smallest number. X 3 pressure 75 force = 75×6 2 force area force 67.2 x 10-4 6.72x10 @ -3 X height 672 x 104 6.72×10⁰ (4) 450 (Total for Question 8 is 3 marks) 0.000 672 pressure = 67-2×104 6.72×105 P 6 2 2 7 7 RA 0 7 2 0 6.72×10 force 4 (Total for Question 9 is 2 marks) area newtons 672x104 7 Turn over ► 10 Given that find a:b:c x3 8 a b 3 2-3 and 2 = 2/ b 4 9:0 2:5 615 a: bic 6:15:20 7:9 3:4 15:20 x 5 6:15:20 (Total for Question 10 is 3 marks) P6 2 2 7 7 RA 08 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 11 (a) Find the value of 81×108 (b) Find the value of 64 (c) Write 3" 92-1 MM √81 × "√ 10² 3×102 3 3 | n 34 ÷ 32(n-1) 3" = 32n-2 (n 3 (^ - (²^²-2)) (n-2n + 2) √64 as a power of 3 2-n 300 P6 2 2 7 7 RA09 20 (2) ㅎ (2) 3 2-n (2) (Total for Question 11 is 6 marks) Turn over ► 12 The table gives information about the weekly wages of 80 people. (a) Complete the cumulative frequency table. Juan says Wage (£w) 200w 250 250 w 300 300<w< 350 350 < W< 400 400w 450 450 w 500 10 Wage (£w) 40 80 200 w 250 200 w< 300 200w 350 200w 400 200 < W< 450 200 w 500 (c) Is Juan correct? You must show how you get your answer. 60% of (b) On the grid opposite, draw a cumulative frequency graph for your completed table. 50% 360 or Frequency 5 10 20 20 15 10 No. Juan is Cumulative frequency "60% of this group of people have a weekly wage of £360 or less." 5 15 80 = not 35 55 70 80 have a less 48 wage or correct. P 6 2 2 7 7 RA 0 1 0 20 (1) (2) (3) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 80 75 70 65 60 55 50 Cumulative 45 frequency 40 35 30 25 20 15 10 5 0 200 250 ← 300 ^ 350 400 Wage (£) 450 500 (Total for Question 12 is 6 marks) P 6 2 2 7 7 RA 0 1 1 2 0 550 11 Turn over ► 13 Liquid A and liquid B are mixed to make liquid C. Liquid A has a density of 70 kg/m³ Liquid A has a mass of 1400 kg Liquid B has a density of 280 kg/m³ Liquid B has a volume of 30 m³ Work out the density of liquid C. Liquid A . Liquid 8: 12 Liquid ( 9800 50 = volume Mass = = 980 = 5 mass density density density. density. 280 x 30 = 1960 10 - x volume = =196 1408 70 = Mass volume = 20 1400 + 8 400 30+20 P 6 2 2 7 7 RA 0 1 2 20 8400 kg = 3 196 (Total for Question 13 is 3 marks) 9800 50 kg/m³ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 14 Sally plays two games against Martin. In each game, Sally could win, draw or lose. In each game they play, the probability that Sally will win against Martin is 0.3 the probability that Sally will draw against Martin is 0.1 Work out the probability that Sally will win exactly one of the two games against Martin. 034 0.6L d 0.3 W 064 at D 2₁ m = 3 L₂ m= - 3/32 :W D उह 061 ALD y = - =—=—= x + c 5 = − 1 — — (9) + C 5 = C = 8 - 3 +C WL WD LW 0.6 x 0.3= 0.18 1x0.3= 0.03 0. 0.6 0.3 = 0.18 15 The straight line L, has equation y = 3x - 4 The straight line L₂ is perpendicular to L, and passes through the point (9,5) Find an equation of line L₂ 0.1 +0.3 = 0.03 DW 0.18 +0.03 +0.18 +0.03 0.42 (Total for Question 14 is 3 marks) (9,5) x y y = = 1/3/7 x + 8 (Total for Question 15 is 3 marks) P6 2 2 7 7 RA 01 3 2 0 13 Turn over ► 16 Shirley wants to find an estimate for the number of bees in her hive. On Monday she catches 90 of the bees. She puts a mark on each bee and returns them to her hive. On Tuesday she catches 120 of the bees. She finds that 20 of these bees have been marked. (a) Work out an estimate for the total number of bees in her hive. १० x 14 90 x = = 20 120 6 90=6 x = 90 x 6 Shirley assumes that none of the marks had rubbed off between Monday and Tuesday. (b) If Shirley's assumption is wrong, explain what effect this would have on your answer to part (a). The high answer would be (It would be an over estimate) x too 540 (1) (Total for Question 16 is 4 marks) P 6 2 2 7 7 RA014 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 17 Make f the subject of the formula d = 3(1-f) f-4 d (f-4) df-4d = 3 - df - 4d + 3 f = 3 df + 3 f 18 x is proportional to √y where y > 0 y is increased by 44% Work out the percentage increase in x. 3(1-f) 3 f (d+3)= f = 3 + 4d = 3 + 4 d 3 + 4 d d +3 to increase by √1.44 y √7.44 √5 1.2 √g ↑ Increase of 20% (Total for Question 17 is 4 marks) 44% 3+4d f = = 2 +3³ x 1.44 x = k √y x₂ = 1.2k √5 X₂=1.22 P 6 2 2 7 7 RA 0 1 5 2 0 20 (Total for Question 18 is 3 marks) % 15 Turn over ► 19 f and g are functions such that (a) Find g(5) (b) Find gf(9) 9 (5)= 16 f (9) = = g (4) = = = f(x) = = = 3 (11) 33 12 √x 6 = 6 = 3 x = 3(2(5) + and g(x)= 3(2x + 1) 12 √9 3 ( 2 (4) + 3 (9) 27 =6x (c) Find g (6) Input for inverse function 3(2x + 1) 6x +3 1) 12 3 +)) = 4 33 (1) 27 = output for original P 6 2 2 7 7 RA 0 1 6 20 -|N 을 (2) (Total for Question 19 is 5 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA function DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 20 Show that √180-2√5 5√√5 - 5 √180 √√5 can be written in the form a+ where a and b are integers. b = √36√5 6√5-255 5√5 - 5 = 6√5 4√5 (5√5 + 5) (5√5 - 5) (5√5 + 5) 20 (5) +20 √5 25 (5) +25√5-25√5 - 25 100 + 20√5 100 100 + 2055 100 100 1 √√√5 5 (Total for Question 20 is 4 marks) + P 6 2 2 7 7 RA 0 17 20 17 Turn over ► 21 DEF is a triangle. 18 D& PQ P is the midpoint of FD. Q is the midpoint of DE. FD = a and FE = b Use a vector method to prove that PQ is parallel to FE. DE DF + FE -a + b = = PQ F - 1/2+ Lª b -~ I a 2 + 1/2 b - b D I FE 1/2 2 2 + 1/1/2016 Q E .. parallel (Total for Question 21 is 4 marks) P 6 2 2 7 7 RA 01 8 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 22 The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius (3x - 1) cm from a sector of the circle with radius (5x - 1) cm. Shape B is a circle of diameter (2 - 2x) cm. 5x-1 (3x - 1) cm 2x cm The area of shape A is equal to the area of shape B. Find the value of x. You must show all your working. Area of A = 45 JT (5x-1)² 360 A a cannot be - (2 - 2x) cm Area of B = πT (1-x)² 45 # ( 15x - 1)² = (3 = -1)²) = (1-x)² I 360 (25x²5x -5x + · + 1) − (9x² − 3x-3x+1) (25x² - 10x + 1) - (9x² - 6x +1) 25x² - 10x + 1 = 9x² +6x - 1 16x² - 4x 2= 45 πT (3x − 1)² 360 = ((5x-1)(5x-1) - (3x-1)(3x - 1)) = (1-x)(1-x) 8x²+12x 2x² + 3x-2 be negative (2x −)(x+2) = ½ 8 x==2 P6 2 2 7 7 RA019 20 r = 1-x = 8(1 = = = = 2 8- 8(1-2x 8(1-2x+x²) 16x + 8x²² O -x-x+x²) + x² +x²) YN (Total for Question 22 is 5 marks) 19 Turn over ► 23 There are four types of cards in a game. Each card has a black circle or a white circle or a black triangle or a white triangle. number of cards with a black shape 20 number of cards with a circle number of cards with a white shape 27 number of cards with a triangle : Express the total number of cards with a black shape as a fraction of the total number of cards with a triangle. B Shapes 3: 5 w shapes 45 Circle Triangle 2:7 16:56 3:5 2:7 8 ports x 9 P6 2 2 7 7 RA 0 2 0 20 9 parts 27 56 (Total for Question 23 is 3 marks) TOTAL FOR PAPER IS 80 MARKS x 8 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA