DULWICH COLLEGE -13+ Paper 3 by smoothmaths | Sep 15, 2020 | 0 comments Welcome to your DULWICH COLLEGE -13+ YEAR 9 ENTRANCE AND SCHOLARSHIP EXAMINATION Mathematics Specimen Paper C 1. Use your calculator to work out the value of: \[\frac{\sqrt{436-13^{2}}}{76-35}\] (a) Write down all of the digits shown on your calculator:0.398539869213.64516130.6709677420.7903225810.483589962 (b) Write your answer to (a) rounded to 3 decimal places:0.3990.6700.7900.7900.499 (c) Write your answer to (a) rounded to 2 significant figures:0.400.390.670.790.49 2. Mike’s Gym had 650 male members and 350 female members in 2008. (a) Calculate the percentage of the members of Mike’s Gym in 2008 who were male.65%15%35%75%45% (b) Given that the number of female members decreased by 10% from 2008 to 2009, calculate the number of female members in 2009.315122228350325 (c) If there were 910 male members of the sports club in 2009, calculate the percentage increase of male members between 2008 and 2009.40%71%45%65%35% 3. (a) Simplify the ratio 20 : 15.4:34:53:42:31:3 (b) Divide £280 in the ratio 20 : 15.£160, £120£100, £180£200, £80£170, £110£150, £130 (c) A recipe for a large cake uses 400 grams of flour and 300 grams of butter. A recipe for a smaller cake uses 300 grams of flour. Calculate how much butter is needed for the smaller cake if the ratio of flour to butter is to be the same in both cakes.225400300125250 4. Write a simplified expression for the perimeter of the shape drawn below. 5a+4b-56a+10b+5-5a-4b-53a+5b-2None of the Above 5. Write the following list of numbers in ascending order (i.e. smallest first): \[\frac{3}{16}\] , 0.18 , \[\frac{1}{5}\] , 18.5%0.18, 18.75%, 3/16, 1/518.75%, 0.18,1/5, 3/161/5, 3/16,18.75%,0.180.18, 1/5, 3/16,18.75%1/5, 0.18, 3/16,18.75% 6. Expand the brackets and simplify where possible: (a) 5(3a + 4) — 4(4a — 2)-a+28-a-12a-28a+28None of the Above (b) (b — 2)(b + 7)b\[^{2}\]+5b-14b\[^{2}\]-2b+145\[b^{2}\]-14b\[^{2}\]+7b+14None of the Above 7. (a) Simplify \[\frac{50a^{3}b^{2} }{15ab^{3}}\]\[\frac{10a^{2}} {3b}\]\[\frac{50a^{3}b^{2} }{15ab^{3}}\]3\[a^{2}\]b\[\frac{10a} {3b^{2}}\]None of the Above b) Factorise fully: (i) 18x — 606(3x-10)3(6x-20)2(9x-30)6x(3-10)None of the Above (ii) 5\[a^{2}\]b — 20\[a^{3}\] \[b^{2}\]5\[a^{2}\]b(1-4ab)5\[a^{2}\]b-4abab(5ab-4)\[a^{2}\]b(5-4ab)None of the Above 8. On the grid above draw the result of: (a) reflecting triangle T in the J-axis. Label your answer A (b) enlarging triangle T by a scale factor of 2 with centre of enlargement (5, 9). Label your answer B. (c) translating triangle T by the vector \[\left(\frac{3}{-7}\right)\] . Label your answer C. (d) reflecting triangle T in the line y = x . Label your answer D. 9. The following diagrams show a shape formed by adding successive regular hexagons to an initial hexagon. (a) Complete the following table showing the number of straight lines (L) for each diagram (D). Diagram (D) Number of straight lines (L) 1 6 2 11 3 4 (b) Write down a formula for L in terms of D.L = 1+5DL = 6+3DL = 2+4DL = 3+5DL = 5D-1 (c) Calculate: (i) L when D = 100.501306402503499 (ii) D when L = 251.5030456040 10. The diagram below shows a regular pentagon ABCDE. Within this regular pentagon a regular pentagram (a 5-pointed star) has been drawn, which in turn contains another regular pentagon FGHIJ. Calculate the size of the angle marked: (a) w,72°60°120°36°45° (b) x ,108°60°120°144°None of the Above (c) y,36°72°30°6040° (d) z ,36°72°30°60°40° 11. Solve the following equations: (a) \[\frac{10x}{7}\] = 3\[\frac{21}{10}\]2.1\[\frac{10}{21}\]\[\frac{10}{7}\]Both a & b (b) 3 — 5x = 8-15183 (c) 4x + 5 = 2x + 3-12314 (d) x(2x — 3) = \[2x^{2}\] — 5(x + 2)-5-1025None of the Above 12. Calculate the value of a half of a third of 0.9.0.150.30.450.50.05 13. (a) Two unbiased dice are rolled and their values added together. Complete the table below to show the possible sum of the scores of the two dice. 1 2 3 4 5 6 1 3 6 2 3 4 5 6) 3 6 7 8 9 4 7 ) 5 6 7 8 6 10 (b) Use the table to calculate the probability of rolling a sum less than 8 when rolling two fair dice.\[\frac{7}{12}\]\[\frac{3}{4}\]\[\frac{1}{4}\]\[\frac{7}{15}\]\[\frac{5}{12}\] (c) If the two numbers shown on the dice are multiplied rather than added, find the probability that the result is a prime number.\[\frac{1}{6}\]\[\frac{1}{3}\]\[\frac{3}{4}\]\[\frac{1}{4}\]\[\frac{1}{2}\] 14. (a) Complete the following table of values for y = 2x + 2 x -1 0 1 Y (b) Plot and label the lines y = 2x + 2 and x = 1 on the axes below. (c) Write down the co-ordinates of where the lines y = 2x + 2 and x = 1 intersect.(1,4)(1,0)(0,2)(-1,0)(0,1) (d) Calculate the area of the triangle formed by the lines y = 2x + 2, x = 1 and the x axis.4sq.units8sq.units2sq.units6sq.unitsNone of the Above 15. The diagrams below show a cuboid and a right-angled triangular prism with dimensions as indicated. (a) Calculate the volume of the cuboid.\[cm^{3}\]120\[cm^{3}\]216\[cm^{3}\]125\[cm^{3}\]64\[cm^{3}\]100\[cm^{3}\] (b) Calculate the volume of the triangular prism, leaving your answer simplified in terms of x.4\[x^{2}\]8\[x^{2}\]2\[x^{2}\]6\[x^{2}\]\[x^{2}\] (c) The volume of the triangular prism is equal to the volume of the cube. Calculate the value off, writing your answer to 3 significant figures.5.48cm10.95cm5.37cm6.48cm5.38cm 16. (a) My running speed is twice my walking speed. I can walk 120 metres in 80 seconds. Calculate how long it takes me to run 60 metres.20seconds40seconds80seconds60seconds30seconds b) Peter’s running speed is three times his walking speed. Peter can walk 2p metres inp seconds. Calculate how long it takes Peter to run 60 metres.10seconds6seconds2seconds20seconds15seconds 17. Sarah’s age is one third that of her father. The difference between their ages is 30 years. Calculate the age of Sarah’s father.45years15years25years20years30years 18. (a) A group of 10 students were asked to name what they thought was their favourite prime number. The answers they gave were: 1, 3, 11, 11, 17, 19, 21, 23, 23, 23 (i) Write down how many students answered the question with a number which was not prime.21345 (ii) Calculate the mode and mean for the 10 answers.mode=23, mean=15.2mode=21, mean=16.8mode=19, mean=13.2mode=23, mean=18.5mode=17, mean=16.5 (b) An 11'th student is now asked for his favourite prime number. The mean of all 11 answers is 25. (i) Calculate the number which the 11'th student gave.123275152132135 (ii) State yes or no, with a reason, whether the number which the 11'th student gave was a prime number.no it is not a prime numberyes it is a prime numberit is neither a prime nor odd numberit is an even numberNone of the Above Reason: 19. The diagram below shows a rectangle containing two circles with the same radius. Each circle has an area of 25π \[cm^{2}\] . (a) Calculate the radius of the circles.5cm12.5cm10cm8cm6cm (b) Calculate the perimeter of shaded area labelled Q in the diagram. Round your answer to 3 significant figures.25.7cm15.7cm20.7cm30.7cm35.7cm 20. The following three numbers are prime: 5, 13, 17. (a) The remainder is the same when each of these prime numbers is divided by 4. Calculate this remainder.10234 (b) Calculate a, b, c, d and e given that 5 = \[1^{2}\] + \[a^{2}\] 13 = \[2^{2}\] +\[b^{2}\] 17 = \[c^{2}\] +\[1^{2}\] 29 = \[d^{2}\] + \[e^{2}\]a=2a=3a=4a=5a=1 b=?b=3b=2b=4b=5b=1 c=?c=4c=5c=6c=3c=2 d=?d=5d=4d=3d=2d=1 e=?e=5e=4e=3e=2e=1 (c) Calculate the next prime number after 29 which is the sum of two square numbers.3731392941 (d) In a similar way, solve 61 = \[x^{2}\] + \[(x-1)^{2}\] .x=6x=5x=7x=4x=8 21. A new way to combine two numbers, written a ⊗ b, is calculated by multiplying the two numbers together and then subtracting both numbers. For example, 3 ⊗ 5 = 3 x 5 — 3 — 5 = 7. (a) Calculate 2 ⊗ 642012168 (b) Calculate 4 ⊗ —2-10-6-8-2-12 (c) Solve x ⊗ 3 = 11x=7x=5x=6x=8x=9 (d) Solve x ⊗ (x + 1) — 8 = x ⊗ xx=9x=8x=3x=106 22. The diagram below (not drawn to scale) shows three circles all centred on 0. The radius of the smallest circle is 6 cm and the radius of the middle circle is 10 cm. The trapezium ABCD has been drawn as shown so that AB just touches the smaller circle and DC just touches the middle circle. (a) Calculate the minimum distance between the smallest and middle circle, marked x in the diagram.4106168 (b) Calculate the length AB.16cm8cm6cm10cm4cm (c) Given that the area of the trapezium is 128 cm2, calculate the length DC.4816641036 (d) Calculate the radius of the larger circle.2614241016 Time is Up!