TONBRIDGE SCHOOLScholarship Examination 2005MATHEMATICSITuesday3rdMay 20059.00a.m.Time allowed: 1hour30 minutesAnswer as many questions as you can.Questions 1 to 5 are worth 8 marks each;Questions 6 to 9 are worth 15 marks each.All answers must be supported by adequate explanation.Calculators may be used in any question.Page 1
1.A money-box contains only 2 pence and 5 pence coins. Altogether there are 36 coins with a total value of 129 pence. Use this information to make two simultaneous equations and solve them to find the number of 5 pence coins.[8 marks]2.Consider the following sequence of fractions:.a)What is the pattern relating a fraction in the sequence to its successor?b)Write down the next two fractions in the sequence.c)Work out the first six fractions in the sequence as decimals correct to three decimalplaces. What do you notice about your answers?[8]3.In appropriate units, the maximum pressure, P, that a vertical pillar of height Hcan support before it collapses is given by the formula , where 3.14159 . . . has its usual meaning andCdepends on the mass of the pillar.a)If C=12.7 and H= 5.2, find P.b)If H= 17.3 and P= 9.3, find C.c)In an experiment, the values of P, H, Care measured as P= 62, H= 15, C= 30. What value of does the formula then give from these measurements?[8]4.A unit fractionis one like with numerator 1.a)Write 1 as the sum of three different unit fractions.b)By multiplying your answer in (a) by a suitable unit fraction, write as the sum of three different unit fractions.c)Use youranswers to (a), (b) to write 1 as the sum of five different unit fractions.d)Write 1 as a sum of seven different unit fractions.[8]5.In the diagram below, APR and BRQ are isosceles triangles and ARB is a straight line.a)If a= 40°and b=70°, find x.b)In general, show that xis always the average (mean) of aand b.[8]13717,,,,...125123CPHp=p=p1416ABPQRxba
Page 26.The diagram below shows an isosceles trapezium ABCD with sides of length 3cm, 5cm, 3cm and 8cm.a)Use Pythagoras’ theorem to show that the diagonal AC has a length which is a whole number of centimetres. (Hint: Drop a perpendicular from C to AD to create two right-angled triangles.)b)Give an example of a rhombus with sides and both diagonals allhaving lengths whichare whole numbers of centimetres. Explain your answer carefully.[15]7.A famous formula of Einstein’s predicts that if a 3kg mass moves at a speed which is a fraction xof the speed of light, then its mass increases to a value given by .a)When x= 0.2, show that y= 3.06kg.b)Calculate the values of ycorresponding to x= 0,0.2, 0.4, 0.6, 0.8, 0.9, 0.95.c)Choosing sensible scales,plot a graph of yagainst x.d)What value of xcorresponds to y= 6?e)What is it (i) about the shape of the graph and (ii) about the formula that suggests that the 3kg mass cannot travel faster than the speed of light?[15]ABDC3 cm5 cm8 cm3 cmy231yx=-
Page 3TURN OVER8.The diagram below shows a circular sector (A) and an equilateral triangle (B). The sector has angle 60°and radius rand the equilateral triangle has sides of length 10cm.a)Find rif A and B have the same area.b)Find rif A and B have the same perimeter.[15]9.The terms F1, F2, F3, . . .of the Fibonacci sequenceare given by F1= 1, F2= 1, F3=2,. . .where each term is the sum of the previous two.a)Verify that F4= 3, F5= 5 and write down the next five terms of the Fibonacci sequence.b)Consider the following statement about the Fibonacci sequence:“If you square two successive terms of the sequence and add these squares together,then the answer is another term in the sequence”.Check this statement using your data in(a). Giventwo successive terms as in the statement, explain carefully how you can predict which term of the sequence arises as the answer.c)Use your answer to (b) to find xand ybelow:i)ii).[15]B10 cm10 cm10 cm60°rA225051xFFF+=221137yyFFF++=
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