SMC2012_web_solutions

# This conjecture by finding the even number in the

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this conjecture by finding the even number in the range from 4 to 100 which can be expressed as the sum of two prime numbers in the largest number of ways. 2. The diagram shows an equilateral triangle, a square and a regular pentagon which all share a common vertex. What is the value of θ ? A 98 B 102 C 106 D 110 E 112 Solution: B The interior angles of an equilateral triangle, a square and a regular pentagon are 60 , 90 and 108 , respectively. Hence 102 258 360 ) 108 90 60 ( 360 = = + + = θ . 3. The price of my favourite soft drink has gone up in leaps and bounds over the past ten years. In four of those years it has leapt up by 5p each year, whilst in the other six years it has bounded up by 2p each year. The drink cost 70p in 2002. How much does it cost now? A £0.77 B £0.90 C £0.92 D £1.02 E £1.05 Solution: D The cost has risen by 5p four times, and by 2p six times. So the total price rise in pence has been 32 ) 2 6 5 4 ( = × + × . Therefore the price now is 02 . 1 £ p 102 p ) 32 70 ( = = + . 4. According to one astronomer, there are one hundred thousand million galaxies in the universe, each containing one hundred thousand million stars. How many stars is that altogether? A 13 10 B 22 10 C 100 10 D 120 10 E 121 10 Solution: B One hundred thousand is 5 10 and one million is 6 10 . So one hundred thousand million is 11 6 5 6 5 10 10 10 10 = = × + . Therefore the total number of stars is 22 11 11 11 11 10 10 10 10 = = × + .

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