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1999_Paper I - STEP I 1999 Section A 1 2 Pure Mathematics...

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STEP I, 1999 2 Section A: Pure Mathematics 1 How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5? What is the average value of these integers? How many integers greater than or equal to zero and less than 4179 are not divisible by 3 or 7? What is the average value of these integers? 2 A point moves in the x - y plane so that the sum of the squares of its distances from the three fixed points ( x 1 , y 1 ), ( x 2 , y 2 ), and ( x 3 , y 3 ) is always a 2 . Find the equation of the locus of the point and interpret it geometrically. Explain why a 2 cannot be less than the sum of the squares of the distances of the three points from their centroid. [The centroid has coordinates (¯ x, ¯ y ) where 3¯ x = x 1 + x 2 + x 3 , y = y 1 + y 2 + y 3 . ] 3 The n positive numbers x 1 , x 2 , . . . , x n , where n > 3, satisfy x 1 = 1 + 1 x 2 , x 2 = 1 + 1 x 3 , . . . , x n - 1 = 1 + 1 x n , and also x n = 1 + 1 x 1 . Show that (i) x 1 , x 2 , . . . , x n > 1, (ii) x 1 - x 2 = - x 2 - x 3 x 2 x 3 , (iii) x 1 = x 2 = · · · = x n . Hence find the value of x 1 . 4 Sketch the following subsets of the x - y plane: (i) | x | + | y | 6 1 ; (ii) | x - 1 | + | y - 1 | 6 1 ; (iii) | x - 1 | - | y + 1 | 6 1 ; (iv) | x | | y - 2 | 6 1 .
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STEP I, 1999 3 5 For this question, you may use the following approximations, valid if θ is small: sin θ θ and cos θ 1 - θ 2 / 2 . A satellite X is directly above the point Y on the Earth’s surface and can just be seen (on the horizon) from another point Z on the Earth’s surface. The radius of the Earth is R and the height of the satellite above the Earth is h .
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