Two A.Maths CE-level Exam Questions－航空百大

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Two A.Maths CE-level Exam Questions

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Please go to this site to find the questions, http://aerodrive.twghwfns.edu.hk/~4s123/quadratic.pdf Please explain clearly, esp.10(b)(i)&(ii),11.(b)(i)&(ii) ,(c)(i)&(ii), Thank you~

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Please explain clearly, esp.10(b)(i)&(ii),11.(b)(i)&(ii) ,(c)(i)&(ii), 10)a) f(x) = -(x - a)^2 + b Since P is the vertex of f(x) the cooridinates of P is (a,b) 10b)i) g(x) = (x - b)^2 + a Since f(x) passes through point Q(b,a) f(x) = -(x - a)^2 + b f(b) = -(b - a)^2 + b a = -(b - a)^2 + b (b - a)^2 = b - a So, g(x) = (x - b)^2 + a g(a) = (a - b)^2 + a g(a) = b - a + a g(a) = b So,g(x) will pass through point P(a,b). 10b)ii) Since f(x) touches x-axis. b = 0 where f(x) opens downwards and cut x-axis at (a,0) hence, g(a) = (a - b)^2 + a 0 = a^2 + a a(a + 1) = 0 a = 0 or a = -1 Case 1 is a = 0, f(x) = -x^2 the g(x) will formed by reflecting f(x) by x-axis, both vertex is (0,0), and g(x) will open upwards f(x) will open downwards. Case 2 is a = -1 g(x) = (x - b)^2 + a g(-1) = 1 - 1 g(-1) = 0 g(x) = (x - b)^2 + a g(0) = -1 So,the g(x) will open upwards have an x-intercept at -1,0 y - intercept at (0,-1) where f(x) is -(x + 1)^2 11a) let f(x) = k(x - 4)^2 + 9 f(x) pass through (10,0) 0 = 36k + 9 k = -1/4 so,f(x) = -1/4(x - 4)^2 + 9 f(x) = -1/4x^2 + 2x + 5 b)i) c2 : y = -1/5x^2 - (h - 20 /10)x + h when x = 10 y = -20 - h + 20 + h y = 0 So,C2 passes through (10,0) C1 : y = f(x) = -1/4x^2 + 2x + 5 C2 : y = -1/5x^2 - (h - 20 /10)x + h By comparing coefficent,we can find the other roots in terms of h. 2007-03-02 16:52:26 補充： C1 : y = f(x) = -1/4x^2 十 2x 十 5C2 : y = -1/5x^2 - (h - 20 /10)x 十 hlet they meet at q,whch is a x-coordinate-1/5x^2 - (h - 20 /10)x 十 h = -1/4x^2 十 2x 十 5-4x^2 - 2(h - 20)x 十 20h = -5x^2 十 40x 十 100x^2 - 2hx 十 20h - 100 = 0q 十 10 = 2hq = 2h - 10 2007-03-02 16:52:53 補充： c)i)OD1 = 5c)ii)since OD2 = h5 10h 10so the range of h is 10 2007-03-02 16:55:26 補充： for cii)2h - 10 [is smaller than] 0 or 2h - 10 [is bigger than or equal to ] = 10h [smaller than] 5 (rejected) or h [bigger than or equal to ] 10so,the range is 10 2007-03-02 16:56:17 補充： the range is 10 [smaller than or equal to h] h [smaller than ] 15