IB Math Pre DP: Final Exam Practice Alei - Desert Academy

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1 IB Math Pre DP: Final Exam Practice Alei - Desert Academy Pre-DP Final Exam Practice Problems The final will cover the 4 main topics that we studied this year 1) Vectors and Parametric Equations, ) Circles and their equations, 3) Quadratic functions, and 4) Probability and Statistics. Note: In addition to working these problems, you should review all your quizzes, tests, and your Semester Final A line L passes through A(1, 1, ) and is parallel to the line r =. 1 + s 3 5 (a) Write down a vector equation for L in the form r = a + tb. The line L passes through point P when t =. Find (i) OP ; (ii) OP. (4). In this question, a unit vector represents a displacement of 1 metre. A miniature car moves in a straight line, starting at the point (, 0). After t seconds, its position, (x, y), is given by the vector equation = t 0 1 (a) How far from the point (0, 0) is the car after seconds? () Find the speed of the car. () (c) Obtain the equation of the car s path in the form ax + by = c. Another miniature vehicle, a motorcycle, starts at the point (0, ), and travels in a straight line with constant speed. The equation of its path is y = 0.6x +, x 0. Eventually, the two miniature vehicles collide. (d) Find the coordinates of the collision point. (e) If the motorcycle left point (0, ) at the same moment the car left point (, 0), find the speed of the motorcycle. 3. The diagram below shows a line passing through the points (1, 3) and (6, 5). y (6,5) () () (5) (Total 14 marks) (1,3) 0 Find a vector equation for the line, giving your answer in the form = a + c t, where t is any real number. b d x (Total 4 marks) B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM 1 of 3

2 IB Math Pre DP: Final Exam Practice Alei - Desert Academy x 4. The vectors x + 1 and x 3 are perpendicular for two values of x. 5 (a) Write down the quadratic equation which the two values of x must satisfy. Find the two values of x. Working: Answers: (a) (Total 4 marks) 5. The line L passes through the origin and is parallel to the vector i + 3j. Write down a vector equation for L. Working: Answer:... (Total 4 marks) 6. The following diagram shows part of the graph of f, where f (x) = x x. (a) Find both x-intercepts. Find the x-coordinate of the vertex. 7. Let f(x) = x + 4x 6. (a) Express f(x) in the form f(x) = (x h) + k. Write down the equation of the axis of symmetry of the graph of f. (c) Express f(x) in the form f(x) = (x p)(x q). 8. The quadratic function f is defined by f(x) = 3x 1x (a) Write f in the form f(x) = 3(x h) k. (4) () (1) () The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x) = 3(x p) + q. B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM of 3

3 IB Math Pre DP: Final Exam Practice Alei - Desert Academy The following diagram shows part of the graph of f (x) = 5 x with vertex V (0, 5). h Its image y = g (x) after a translation with vector has vertex T (3, 6). k (a) (c) Write down the value of (i) h; (ii) k. Write down an expression for g (x). On the same diagram, sketch the graph of y = g ( x). () () () In addition to understanding these problems, be sure to review circles and probability/statistics. B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM 3 of 3

4 IB Math Pre DP: Final Exam Practice MarkScheme Alei - Desert Academy Pre-DP Final Exam Practice Problems - MarkScheme 1. (a) correct equation in the form r = a + tb A N 1 1 r = 1 + t 3 (i) attempt to substitute t = into the equation 1 1 e.g. 6, OP = 5 3 A1 N (ii) correct substitution into formula for magnitude A1 e.g ( ), OP = 38 A1 N (a) At t =, + = 0 1 Distance from (0, 0) = 0.7 = B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM 1 of = 3.94 m = 1. m s 1 (c) x = t and y = t x 0.7y = (d) y = 0.6x + and x 0.7y = x = 5.86 and y = 5.5 or x = and y = (e) The time of the collision may be found by solving = + t for t t = 5.5 s [ie collision occurred 5.5 seconds after the vehicles set out]. Distance d travelled by the motorcycle is given by d = = (5.86) + (3.5) 5.5 = = 6.84 m d 6.84 Speed of the motorcycle = = t 5.5 = 1.4 m s Direction vector = 5 3 [14]

5 IB Math Pre DP: Final Exam Practice MarkScheme Alei - Desert Academy = 1 5 = + t 3 OR 6 5 = + t 5 4. (a) x + x 1 = 0 x 3 5 x(x + 1) + (x 3)(5) = 0 x + 7x 15 = 0 METHOD 1 x + 7x 15 = (x 3)(x + 5) = 0 (A) (A) (C4) (C3) [4] 3 x = or x = 5 (C1) METHOD 7 ± x = 7 4()( 15) () x = 3 or x = 5 (C1) [4] 5. Vector equation of a line r = a + λt 0 a =, t = 0 3 r = λ(i + 3j) (C4) 6. (a) evidence of attempting to solve f (x) = 0 evidence of correct working A1 1± 9 e.g. ( x + 1)( x ), intercepts are ( 1, 0) and (, 0) (accept x = 1, x = ) A1A1N1N1 evidence of appropriate method x1 + x b e.g. xv =, xv =, reference to symmetry a x v = 0.5 A1 N 7. (a) evidence of obtaining the vertex b e.g. a graph, x =, completing the square a f(x) = (x + 1) 8 A N3 x = 1 (equation must be seen) A1 N1 (c) f(x) = (x 1)(x + 3) A1A1 N [4] intercepts are ( 1, 0) and (, 0) (accept x = 1, x = ) A1A1N1N1 B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM of 3

6 IB Math Pre DP: Final Exam Practice MarkScheme Alei - Desert Academy evidence of appropriate method x1 + x b e.g. xv =, xv =, a (a) For a reasonable attempt to complete the square, (or expanding) 8. e.g. 3x 1x + 11 = 3(x 4x + 4) f(x) = 3(x ) 1 (accept h =, k = 1) A1A1 N3 METHOD 1 Vertex shifted to ( + 3, 1 + 5) = (5, 4) M1 so the new function is 3(x 5) + 4 (accept p = 5, q = 4) A1A1 N METHOD g(x) = 3((x 3) h) + k + 5 = 3((x 3) ) M1 = 3(x 5) + 4 (accept p = 5, q = 4) A1A1 N 9. (a) (i) h = 3 A1 N1 (ii) k = 1 A1 N1 g (x) = f (x 3) + 1, 5 (x 3) + 1, 6 (x 3), x + 6x 3 A N (c) y V T x Note: Award M1 for attempt to reflect through y-axis, A1 for vertex at approximately ( 3, 6). M1A1 N B: C:\Documents and Settings\Bob\My Documents\Dropbox\Desert\PreDP\LP_PreDPS.doc on 05/14/01 at 10:48 PM 3 of 3