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WORKSHEET - Mathematics (Advanced) 1 Questions WORKSHEET - Mathematics (Advanced) 1. Geometry and Calculus, 2UA 2017 HSC 4 MC 10. Geometrical Applications of Differentiation Curve Sketching and The Primitive Function The function is defined for Teacher: James Butterworth On this interval, Exam Equivalent Time: 94.5 minutes (based on HSC allocation of 1.5 Which graph best represents ? minutes approx. per mark) (A) y (B) y x a b x a b x (C) y (D) y HISTORICAL CONTRIBUTION x x T10 Geometry and Calculus is the single largest topic within the Mathematics a b a b course, contributing 15.0% to each paper, on average, in the last 10 years. x This topic has been split into three sub-categories: 1-Maxima and Minima (5.3%), 2-Curve Sketching and The Primitive Function (7.6%), and 3-Tangents and Normals (2.1%). 2017 HSC ANALYSIS - What to expect and common pitfalls Curve Sketching and The Primitive Function (7.6%). The most popular curve is easily the cubic, asked 7 times since 2003, including 2015. Note that we have to go back 10 years before cubics have been left off the HSC two years in a row in this topic area. Cubics were left off in the 2016 exam... enough said. Sketches of polynomials of degree 4 are the second most popular, asked in 2007, 2008, 2012 and 2016. Focus Area: An examiner favourite in this topic area requires students to switch between f(x) and f´(x) for a given function, either graphically or by using an equation. This has been tested every year in recent times, except not in 2015 or 2016, in questions worth 1-5 marks. Markers' Comments of note: Many students do not have a clear understanding of concavity and the second derivative. When drawing a graph within a given domain, clearly identify the extremes! 2. Geometry and Calculus, 2UA 2012 HSC 4 MC 3. Geometry and Calculus, 2UA 2013 HSC 8 MC The diagram shows the graph . The diagram shows points , , and on the graph . Which of the following statements is true? (A) (B) (C) At which point is and ? (D) (A) (B) (C) (D) 4. Geometry and Calculus, 2UA 2017 HSC 9 MC 7. Geometry and Calculus, 2UA 2005 HSC 4b The graph of is shown. A function is defined by . y (i) Find all solutions of (2 marks) (ii) Find the coordinates of the turning points of the graph of , and determine their nature. (3 marks) 4 (iii) Hence sketch the graph of , showing the turning points and the points where the curve meets the -axis. (2 marks) (iv) For what values of is the graph of concave down? (1 mark) y = f ′(x) 8. Geometry and Calculus, 2UA 2006 HSC 5a A function is defined by . O 2 x (i) Find the coordinates of the turning points of and determine their nature. ( 3 marks) (ii) Find the coordinates of the point of inflexion. (1 mark) The curve has a maximum value of 12. (iii) Hence sketch the graph of , showing the turning points, the point of inflexion What is the equation of the curve ? and the points where the curve meets the -axis. (3 marks) (A) (iv) What is the minimum value of for ? (1 mark) (B) 9. Geometry and Calculus, 2UA 2007 HSC 6b (C) Let . (D) (i) Find the coordinates of the points where the curve crosses the axes. (2 marks) 5. Geometry and Calculus, 2UA 2013 HSC 12a (ii) Find the coordinates of the stationary points and determine their nature. (4 marks) (iii) Find the coordinates of the points of inflexion. (1 mark) The cubic has a point of inflexion at . (iv) Sketch the graph of , indicating clearly the intercepts, stationary points and Show that . (2 marks) points of inflexion. (3 marks) 6. Geometry and Calculus, 2UA 2011 HSC 7a 10. Geometry and Calculus, 2UA 2015 HSC 13c Let . Consider the curve . (i) Find the coordinates of the stationary points of , and determine their nature. (i) Find the stationary points and determine their nature. (4 marks) (3 marks) (ii) (ii) Hence, sketch the graph showing all stationary points and the -intercept. Given that the point lies on the curve, prove that there is a point of (2 marks) inflexion at . (2 marks) (iii) Sketch the curve, labelling the stationary points, point of inflexion and -intercept. (2 marks) 11. Geometry and Calculus, 2UA 2016 HSC 13a Worked Solutions Consider the function 1. Geometry and Calculus, 2UA 2017 HSC 4 MC i. Find the two stationary points and determine their nature. (4 marks) ii. Sketch the graph of the function, clearly showing the stationary points and the and intercepts. (2 marks) 12. Geometry and Calculus, 2UA 2014 HSC 11f The gradient function of a curve is given by . The curve passes through the point . 2. Geometry and Calculus, 2UA 2012 HSC 4 MC Find the equation of the curve. (2 marks) 13. Geometry and Calculus, 2UA 2014 HSC 14a Find the coordinates of the stationary point on the graph , and determine its nature. (3 marks) 14. Geometry and Calculus, 2UA 2017 HSC 13b 3. Geometry and Calculus, 2UA 2013 HSC 8 MC Consider the curve . ♦ Mean mark 48% (i) Find the stationary points of the curve and determine their nature. (4 marks) (ii) Sketch the curve, labelling the stationary points. (2 marks) (iii) Hence, or otherwise, find the values of for which is positive. (1 mark) Copyright © 2004-16 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW) 4. Geometry and Calculus, 2UA 2017 HSC 9 MC 6. Geometry and Calculus, 2UA 2011 HSC 7a (i) 5. Geometry and Calculus, 2UA 2013 HSC 12a MARKER'S COMMENT: Graphs should be (ii) large (around ½ page), axes drawn with a ruler, with intercepts and turning points clearly shown. The scale can be different on each axe for clarity, as shown in the Worked Solution. (iii) 7. Geometry and Calculus, 2UA 2005 HSC 4b (i) (iv) (ii) 8. Geometry and Calculus, 2UA 2006 HSC 5a (i) (iv) (ii) 9. Geometry and Calculus, 2UA 2007 HSC 6b (i) (iii) (ii) (iv) 10. Geometry and Calculus, 2UA 2015 HSC 13c (i) (iii) (ii) (iii) 11. Geometry and Calculus, 2UA 2016 HSC 13a 12. Geometry and Calculus, 2UA 2014 HSC 11f i. 13. Geometry and Calculus, 2UA 2014 HSC 14a ii. 14. Geometry and Calculus, 2UA 2017 HSC 13b (i) x -2 1 Copyright © 2014-2017 M2 Mathematics Pty Ltd (SmarterMaths.com.au) (ii) y (-2, 27) 27 7 x -2 -1 (1, 0) (iii) Questions WORKSHEET - Mathematics (Advanced) 1. Geometry and Calculus, 2UA 2004 HSC 4b 10. Geometrical Applications of Differentiation Curve Sketching and The Primitive Function Consider the function . Teacher: James Butterworth (i) Find the coordinates of the stationary points of the curve and determine Exam Equivalent Time: 100.5 minutes (based on HSC allocation of 1.5 their nature. (3 marks) minutes approx. per mark) (ii) Sketch the curve showing where it meets the axes. (2 marks) (iii) Find the values of for which the curve is concave up. (2 marks) 2. Geometry and Calculus, 2UA 2010 HSC 6a Let . (i) Show that the graph has no stationary points. (2 marks) (ii) Find the values of for which the graph is concave down, and the values for which it is concave up. (2 marks) (iii) Sketch the graph , indicating the values of the and intercepts. (2 marks) 3. Geometry and Calculus, 2UA 2014 HSC 15c The line is a tangent to the curve at a point . (i) Sketch the line and the curve on one diagram. (1 mark) HISTORICAL CONTRIBUTION (ii) Find the coordinates of . (3 marks) T10 Geometry and Calculus is the single largest topic within the Mathematics (iii) Find the value of . (1 mark) course, contributing 15.0% to each paper, on average, in the last 10 years. This topic has been split into three sub-categories: 1-Maxima and Minima (5.3%), 2-Curve Sketching and The Primitive Function (7.6%), and 3-Tangents and Normals (2.1%). 4. Geometry and Calculus, 2UA 2008 HSC 8a Let . 2017 HSC ANALYSIS - What to expect and common pitfalls Curve Sketching and The Primitive Function (7.6%). The most popular curve is easily the (i) Find the coordinates of the points where the graph of crosses the axes. (2 cubic, asked 7 times since 2003, including 2015. Note that we have to go back 10 years marks) (ii) Show that is an even function. (1 mark) before cubics have been left off the HSC two years in a row in this topic area. Cubics were left off in the 2016 exam... enough said. (iii) Find the coordinates of the stationary points of and determine their nature. (4 marks) Sketches of polynomials of degree 4 are the second most popular, asked in 2007, 2008, (iv) Sketch the graph of . (1 mark) 2012 and 2016. Focus Area: An examiner favourite in this topic area requires students to switch between f(x) and f´(x) for a given function, either graphically or by using an equation. This has been 5. Geometry and Calculus, 2UA 2010 HSC 8d tested every year in recent times, except not in 2015 or 2016, in questions worth 1-5 marks. Let , where is a constant. Markers' Comments of note: Many students do not have a clear understanding of concavity and the second derivative. When drawing a graph within a given domain, clearly identify the Find the values of for which is an increasing function. (2 marks) extremes! 6. Geometry and Calculus, 2UA 2011 HSC 9c 8. Geometry and Calculus, 2UA 2004 HSC 9c The graph in the diagram has a stationary point when , a point of inflexion Consider the function , for .
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