the concept of a unit vector; [Note: Vectors parallel to a unit vector a can be written in the form ka‚ where k is the length of the vector.] 10. find the unit vector of a directed line segment; 11. determine whether three points with given coordinates are collinear; 12. use the ratio theorem to find the position vector of a point that divides a line segment in a given ratio; 13. understand that the scalar product can be expressed as or ; 14. understand that the vector product can
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with equation 2y = 11x + 3. (3) 7. (a) Factorise completely x3 - 4x. (3) (b) Sketch the curve with equation y = x3 - 4x‚ showing the coordinates of the points where the curve crosses the x-axis. (3) (c) On a separate diagram‚ sketch the curve with equation y = (x - 1)3 - 4(x - 1)‚ showing the coordinates of the points where the curve crosses the x-axis. (3) 8. The straight line l1 has equation y = 3x - 6. The straight line l2 is perpendicular to l1
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J. Wijayakulasooriya ABSTRACT 2D vector plotters are designed to take a pointer to any given coordinate in the xy plane. An application of this plotting device is in PCB drilling. In PCB drilling the location of the drill holes are fed into the system which will drill the PCB at the specified coordinates. The main objective of this project is to be able to take a pointer to these set of coordinates along the shortest path. The parameter measured is the length of the path taken. Path planning and
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plot: creates 2d line plot axis: changes aspect ratio of x and y axis x label: annoted the x axis y label: annoted the y axis title: puts the title on the plot title of prog: title(’circle of unit radius’) print: prints the hardcopy of the plot EX: draw a circle of unit radius x and y co ordinates 100 points of the circle the parametric eq is x=cos(t) y=sin(t) theta=linspace(0‚2*pi‚100); axis=’equal’; xlabel(’x’) ylabel(’y’) 1. plot y=sinx range 0 x=linspace(0
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………………………………………….. (c) ………………………………………….. (Total 4 marks) 1 2. The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A‚ and has a vertex at B. y A x O B (a) Factorize x2 – 2x – 8. (b) Write down the coordinates of each of these points (i) A; (ii) B. Working: Answers: (a) ………………………………………….. (b) (i) …………………………………….. (ii) …………………………………….. (Total 4 marks) 2 3. The diagram below shows a path x m wide around a rectangular lawn which measures 10 m by
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plane is not parallel to the axis‚ base‚ or generatrix of the intersected cone. 3. n. The locus of points for which the sum of the distances from each point to two fixed points is equal. 4. n. Ellipsis. Century Dictionary and Cyclopedia 1. n. In geometry‚ a plane curve such that the sums of the distances of each point in its periphery from two fixed points‚ the foci‚ are equal. It is a conic section (see conic) formed by the intersection of a cone by a plane which cuts obliquely the axis and the
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SAVE A TREE – PLEASE DO NOT PRINT ME IB Math Studies – Chapter 16 and 17 – Exponential Functions – Review Questions 1.The diagrams below are sketches of some of the following functions. (i)y = ax(ii)y = x2 – a (iii)y = a – x2 (iv)y = a – x(v)y = x – a Complete the table to match each sketch to the correct function. Sketch Function (a) (b) (c) (d) Working: (Total 8 marks) 2.The following diagrams show the graphs of five functions. Each of the following sets represents the range of
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Chapter 1 Review of Straight Lines We start with a brief review of properties of straight lines‚ since these properties are fundamentally important to our understanding of more advanced concepts (tangents‚ slopes‚ derivatives). Skills described in this introductory material will be required in many contexts. 1.1 Geometric ideas: lines‚ slopes‚ equations Straight lines have some important geometric properties‚ namely: The slope of a straight line is the same everywhere along its length.
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work. Y = -2/3x + 30 x-intercept 0=-2/3x+30 -30=-2/3X (3/2)(-30)=(-2/3x)(3/2) -45=-x 45=x x-intercept: (45‚0) y-intercept y=-2/3(0)+30 y=30 y-intercept: (0‚30) 2. Graph the given equation. • Label each axis of the coordinate plane with descriptive labels. • Label each intercept as “x-intercept” or “y-intercept” and include the ordered pair. 3. Identify the points on the graph that most accurately represent
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called “The Father of Modern Philosophy”‚ this title is supported by his contributions to philosophy and mathematics. The coordinate system‚ used today‚ is accredited to him along with many other mathematical contributions. He also had many contributions to philosophy‚ including his most famous‚ Meditations on First Philosophy. Every time you graph an equation on a Cartesian coordinate system‚ you are using the work of Rene Descartes. Born in La Haye on March 31‚ 1596‚ Descartes was sent to the Jesuit
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