ࡱ> {9}|Oh+'0  0 < H T`hpx Question 1 uesUserionserserNormal2Userl24erMicrosoft Word 8.0@F#@Vc@D>Zz@Zz92(ᅊ^)d;|j9L9U2(V[qAY)[% FMicrosoft Equation 3.0 DS Equation EQUATION.39q  The Curriculum Pression6/ Jj  Question 1 Title 6> _PID_GUIDAN{AB05C4A0-8DD8-11D2-8A1B-444553540000}[%1. Determine a)  EMBED Equation.3 , giving your answer in terms of natural logarithms [5] b)  EMBED Equation.3  [4] 2. a) Evaluate correct to 2 decimal places  EMBED Equation.3  [6] b) Evaluate  EMBED Equation.3  [6] c) Evaluate  EMBED Equation.3  [5] 3. a) Sketch the graph of y = 2x2 + 1 from x = -2 to x = 3. [3] b) Determine by integration the area enclosed by the curve, the x-axis and the lines x = -1 and x = 2. [4] 4. The work in Joules done by a force F newtons in moving a body from x1 to x2 is given by  EMBED Equation.3  The force acting on a body at a distance x metres from a fixed point is given by F = 3x + 2x2. Determine the work done when the body moves from the position where x = 1 m to x = 4 m. [4] 5. a) Sketch the curve y = 3ex [2] b) Determine the area enclosed by the curve, the x-axis and lines x = 1 and x = 3, giving your answer in terms of e. [3] c) The points P and Q on the curve y = 3ex have the x coordinates 1 and 3. Using your answer to b), or otherwise, determine the area enclosed between the curve and the line PQ [5] 6. a) Sketch a graph of y = x2 + x 6 showing clearly where the graph cuts the coordinate axes. [3] b) Determine the finite area enclosed between the curve and the x-axis. [6] 7. a) Below is shown a sketch of y =  EMBED Equation.3  and 2y + 3x = 39 for x > 0    i) Write down the coordinates of points Q and R [3] ii) Find the coordinates of point S. [5] b) Show that  EMBED Equation.3  dx = (16 5 ln2) [7] c) Find the final area enclosed between y =  EMBED Equation.3 , 2y + 3x = 39 and the x-axis, giving your answer in terms of natural logarithms [4] 8. a) Determine the coordinates of the points of intersection of the parabolas y2 = 3x and x2 = 3y [5] b) Find the area enclosed between these curves [8] 9. a) Sketch the curves y = x2 + 3 and y = 6 2x showing the intersection points of the curves with each other and the coordinate axes [7] b) Determine the area enclosed between the curves [6] 10. a) Express (x + 3) (x + 1) (x 2) in the form x3 + Ax2 + Bx + C [3] b) Determine  EMBED Equation.3  [3] c) Sketch the graph of y = x3 + 2x2 5x 6 [3] d) Determine the area enclosed by the curve and the x-axis. [5] 11. The diagram below shows the curve y2 = (4x)3 and points P(4,64) and Q(1,-8) on this curve  a) Find the equation of the line L through P and Q [3] b) Find the coordinates of the point R at which L crosses the x-axis [2] c) Find the finite area enclosed between the curve y2 = (4x)3 and the line L [16] 12. a) Sketch the curve y = 15 2x x2 and the line 7x + y = 19 on the same diagram, showing the intersection points of each with the coordinate axes. [11] b) Find the finite area enclosed between the curve and the line [8] 13. a) Sketch, on the same graph, the curves y = 2ex and y = e-x + 1, stating for each the coordinates of any intersections with the coordinate axes and the equation of its asymptote. [5] b) Show that the curves only intersect when x = 0 [5] c) Find the finite area enclosed between the two curves, the lines x = -1 and x = 1, and the x-axis, giving your answer in terms of e. [7] 14.a) The gradient of the curve C is given by  EMBED Equation.3 = 3x2 + 4x + k, where k is a constant. Given that the points (0,4) and (1,10) are on the curve, find its equation in a form not involving k. [7] b) Show that the tangent to the curve at (1,10) passes through the origin. [5] c) Find the area enclosed between the curve, this tangent and the y-axis. [7] 15. The rate of change of the temperature ( of a bath at time t seconds after it has been run is given by  EMBED Equation.3 where k is a constant. a) Given that, initially, the bath water is at 70oC, show that ( = 20 + 50e-kt [5] b) Given that k = 0.003, find, to 3 significant figures i) the temperature of the bath after 10 minutes [3] ii) the time taken for the bath to cool down to 30oC [3] 16. The curve C has equation  EMBED Equation.3  a) Express the equation of C in the form y = Ax-1 + B + Cx, where A, B and C are constants to be determined [3] b) The sketch below shows the curve C  i) Find the x coordinates of points A and B, giving your answer in the form of  EMBED Equation.3  [7] ii) Point C has y coordinate 13. Find its x coordinate [5] iii) Find the shaded area, giving your answer to 3 significant figures. [6] 17. The curve C has equation y = 2x3. The portion of the curve enclosed between the lines x = 1 and x = 2 is rotated through 360( about the x-axis. Find the volume of the solid generated, giving your answer in terms of (. [6] 18. The curve C has equation y = 2 EMBED Equation.3 + 4 x > 0 a) Show that y2 = 4x + 16 EMBED Equation.3 + 16 [2] b) The portion of the curve C enclosed between the lines x = 9 and x = 4 is rotated through 360( about the x-axis. Find the exact volume generated, giving your answer in terms of (. [6] 19. The curve C has equation y =  EMBED Equation.3 (x a) x > 0 where a is a positive constant. The points P and Q lie on C, and have coordinates a and 4a respectively. Find the area of the region enclosed between C and the line segment PQ. [13] 20. The curve C1 has the equation y = f(x). a) Given that f ((x) =  EMBED Equation.3 , and that f(1) = 0, find f(x). [4] b) Show that f(x) can be expressed in the form f(x) (  EMBED Equation.3  for x > 0 [2] The curve C2 has equation y =  EMBED Equation.3  x > 0 Curves C1 and C2 intersect at points A and B. The x-coordinate of A is a, and the x-coordinate of B is b, where a < b. c) Show that a and b are the solutions of the equation  EMBED Equation.3  [4] d) By setting  EMBED Equation.3 , or otherwise, find the values of a and b, giving your answers in surd form. [7] 21. The curve C passes through the point (a, 2) and its gradient at the point (x, y) is given by  EMBED Equation.3  where a is a positive constant a) Use integration to show that the equation of C is y =  EMBED Equation.3  [4] b) The portion of C enclosed between the lines x = 1 and x = 2 is rotated through 360( about the x-axis. Find the volume generated, giving your answer in terms of ( and a. 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Determine a)  EMBED Equation.3 , giving your answer in terms of natural logarithms [5] b)  EMBED Equation.3  [4] 2. a) Evaluate correct to 2 decimal places  EMBED Equation.3  [6] b) Evaluate  EMBED Equation.3  [6] c) Evaluate  EMBED Equation.3  [5] 3. a) Sketch the graph of y = 2x2 + 1 from x = -2 to x = 3. [3] b) Determine by integration the area enclosed by the curve, the x-axis and the lines x = -1 and x = 2. [4] 4. The work in Joules done by a force F newtons in moving a body from x1 to x2 is given by  EMBED Equation.3  The force acting on a body at a distance x metres from a fixed point is given by F = 3x + 2x2. Determine the work done when the body moves from the position where x = 1 m to x = 4 m. [4] 5. a) Sketch the curve y = 3ex [2] b) Determine the area enclosed by the curve, the x-axis and lines x = 1 and x = 3, giving your answer in terms of e. [3] c) The points P and Q on the curve y = 3ex have the x coordinates 1 and 3. Using your answer to b), or otherwise, determine the area enclosed between the curve and the line PQ [5] 6. a) Sketch a graph of y = x2 + x  6 showing clearly where the graph cuts the coordinate axes. [3] b) Determine the finite area enclosed between the curve and the x-axis. [6] 7. a) Below is shown a sketch of y =  EMBED Equation.3  and 2y + 3x = 39 for x > 0    i) Write down the coordinates of points Q and R [3] ii) Find the coordinates of point S. [5] b) Show that  EMBED Equation.3  dx = (16  5 ln2) [7] c) Find the final area enclosed between y =  EMBED Equation.3 , 2y + 3x = 39 and the x-axis, giving your answer in terms of natural logarithms [4] 8. a) Determine the coordinates of the points of intersection of the parabolas y2 = 3x and x2 = 3y [5] b) Find the area enclosed between these curves [8] 9. a) Sketch the curves y = x2 + 3 and y = 6  2x showing the intersection points of the curves with each other and the coordinate axes [7] b) Determine the area enclosed between the curves [6] 10. a) Express (x + 3) (x + 1) (x  2) in the form x3 + Ax2 + Bx + C [3] b) Determine  EMBED Equation.3  [3] c) Sketch the graph of y = x3 + 2x2  5x  6 [3] d) Determine the area enclosed by the curve and the x-axis. [5] 11. The diagram below shows the curve y2 = (4x)3 and points P(4,64) and Q(1,-8) on this curve  a) Find the equation of the line L through P and Q [3] b) Find the coordinates of the point R at which L crosses the x-axis [2] c) Find the finite area enclosed between the curve y2 = (4x)3 and the line L [16] 12. a) Sketch the curve y = 15  2x  x2 and the line 7x + y = 19 on the same diagram, showing the intersection points of each with the coordinate axes. [11] b) Find the finite area enclosed between the curve and the line [8] 13. a) Sketch, on the same graph, the curves y = 2ex and y = e-x + 1, stating for each the coordinates of any intersections with the coordinate axes and the equation of its asymptote. [5] b) Show that the curves only intersect when x = 0 [5] c) Find the finite area enclosed between the two curves, the lines x = -1 and x = 1, and the x-axis, giving your answer in terms of e. [7] 14.a) The gradient of the curve C is given by  EMBED Equation.3 = 3x2 + 4x + k, where k is a constant. Given that the points (0,4) and (1,10) are on the curve, find its equation in a form not involving k. [7] b) Show that the tangent to the curve at (1,10) passes through the origin. [5] c) Find the area enclosed between the curve, this tangent and the y-axis. [7] 15. The rate of change of the temperature ( of a bath at time t seconds after it has been run is given by  EMBED Equation.3 where k is a constant. a) Given that, initially, the bath water is at 70oC, show that ( = 20 + 50e-kt [5] b) Given that k = 0.003, find, to 3 significant figures i) the temperature of the bath after 10 minutes [3] ii) the time taken for the bath to cool down to 30oC [3] 16. The curve C has equation  EMBED Equation.3  a) Express the equation of C in the form y = Ax-1 + B + Cx, where A, B and C are constants to be determined [3] b) The sketch below shows the curve C  i) Find the x coordinates of points A and B, giving your answer in the form of  EMBED Equation.3  [7] ii) Point C has y coordinate 13. Find its x coordinate [5] iii) Find the shaded area, giving your answer to 3 significant figures. [6] 17. The curve C has equation y = 2x3. The portion of the curve enclosed between the lines x = 1 and x = 2 is rotated through 360( about the x-axis. Find the volume of the solid generated, giving your answer in terms of (. [6] 18. The curve C has equation y = 2 EMBED Equation.3 + 4 x > 0 a) Show that y2 = 4x + 16 EMBED Equation.3 + 16 [2] b) The portion of the curve C enclosed between the lines x = 9 and x = 4 is rotated through 360( about the x-axis. Find the exact volume generated, giving your answer in terms of (. [6] 19. The curve C has equation y =  EMBED Equation.3 (x  a) x > 0 where a is a positive constant. The points P and Q lie on C, and have coordinates a and 4a respectively. Find the area of the region enclosed between C and the line segment PQ. [13] 20. The curve C1 has the equation y = f(x). a) Given that f ((x) =  EMBED Equation.3 , and that f(1) = 0, find f(x). [4] b) Show that f(x) can be expressed in the form f(x) (  EMBED Equation.3  for x > 0 [2] The curve C2 has equation y =  EMBED Equation.3  x > 0 Curves C1 and C2 intersect at points A and B. The x-coordinate of A is a, and the x-coordinate of B is b, where a < b. c) Show that a and b are the solutions of the equation  EMBED Equation.3  [4] d) By setting  EMBED Equation.3 , or otherwise, find the values of a and b, giving your answers in surd form. [7] 21. The curve C passes through the point (a, 2) and its gradient at the point (x, y) is given by  EMBED Equation.3  where a is a positive constant a) Use integration to show that the equation of C is y =  EMBED Equation.3  [4] b) The portion of C enclosed between the lines x = 1 and x = 2 is rotated through 360( about the x-axis. Find the volume generated, giving your answer in terms of ( and a. 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