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1EDEXCEL C4 June 2014 Q5
Figure shows a sketch of the curve C with parametric equations x = 4cos(t + pi/6), y = 2sint, 0 <= t < 2pi. (a) Show that x + y = 2sqrt3 cost. (3) (b) Show that a cartesian equation of C is (x + y)^2 + ay^2 = b, where a and b are integers to be determined. (2)
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2EDEXCEL C4 January 2013 Q5
Figure shows a sketch of part of the curve C with parametric equations x = 1 - t/2, y = 2^t - 1. The curve crosses the y-axis at the point A and crosses the x-axis at the point B. (a) Show that A has coordinates (0, 3). (2) (b) Find the x coordinate of the point B. (2) (c) Find an equation of the normal to C at the point A. (5) The region R is bounded by the curve C, the line x = -1 and the x-axis. (d) Use integration to find the exact area of R. (6)
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3EDEXCEL C4 June 2012 Q6
Figure shows a sketch of the curve C with parametric equations x = sqrt3 sin2t, y = 4cos^2 t, 0 <= t <= pi. (a) Show that dy/dx = k sqrt3 tan2t, where k is a constant to be determined. (5) (b) Find an equation of the tangent to C at the point where t = pi/3. Give your answer in the form y = ax + b. (4) (c) Find a cartesian equation of C. (3)
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4EDEXCEL C4 January 2010 Q7
Figure shows a sketch of the curve C with parametric equations x = 5t^2 - 4, y = t(9 - t^2). The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate at the point B. (3) The region R is enclosed by the loop of the curve. (b) Use integration to find the area of R. (6)
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5EDEXCEL C4 June 2009 Q5
Figure shows a sketch of the curve with parametric equations x = 2cos2t, y = 6sint, 0 <= t <= pi/2. (a) Find the gradient of the curve at the point where t = pi/3. (4) (b) Find a cartesian equation of the curve in the form y = f(x), -k <= x <= k, stating the value of the constant k. (4) (c) Write down the range of f(x). (2)
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6EDEXCEL C4 January 2009 Q7
The curve C has parametric equations x = t^3 - 8t, y = t^2, where t is a parameter. Given that the point A has parameter t = -1, (a) find the coordinates of A. (1) The line l is the tangent to C at A. (b) Show that an equation for l is 2x - 5y - 9 = 0. (5) The line l also intersects the curve at the point B. (c) Find the coordinates of B. (6)
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7EDEXCEL C4 June 2008 Q8
Figure shows the curve C with parametric equations x = 8cost, y = 4sin2t, 0 <= t <= pi/2. The point P lies on C and has coordinates (4, 2sqrt3). (a) Find the value of t at the point P. (2) The line l is a normal to C at P. (b) Show that an equation for l is y = -x sqrt3 + 6sqrt3. (6) The finite region R is enclosed by the curve C, the x-axis and the line x = 4. (c) Show that the area of R is given by the integral from pi/3 to pi/2 of 64 sin^2 t cos t dt. (4) (d) Use this integral to find the area of R, giving your answer in the form a + b sqrt3. (4)
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8EDEXCEL C4 January 2008 Q7
The curve C has parametric equations x = ln(t + 2), y = 1/(t + 1), t > -1. The finite region R between the curve C and the x-axis, bounded by the lines x = ln2 and x = ln4. (a) Show that the area of R is given by the integral from 0 to 2 of 1/((t+1)(t+2)) dt. (4) (b) Hence find an exact value for this area. (6) (c) Find a cartesian equation of the curve C, in the form y = f(x). (4) (d) State the domain of values for x for this curve. (1)
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9Edexcel A Level June 2022 Paper 2 Q16
Figure 6 shows a sketch of the curve C with parametric equations x = 2tant + 1, y = 2sec^2 t + 3, -pi/4 <= t <= pi/3. The line l is the normal to C at the point P where t = pi/4. (a) Using parametric differentiation, show that an equation for l is y = -1/2 x + 17/2. (5) (b) Show that all points on C satisfy the equation y = 1/2 (x - 1)^2 + 5. (2) The straight line with equation y = -1/2 x + k, where k is a constant, intersects C at two distinct points. (c) Find the range of possible values for k. (5)
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10Edexcel A Level June 2019 Paper 2 Q4
The curve C1 with parametric equations x = 10cost, y = 4sqrt2 sint, 0 <= t < 2pi, meets the circle C2 with equation x^2 + y^2 = 66 at four distinct points as shown in Figure 2. Given that one of these points, S, lies in the 4th quadrant, find the Cartesian coordinates of S. (6)
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11Edexcel A Level June 2023 Paper 2 Q9
The curve C has parametric equations x = t^2 + 6t - 16, y = 6ln(t + 3), t > -3. (a) Show that a Cartesian equation for C is y = A ln(x + B), x > -B, where A and B are integers to be found. (3) The curve C cuts the y-axis at the point P. (b) Show that the equation of the tangent to C at P can be written in the form ax + by = c ln5, where a, b and c are integers to be found. (4)
12Edexcel A Level June 2024 Paper 2 Q10
Figure 4 shows a sketch of the curve C with parametric equations x = (t + 3)^2, y = 1 - t^3, -2 <= t <= 1. The point P with coordinates (4, 2) lies on C. (a) Using parametric differentiation, show that the tangent to C at P has equation 3x + 4y = 20. (5) The curve C is used to model the profile of a slide at a water park. Units are in metres, with y being the height of the slide above water level. (b) Find, according to the model, the greatest height of the slide above water level. (1)
