Edexcel past paper questions. Core Mathematics 4. Parametric Equations

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1 Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran kvkumaran@gmail.com C4 Maths Parametric equations Page 1

2 Co-ordinate Geometry A parametric equation of a curve is one which does not give the relationship between x and y directly but rather uses a third variable, typically t, to do so. The third variable is known as the parameter. A simple example of a pair of parametric equations: x = 5t + 3 y = t 2 + 2t Converting to Cartesian You need to be able to find the Cartesian equation of the curve from parametric equations, that is the equation that relates x and y directly. To do this you need to eliminate the parameter. The easiest way to do this is to rearrange on parametric equation to get the parameter as the subject and then substitute this into the other equation. A circle with an origin (a, b) has the parametric equation: x = a + rcosθ y = b + rsinθ You can use the result sin 2 θ + cos 2 θ = 1 to derive these. As before, θ is the parameter instead of t in the equations. You need to be able to recognise these as parametric equations of circles in the exam. C4 Maths Parametric equations Page 2

3 C4 Maths Parametric equations Page 3

4 Example The curve C is described by the parametric equations x = 5cost, y = cos2t, 0 t π a) Find a Cartesian equation for the curve C. b) Draw a sketch of the curve C. a) From C3 again you should remember that: Therefore: cos2t cos 2 t sin 2 t 2cos 2 t 1 y = 2cos 2 t 1 If x = 5cost then cost = x 5 So finally y = 2 2x 25-1 b) This is simply a quadratic that is symmetrical about the y axis and intercepts with the y axis at C4 Maths Parametric equations Page 4

5 Example C4 Maths Parametric equations Page 5

6 The parametric equations of a curve are x = 14 sin t, y = 14t cos t π dy where 0 < t <. Find in terms of t, and hence show that the gradient of the 2 dx curve is zero where tan t 1 t dy Since the curve is given parametrically we can use the chain rule to find dx dy dy dt dx dt dx So by differentiating the parametric: x = 14 sin t dx dy = 14 cos t dt y = 14t cos t (a Product) = 14 cos 14t sin t dt dy dy dt dx dt dx dy dx 14 cost - 14t sin t 1 ttant 14cost When the gradient is zero 1 ttan t 0 tan t 1 t This equation could be solved by iterative methods (C3). C4 Maths Parametric equations Page 6

7 Example 4 A curve is given by the parametric equations x = 7 sin 3 t, y = 6 cos 2t, 0 < t < 4 Show that dx dy By chain rule 7 sin t 8 dx dx dt dy dt dy dx 21sin 2 tcost dt dy 12sin2t don't forget the 2. dt By C3 trig identities sin 2t = 2 sin t cos t 2 dx dx dt 21sin t cos t dy dt dy 24sin tcost dx 7 sin t dy 8 The final example in this section deals with tangents and normals to curves. Example 5 The curve C is described by the parametric equations x = tan t y = sin 2t t 2 2 π a) Find the gradient of the curve at the point P where t= 3 b) Find the equation of the normal to the curve at P. a) Find the gradient of the curve at the point P where t= 3 C4 Maths Parametric equations Page 7

8 Using chain rule: dx dy sec 2 t 2cos2t dt dt dx dx dt dy dt dy Let t= 3 π dy 2cos 2t 2cos 2 tcos2t 2 dx sec t 2 Grad = 2 cos cos b) Find the equation of the normal to the curve at P. We are asked for the equation of the normal therefore the gradient will be 4 (why?). Using t= 3 the x and y coordinates are 3 and 23 respectively. Using y = mx + c 3 = c 2 c = Therefore the equation of the normal is y 4x Differentiating a x C4 Maths Parametric equations Page 8

9 This function describes growth and decay, and its derivative gives a measure of the rate of change of this growth/decay. Since y = a x, taking logs of both sides gives ln y = ln a x = x ln a. Using implicit differentiation to differentiate ln y: 1 dy = ln a y dx dy dx = y ln a = ax ln a This result needs to be learn, and is not given in the formula sheet. C4 Parametric differentiation past paper questions C4 Maths Parametric equations Page 9

10 1. A curve has parametric equations x = 2 cot t, y = 2 sin 2 t, 0 < t 2 (a) Find an expression for y dx d in terms of the parameter t. (4) (b) Find an equation of the tangent to the curve at the point where t = 4 (4) (c) Find a cartesian equation of the curve in the form y = f(x). State the domain on which the curve is defined. (4) (C4 June 2005, Q6.) 2. Figure 2 y O x The curve shown in Figure 2 has parametric equations x = sin t, y = sin t, 6 < t <. 2 2 (a) Find an equation of the tangent to the curve at the point where t = 6 (6) (b) Show that a cartesian equation of the curve is C4 Maths Parametric equations Page 10

11 y = 3. A curve has parametric equations 3 1 x + (1 x 2 ), 1 < x < (3) (C4 June 2006, Q4.) x = 7 cos t cos 7t, y = 7 sin t sin 7t, < t <. 8 3 (a) Find an expression for dy dx in terms of t. You need not simplify your answer. (3) (b) Find an equation of the normal to the curve at the point where t = 6 Give your answer in its simplest exact form. 4. A curve has parametric equations (6) (C4 Jan 2007, Q3.) x = tan 2 t, y = sin t, 0 < t < 2 (a) Find an expression for dy dx in terms of t. You need not simplify your answer. (3) (b) Find an equation of the tangent to the curve at the point where t = 4 Give your answer in the form y = ax + b, where a and b are constants to be determined. (5) (c) Find a cartesian equation of the curve in the form y 2 = f(x). (4) 5. (C4 June 2007, Q6.) C4 Maths Parametric equations Page 11

12 Figure 3 The curve C shown in Figure 3 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) (C4 Jan 2009, Q7.) C4 Maths Parametric equations Page 12

13 6. Figure 2 Figure 2 shows a sketch of the curve with parametric equations x = 2 cos 2t, y = 6 sin t, 0 t 2 (a) Find the gradient of the curve at the point where t = 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. (c) Write down the range of f(x). (4) (2) (C4 June 2009, Q5.) C4 Maths Parametric equations Page 13

14 7. A curve C has parametric equations x = sin 2 t, y = 2 tan t, 0 t < 2 (a) Find y dx d in terms of t. (4) The tangent to C at the point where t = 3 cuts the x-axis at the point P. (b) Find the x-coordinate of P. (6) (C4 June 2010, Q4.) 8. The curve C has parametric equations Find x = ln t, y = t 2 2, t > 0. (a) An equation of the normal to C at the point where t = 3, (b) A Cartesian equation of C. (6) (3) (C4 Jan 2011, Q6.) C4 Maths Parametric equations Page 14

15 9. Figure 3 Figure 3 shows part of the curve C with parametric equations x = tan, y = sin, 0 < 2 The point P lies on C and has coordinates 1 3, 3 2 (a) Find the value of at the point P. (2) The line l is a normal to C at P. The normal cuts the x-axis at the point Q. (b) Show that Q has coordinates (k 3, 0), giving the value of the constant k. (6) (C4 June 2011, Q7.) C4 Maths Parametric equations Page 15

16 10. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = 4 sin (a) Find an expression for 6 y dx t, y = 3 cos 2t, 0 t < 2. d in terms of t. (3) (b) Find the coordinates of all the points on C where y dx d = 0. (5) (C4 Jan 2012, Q5.) C4 Maths Parametric equations Page 16

17 11. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = 3 sin 2t, y = 4 cos 2 t, 0 t. (a) Show that d y dx = k 3 tan 2t, where k is a constant to be determined. (5) (b) Find an equation of the tangent to C at the point where t = 3 Give your answer in the form y = ax + b, where a and b are constants. (c) Find a Cartesian equation of C. (4) (3) (C4 June 2012, Q6.) C4 Maths Parametric equations Page 17

18 12. Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations x = t, 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). (b) Find the x-coordinate of the point B. (c) Find an equation of the normal to C at the point A. (2) (2) (5) (C4 Jan 2013, part of Q5.) C4 Maths Parametric equations Page 18

19 13. A curve C has parametric equations x = 2sin t, y = 1 cos 2t, t 2 2 (a) Find d y dx at the point where t = 6 (b) Find a cartesian equation for C in the form (4) stating the value of the constant k. y = f(x), k x k, (3) (c) Write down the range of f(x). (2) (C4 June 2013, Q4) 14. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations 3 x 27sec t, y 3tan t, 0 t 3 C4 Maths Parametric equations Page 19

20 (a) Find the gradient of the curve C at the point where t = 6 (4) (b) Show that the cartesian equation of C may be written in the form y stating values of a and b ( x 9), a x b (3) (C4 June 2013_R, part of Q7) 15. Figure 3 Figure 3 shows a sketch of the curve C with parametric equations x 4cos t, y = 2sin t, 0 t 2π 6 (a) Show that (b) Show that a cartesian equation of C is x + y = 2 3 cos t (3) (x + y) 2 + ay 2 = b where a and b are integers to be determined. (2) (C4 June 2014, Q5) C4 Maths Parametric equations Page 20

21 16. Figure 3 The curve shown in Figure 3 has parametric equations x = t 4 sin t, y = 1 2 cos t, 2 2 t 3 3 The point A, with coordinates (k, 1), lies on the curve. Given that k > 0 (a) find the exact value of k, (b) find the gradient of the curve at the point A. There is one point on the curve where the gradient is equal to 1 2 (2) (4) (c) Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal places. [Solutions based entirely on graphical or numerical methods are not acceptable.] (6) (C4 June 2014_R, Q8) C4 Maths Parametric equations Page 21

22 17. A curve C has parametric equations x = 4t + 3, y = 4t , t 0. 2t (a) Find the value of simplest form. dy dx at the point on C where t = 2, giving your answer as a fraction in its (3) (b) Show that the Cartesian equation of the curve C can be written in the form y = x 2 ax b, x 3, x 3 where a and b are integers to be determined. (3) (C4 June 2015, Q5) C4 Maths Parametric equations Page 22

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