6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary

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1 6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3 1 (1 4 ) d. () (Total 7 marks). (a) Show that, for = ln k, where k is a positive constant, 4 k 1 cosh =. k Given that f() = p tanh, where p is a constant, (b) find the value of p for which f() has a stationary value at = ln, giving your answer as an eact fraction. (Total 7 marks)

2 3. Figure 1 y O Figure 1 shows a sketch of the curve with parametric equations = a cos 3 t, y = a sin 3 t, t, where a is a positive constant. The curve is rotated through radians about the -ais. Find the eact value of the area of the curved surface generated. (Total 7 marks) 4. I n = n e d, n. (a) Prove that, for n 1, I n = 1 ( n e ni n 1 ). (b) Find, in terms of e, the eact value of 1 e d. (Total 8 marks)

3 5. The point P(ap, ap) lies on the parabola M with equation y = 4a, where a is a positive constant. (a) Show that an equation of the tangent to M at P is py = + ap. The point Q(16ap, 8ap) also lies on M. (b) Write down an equation of the tangent to M at Q. () The tangent at P and the tangent at Q intersect at the point V. (c) Show that, as p varies, the locus of V is a parabola N with equation 4y = 5a. (d) Find the coordinates of the focus of N, and find an equation of the directri of N. () (e) Sketch M and N on the same diagram, labelling each of them. () (Total 13 marks)

4 6. Figure y R O 1 Figure shows a sketch of the curve with equation y = arcosh, 1. The region R, as shown shaded in Figure, is bounded by the curve, the -ais and the line =. Show that the area of R is 7 4 ln ( + 3) 3. (Total 1 marks) 7. The curve C has parametric equations = t + sin t, y = 1 cos t, t <. The arc length s of the curve C is measured from the origin O. Show that (a) s = 4 sin t, (b) an intrinsic equation of C is s = 4 sin. Hence, or otherwise, (c) find the radius of curvature of C at the point for which t = 3. () (Total 1 marks)

5 8. (a) Show that, for < 1, 1 (1 ) 1 (1 ) ln = ln. (b) Using the definition of cosh or sech in terms of eponentials, show that, for < 1, (c) Solve the equation arsech = ln 1 (1 ). 3 tanh 4 sech + 1 =, giving eact answers in terms of natural logarithms. 6675/1 Edecel GCE END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS Pure Mathematics P5 Advanced/Advanced Subsidiary Thursday 1 January 6 Afternoon Time: 1 hour 3 minutes 1. Evaluate 1 4 ( 1 17) d, giving your answer as an eact logarithm.

6 . The hyperbola H has equation 16 y 4 = 1. Find (a) the value of the eccentricity of H, (b) the distance between the foci of H. () () The ellipse E has equation 16 y + 4 = 1. (c) Sketch H and E on the same diagram, showing the coordinates of the points where each curve crosses the aes. 3. A curve is defined by where t is a parameter. = t + sin t, y = 1 cos t, Find the length of the curve from t = to t =, giving your answer in surd form. (7) 4. (a) Using the definition of cosh in terms of eponentials, prove that (b) Hence, or otherwise, solve the equation 4 cosh 3 3 cosh = cosh 3. cosh 3 = 5 cosh, giving your answer as natural logarithms. 5. The curve C has equation y = ln (sec ), <. 3 Taking s = at the point where = 3, find an equation for C in the form s = f(), where s and are intrinsic coordinates. (8)

7 6. The curve C has equation y = cosh 3. (a) Show that the radius of curvature of C may be written as = (9c 9c 1) 3c(3c ), where c = cosh. (b) Find, to significant figures, the radius of curvature of C at the point where = ln. 7. Given that (a) show that I n = I n = 4 n 8n In 1, n 1. n 3 ( 4 ) d, n, Given that ( 4 ) d, (b) use the result in part (a) to find the eact value of 8. (a) Show that artanh sin = ln (1 + ). 4 d y (b) Given that y = artanh (sin ), show that = sec. d 4 ( 4 ) d. () (c) Find the eact value of 4 sin artanh (sin ) d.

8 9. The parabola C has equation y = 4a, where a is a constant. (a) Show that an equation for the normal to C at the point P(ap, ap) is y + p = ap + ap 3. The normals to C at the points P(ap, ap) and Q(aq, aq), p q, meet at the point R. (b) Find, in terms of a, p and q, the coordinates of R. The points P and Q vary such that pq = 3. (c) Find, in the form y = f(), an equation of the locus of R. Edecel GCE Further Pure Mathematics FP Advanced Level Wednesday 1 June 6 Afternoon Time: 1 hour 3 minutes 1. Find the values of for which 5 cosh sinh = 11, giving your answers as natural logarithms.. The point S, which lies on the positive -ais, is a focus of the ellipse with equation Given that S is also the focus of a parabola P, with verte at the origin, find (a) a cartesian equation for P, (b) an equation for the directri of P. 4 + y = 1. (1)

9 3. The radius of curvature of a curve C, at any point on C, is e sin cos, where is the angle between the tangent to C at P and the positive ais, and. Taking s = at =, find an intrinsic equation for C. 4. The curve C has equation y = arctan, y <. Find, in surd form, the value of the radius of curvature of C at the point where = The curve with equation y = + tanh 4,, has a maimum turning point A. (a) Find, in eact logarithmic form, the -coordinate of A. (b) Show that the y-coordinate of A is 41 {3 ln( + 3)}. 6. Figure 1 y C O The curve C, shown in Figure 1, has parametric equations = t ln t, y = 4t, 1 t 4. (a) Show that the length of C is 3 + ln 4. The curve is rotated through radians about the -ais. (b) Find the eact area of the curved surface generated. (7)

10 7. Figure y R O Figure shows a sketch of part of the curve with equation y = arsinh. 3 The region R, shown shaded in Figure, is bounded by the curve, the -ais and the line = 3. Show that the area of R is 9 ln (3 + 1) 91 ( + 71). (1) 8. I n = n cosh d, n. (a) Show that, for n, (b) Hence show that I n = n sinh n n 1 cosh + n(n 1)I n. I 4 = f() sinh + g() cosh + C, where f() and g() are functions of to be found, and C is an arbitrary constant. (c) Find the eact value of 1 4 cosh d, giving your answer in terms of e. N6318A 4

11 9. The ellipse E has equation a and c >. y b + = 1 and the line L has equation y = m + c, where m > (a) Show that, if L and E have any points of intersection, the -coordinates of these points are the roots of the equation (b + a m ) + a mc + a (c b ) =. () Hence, given that L is a tangent to E, (b) show that c = b + a m. () The tangent L meets the negative -ais at the point A and the positive y-ais at the point B, and O is the origin. (c) Find, in terms of m, a and b, the area of triangle OAB. (d) Prove that, as m varies, the minimum area of triangle OAB is ab. (e) Find, in terms of a, the -coordinate of the point of contact of L and E when the area of triangle OAB is a minimum. Edecel GCE Further Pure Mathematics FP Advanced/Advanced Subsidiary Friday June 7 Morning Time: 1 hour 3 minutes Evaluate d, giving your answer as an eact logarithm. 1 y y. The ellipse D has equation + =1 and the ellipse E has equation + = (a) Sketch D and E on the same diagram, showing the coordinates of the points where each curve crosses the aes. The point S is a focus of D and the point T is a focus of E. (b) Find the length of ST.

12 3. The curve C has equation 1 y = > 4 ( 1 n ),. Find the length of C from =.5 to =, giving your answer in the form a + b1n, where a and b are rational numbers. (7) 4. (a) Starting from the definitions of cosh and sinh in terms of eponentials, prove that cosh(a B) = cosh A cosh B sinh A sinh B. (b) Hence, or otherwise, given that cosh( 1) = sinh, show that e + 1 tanh =. e + e 1 5. The curve C has parametric equations = t sin t, y = cos t, t < π. (a) Find, in terms of cos t only, an epression for the radius of curvature of C. (b) Write down the least value of y and hence find the radius of curvature of C at the point where y has this least value. () 6. Given that 8 n 3 In = ( 8 ) d, n, 1 (a) show that I n 4n = I 3n + 4 n 1, n 1. (b) Hence find the eact value of 8 3 ( + 5)( 8 ) d. 1

13 7. y C R O 4 Figure 1 Figure 1 shows part of the curve C with equation y = arsinh,. (a) Find the gradient of C at the point where = 4. The region R, shown shaded in Figure 1, is bounded by C, the -ais and the line = 4. (b) Using the substitution = sinh θ, or otherwise, show that the area of R is k1n + 5 5, where k is a constant to be found. (1) 8. The points P(ap, ap) and Q(aq, aq), p = q, lie on the parabola C with equation y = 4a, where a is a constant. (a) Show that an equation for the chord PQ is (p + q) y = ( + apq). The normals to C at P and Q meet at the point R. (b) Show that the coordinates of R are (a(p + q + pq + ), apq(p + q)). (7) Given that the points P and Q vary such that PQ always passes through the point (5a, ), (c) find, in the form y = f(), an equation for the locus of R.

14 6675/1 Edecel GCE Further Pure Mathematics FP Advanced/Advanced Subsidiary Wednesday 18 June 8 Morning Time: 1 hour 3 minutes 1. Show that d [ ln(tanh ) ] = cosech, >. d. Find the values of for which 8 cosh 4 sinh = 13, giving your answers as natural logarithms. 3. Show that d = 31n ( 9) 3 (7) 4. The curve C has equation The point P on C has -coordinate. y = arsinh ( 3 ),. (a) Show that an equation of the tangent to C at P is y = + ln (3 + ). The tangent to C at the point Q is parallel to the tangent to C at P. (b) Find the -coordinate of Q, giving your answer to decimal places.

15 5. Given that I = n n, e sin d d, n Àπ (a) show that, for n, I n = nn ( 1) I n + 1 n. (8) (b) Find the eact value of I y C R O Figure 1 shows the curve C with equation Figure 1 y = 1 cosh arctan (sinh ),. 1 The shaded region R is bounded by C, the -ais and the line =. (a) Find cosh arctan(sinh ) ) d d.. (b) Hence show that, to significant figures, the area of R is.34 ()

16 7. The hyperbola H has equation y = (a) Show that an equation for the normal to H at a point P (4 sec t, 3 tan t) is 4 sin t + 3y = 5 tan t. The point S, which lies on the positive -ais, is a focus of H. Given that PS is parallel to the y-ais and that the y-coordinate of P is positive, (b) find the values of the coordinates of P. Given that the normal to H at this point P intersects the -ais at the point R, (c) find the area of triangle PRS. 8. The curve C has parametric equations = 3(t + sin t), y = 3(1 cos t), t < π. (a) Show that dy t = tan. d The arc length s of C is measured from the origin O. t (b) Show that s =1sin. (c) Hence write down the intrinsic equation of C in the form s = f (ψ). (1) The point P lies on C and the arc OP of C has length L. The arc OP is rotated through π radians about the -ais. (d) Show that the area of the curved surface generated is given by π L (7)

17 Edecel GCE Further Pure Mathematics FP Advanced/Advanced Subsidiary Friday 19 June 9 Afternoon Time: 1 hour 3 minutes 1. y = (arsinh ) Find the eact value of d y d at = 1, giving your answer in the form a ln b, where a and b are real numbers.. The ellipse E has equation = 1, where a >. a + y 8 The eccentricity of E is 1. (a) Calculate the value of a. () The ellipse E cuts the y-ais at the points D and D'. The foci of E are S and S'. 3. (b) Calculate the area of the quadrilateral SDS'D'. 1 ( ) I = 1 coshd, n. n (a) Prove that, for n, I n = n(n 1) I n n. (b) Find an eact epression for I 4, giving your answer in terms of e. n 4. f () = 15 sinh 17 cosh + 6 The curve with equation y = f () has a stationary point P. (a) Find the eact -coordinate of P, giving your answer in terms of ln. (b) Determine the nature of the stationary point.

18 5. A curve has parametric equations = t 3, y = 3t, t 1. The curve is rotated through π radians about the -ais. ( + 1) 4π Prove that the area of the curved surface generated is. 5 (9) 6. Using the substitution u = cosh θ, find the value of ln 4 dθ, sinhθ( coshθ 1) ln coshθ + 1 giving your answer as an eact fraction. (1) 7. The curve C has cartesian equation y = ln (sin ), < < π. The intrinsic equation of C is s = f (ψ), where s increases as ψ decreases. π (a) Show that ψ =. The point with intrinsic coordinates (, π lies on C. 4 (b) Show that s = ln ( + 1 secψ + tanψ π (c) Find the radius of curvature of C at the point where ψ =. 6. ( ( 8. The parabola C has equation y = 4a, where a is a positive constant. The point P on C has coordinates (ap, ap). (a) Show that an equation of the normal to C at P is y + p = ap + ap 3. The normal to C at P meets the curve again at Q. ( + p (b) Show that the y-coordinate of Q is a. p (c) Show that, as p varies, the least distance from P to Q is 6 3a. ( (7)

19 6668/1 Edecel GCE Further Pure Mathematics FP Advanced/Advanced Subsidiary Thursday 4 June 1 Morning Time: 1 hour 3 minutes 1. (a) Epress 3 (3r 1)(3r+ ) in partial fractions. () (b) Using your answer to part (a) and the method of differences, show that n 3 = r (3r 1)(3r ) = 1 + 3n (3n + ) (c) Evaluate 1 3, giving your answer to 3 significant figures. = (3 1)(3 + ) r 1 r r (). The displacement metres of a particle at time t seconds is given by the differential equation d + + cos = dt When t =, = and d 1 dt =. Find a Taylor series solution for in ascending powers of t, up to and including the term in t (a) Find the set of values of for which + 4 > + 3 (b) Deduce, or otherwise find, the values of for which + 4 > + 3 (1)

20 4. z = 8+ ( 8 3)i (a) Find the modulus of z and the argument of z. Using de Moivre s theorem, (b) find 3 z, () 5. (c) find the values of w such that ab,. θ = π r = 4 w = z, giving your answers in the form a + ib, where S r = sin 3θ O θ = Figure 1 Figure 1 shows the curves given by the polar equations r =, θ, and r = sin 3θ, θ. (a) Find the coordinates of the points where the curves intersect. The region S, between the curves, for which r > and for which r < (1.5 + sin 3θ), is shown shaded in Figure 1. (b) Find, by integration, the area of the shaded region S, giving your answer in the form aπ + b 3, where a and b are simplified fractions. (7)

21 6. A comple number z is represented by the point P in the Argand diagram. (a) Given that z 6 = z, sketch the locus of P. (b) Find the comple numbers z which satisfy both z 6 = z and z 3 4i = 5. 3 The transformation T from the z-plane to the w-plane is given by w =. z (c) Show that T maps z 6 = z onto a circle in the w-plane and give the cartesian equation of this circle. () 7. (a) Show that the transformation z = y 1 transforms the differential equation dy 1 4ytan y d = (I) into the differential equation dz ztan 1 d = (II) (b) Solve the differential equation (II) to find z as a function of. (c) Hence obtain the general solution of the differential equation (I). (1) 8. (a) Find the value of λ for which y = λ sin 5 is a particular integral of the differential equation d y d 5y 3cos 5 + = (b) Using your answer to part (a), find the general solution of the differential equation d y d 5y 3cos 5 + = Given that at =, y = and d y 5 d =, (c) find the particular solution of this differential equation, giving your solution in the form y = f(). (d) Sketch the curve with equation y = f() for π. ()

22 6668/1 Edecel GCE Further Pure Mathematics FP Advanced/Advanced Subsidiary Thursday 3 June 11 Morning Time: 1 hour 3 minutes 1. Find the set of values of for which (7). (a) Show that, where k is a constant to be found. Given that, at and, (b) find a series solution for y in ascending powers of, up to and including the term in. 3. Find the general solution of the differential equation giving your answer in the form. (8)

23 4. Given that, (a) find the values of the constants A, B and C. () (b) Show that () (c) Using the result in part (b) and the method of differences, show that 5. The point P represents the comple number z on an Argand diagram, where The locus of P as z varies is the curve C. (a) Find a cartesian equation of C. (b) Sketch the curve C. () () A transformation T from the z-plane to the w-plane is given by The point Q is mapped by T onto the point R. Given that R lies on the real ais, (c) show that Q lies on C.

24 6. A C R O N Figure 1 The curve C shown in Figure 1 has polar equation At the point A on C, the value of r is. The point N lies on the initial line and AN is perpendicular to the initial line. The finite region R, shown shaded in Figure 1, is bounded by the curve C, the initial line and the line AN. Find the eact area of the shaded region R. (9) 7. (a) Use de Moivre s theorem to show that Hence, given also that (b) find all the solutions of in the interval Give your answers to 3 decimal places.

25 8. The differential equation describes the motion of a particle along the -ais. (a) Find the general solution of this differential equation. (8) (b) Find the particular solution of this differential equation for which, at and. On the graph of the particular solution defined in part (b), the first turning point for is the point A. (c) Find approimate values for the coordinates of A. ()