Extra FP3 past paper - A

Transcription

1 Mark schemes for these "Extra FP3" papers at Extra FP3 past paper - A More FP3 practice papers, with mark schemes, compiled from Solomon and AQA papers, at the end of this booklet This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 8. x y.[] An ellipse has equation =. (a) Sketch the ellipse. (b) Find the value of the eccentricity e. (c) State the coordinates of the foci of the ellipse. () () () [P5 June 00 Qn ] 3. [3] Solve the equation l0 cosh x + sinh x =. Give each answer in the form ln a where a is a rational number. (7) [P5 June 00 Qn 3] 4. [4] I n = x n cosx dx, n 0. (a) Prove that I n = 0 n n(n )I n, n. (b) Find an exact expression for I 6. (5) [P5 June 00 Qn 4] Extra FP3 papers -

2 5. [5] (a) Given that y = arctan 3x, and assuming the derivative of tan x, prove that d y 3 =. dx 9x (b) Show that x arctan3x dx = 9 (4 33). (6) [P5 June 00 Qn 6] 6. [6] Figure y O x The curve C shown in Fig. has equation y = 4x, 0 x. The part of the curve in the first quadrant is rotated through radians about the x-axis. (a) Show that the surface area of the solid generated is given by 4 (b) Find the exact value of this surface area. 0 ( x ) dx. (c) Show also that the length of the curve C, between the points (, ) and (, ), is given by Extra FP3 papers -

3 0 x dx. x (d) Use the substitution x = sinh to show that the exact value of this length is [ + ln( + )]. (6) [P5 June 00 Qn 8] 7. [7] Prove that sinh (i ) = sinh. [P6 June 00 Qn ] [8] A = (a) Verify that is an eigenvector of A and find the corresponding eigenvalue. (b) Show that 9 is another eigenvalue of A and find the corresponding eigenvector. (5) (c) Given that the third eigenvector of A is, write down a matrix P and a diagonal matrix D such that P T AP = D. (5) [P6 June 00 Qn 5] Extra FP3 papers - 3

4 9. [9] The plane Π passes through the points A(,, ), B(4,, ) and C (,, 0). (a) Find a vector equation of the line perpendicular to Π which passes through the point D (,, 3). (b) Find the volume of the tetrahedron ABCD. (c) Obtain the equation of Π in the form r.n = p. The perpendicular from D to the plane Π meets Π at the point E. (d) Find the coordinates of E. (e) Show that DE = The point D is the reflection of D in. (f) Find the coordinates of D. () [P6 June 00 Qn 7] End of paper Extra FP3 papers - 4

5 Extra FP3 past paper - B This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 7.. [0] Find the values of x for which 4 cosh x + sinh x = 8, giving your answer as natural logarithms. (6) [P5 June 003 Qn ]. [] (a) Prove that the derivative of artanh x, < x <, is x (b) Find artanh x dx.. [P5 June 003 Qn ] Extra FP3 papers - 5

6 3. [] Figure y R O Figure shows the cross-section R of an artificial ski slope. The slope is modelled by the curve with equation 0 y, 0 x 5. (4x 9) Given that unit on each axis represents 0 metres, use integration to calculate the area R. Show your method clearly and give your answer to significant figures. (7) 5 x [P5 June 003 Qn 3] 4. [5] I n = 0 x n e x dx and J n = 0 x n e x dx, n 0. (a) Show that, for n, (b) Find a similar reduction formula for J n. 5 (c) Show that J. e I n e ni. n () (d) Show that x n cosh x dx = (I n + J n ). () 0 (e) Hence, or otherwise, evaluate 0 x cosh x dx, giving your answer in terms of e. [P5 June 003 Qn 7] Extra FP3 papers - 6

7 x y 5. [6] The hyperbola C has equation. b a (a) Show that an equation of the normal to C at the point P (a sec t, b tan t) is ax sin t + by = (a + b ) tan t. (6) The normal to C at P cuts the x-axis at the point A and S is a focus of C. Given that the eccentricity of C is 3, and that OA = 3OS, where O is the origin, (b) determine the possible values of t, for 0 t. (8) [P5 June 003 Qn ] 6. [7] Referred to a fixed origin O, the position vectors of three non-collinear points A, B and C are a, b and c respectively. By considering AB AC, prove that the area of ABC can be expressed in the form a b b c c a. (5) [P6 June 003 Qn ] [8] M 6 9 (a) Find the eigenvalues of M. A transformation T: R R is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T. (b) Find a cartesian equation of this line. [P6 June 003 Qn 3] Extra FP3 papers - 7

8 8. [0] The plane Π passes through the P, with position vector i + j k, and is perpendicular to the line L with equation r = 3i k +(i + 5j + 3k). (a) Show that the Cartesian equation of Π is x 5y 3z = 6. The plane Π contains the line L and passes through the point Q, with position vector i + j + k. (b) Find the perpendicular distance of Q from Π. (c) Find the equation of Π in the form r = a + sb + tc. [P6 June 003 Qn 7] End of paper Extra FP3 papers - 8

9 Extra FP3 past paper - C This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 79. Extra FP3 papers - 9

10 . [3] Figure P y Q -ka O ka x A rope is hung from points P and Q on the same horizontal level, as shown in Fig.. The curve formed by the rope is modelled by the equation x y a cosh, a where a and k are positive constants. ka x ka, (a) Prove that the length of the rope is a sinh k. Given that the length of the rope is 8a, (5) (b) find the coordinates of Q, leaving your answer in terms of natural logarithms and surds, where appropriate. [P5 June 003 Qn 5]. [4] The curve C has equation (a) Prove that dy dx y = arcsec e x, x > 0, 0 < y <.. x (e ) (b) Sketch the graph of C. The point A on C has x-coordinate ln. The tangent to C at A intersects the y-axis at the point B. (c) Find the exact value of the y-coordinate of B. [P5 June 003 Qn 6] (5) () Extra FP3 papers - 0

11 3. [9] (a) Show that det A =(u ). 3 A 5 3, u u. () (b) Find the inverse of A. (6) The image of the vector a b c when transformed by the matrix 3 3 is (c) Find the values of a, b and c. [P6 June 003 Qn 6] 4. [3] An ellipse, with equation x y =, has foci S and S. 9 4 (a) Find the coordinates of the foci of the ellipse. (b) Using the focus-directrix property of the ellipse, show that, for any point P on the ellipse, SP + SP = 6. [P5 June 004 Qn 3] Extra FP3 papers -

12 5. [5] Figure Figure shows the curve with parametric equations x = a cos 3, y = a sin 3, 0 <. (a) Find the total length of this curve. (7) The curve is rotated through radians about the x-axis. (b) Find the area of the surface generated. (5) [P5 June 004 Qn 7] 6. [6] The points A, B and C lie on the plane Π and, relative to a fixed origin O, they have position vectors a = 3i j + 4k, b = i + j, c = 5i 3j + 7k respectively. (a) Find AB AC. (b) Find an equation of Π in the form r.n = p. () The point D has position vector 5i + j + 3k. (c) Calculate the volume of the tetrahedron ABCD. [P6 June 004 Qn 3] Extra FP3 papers -

13 7. [7] The matrix M is given by M = 3 a 4 0 b p, c where p, a, b and c are constants and a > 0. Given that MM T = ki for some constant k, find (a) the value of p, (b) the value of k, (c) the values of a, b and c, (d) det M. () () (6) () [P6 June 004 Qn 5] Extra FP3 papers - 3

14 8. [3] Figure y O x Figure shows a sketch of the curve with parametric equations x = a cos 3 t, y = a sin 3 t, 0 t, where a is a positive constant. The curve is rotated through radians about the x-axis. Find the exact value of the area of the curved surface generated. (7) [FP/P5 June 005 Qn 3] End of paper Extra FP3 papers - 4

15 Extra FP3 past paper - D This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 78.. [] Using the definitions of cosh x and sinh x in terms of exponentials, (a) prove that cosh x sinh x =, (b) solve cosech x coth x =, giving your answer in the form k ln a, where k and a are integers. [P5 June 004 Qn ]. [] 4x + 4x + 7 (ax + b) + c, a > 0. (a) Find the values of a, b and c. (b) Find the exact value of x 4x 7 dx. [P5 June 004 Qn ] 3. [4] Given that y = sinh n x cosh x, (a) show that d y = (n ) sinh n x + n sinh n x. dx Extra FP3 papers - 5

16 arsinh The integral I n is defined by I n = 0 sinh n x dx, n 0. (b) Using the result in part (a), or otherwise, show that (c) Hence find the value of I 4. ni n = (n )I n, n () [P5 June 004 Qn 5] 4. [8] The transformation R is represented by the matrix A, where 3 A. 3 (a) Find the eigenvectors of A. (b) Find an orthogonal matrix P and a diagonal matrix D such that A = PDP. (5) (5) (c) Hence describe the transformation R as a combination of geometrical transformations, stating clearly their order. [P6 June 004 Qn 6] 5. [3] I n = x n e x dx, n 0. (a) Prove that, for n, I n = (x n e x ni n ). (b) Find, in terms of e, the exact value of x 0 e x dx. (5) [FP/P5 June 005 Qn 4] Extra FP3 papers - 6

17 6. [35] (a) (i) Explain why, for any two vectors a and b, a.b a = 0. () (ii) Given vectors a, b and c such that a b = a c, where a 0 and b c, show that b c = a, where is a scalar. () (b) A, B and C are matrices. (i) Given that AB = AC, and that A is not singular, prove that B = C. () (ii) Given that AB = AC, where A = whose elements are all non-zero. 3 6 and B = 0 5, find a matrix C [FP3/P6 June 005 Qn ] x 7. [39] The hyperbola H has equation 6 y 4 =. Find (a) the value of the eccentricity of H, (b) the distance between the foci of H. () () x The ellipse E has equation 6 + y 4 =. (c) Sketch H and E on the same diagram, showing the coordinates of the points where each curve crosses the axes. [FP/P5 January 006 Qn ] 8. [40] A curve is defined by where t is a parameter. x = t + sin t, y = cos t, Find the length of the curve from t = 0 to t =, giving your answer in surd form. (7) [FP/P5 January 006 Qn 3] Extra FP3 papers - 7

18 9. [7] M = 0 p 3 p q, where p and q are constants. Given that is an eigenvector of M, (a) show that q = 4p. Given also that λ = 5 is an eigenvalue of M, and p < 0 and q < 0, find (b) the values of p and q, (c) an eigenvector corresponding to the eigenvalue λ = 5. [FP3 June 008 Qn ] End of paper Extra FP3 papers - 8

19 Extra FP3 past paper - E This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 73. x. [9] (a) Find dx. ( 4x ) (b) Find, to 3 decimal places, the value of x ( 4x ) dx. (5) () (Total 7 marks) [FP/P5 June 005 Qn ]. [30] (a) Show that, for x = ln k, where k is a positive constant, 4 k cosh x =. k Given that f(x) = px tanh x, where p is a constant, (b) find the value of p for which f(x) has a stationary value at x = ln, giving your answer as an exact fraction. (Total 7 marks) [FP/P5 June 005 Qn ] Extra FP3 papers - 9

20 3. [33] Figure y R O x Figure shows a sketch of the curve with equation y = x arcosh x, x. The region R, as shown shaded in Figure, is bounded by the curve, the x-axis and the line x =. Show that the area of R is 7 ln ( + 3). 4 3 (Total 0 marks) [FP/P5 June 005 Qn 6] 4. [36] The line l has equation and the line l has equation r = i + 6j k + (i + 3k) r = 3i + pj + (i j + k), where p is a constant. The plane Π contains l and l. (a) Find a vector which is normal to Π. (b) Show that an equation for Π is 6x + y 4z = 6. () () Extra FP3 papers - 0

21 (c) Find the value of p. () The plane Π has equation r.(i + j + k) =. (d) Find an equation for the line of intersection of Π and Π, giving your answer in the form (r a) b = 0. (5) [FP3/P6 June 005 Qn 3] 5. [37] A = k (a) Show that det A = 0 4k. (b) Find A. () (6) Given that k = 3 and that 0 is an eigenvector of A, (c) find the corresponding eigenvalue. () Given that the only other distinct eigenvalue of A is 8, (d) find a corresponding eigenvector. [FP3/P6 June 005 Qn 7] Extra FP3 papers -

22 6. [4] Given that (a) show that I n = I n = 4 x n 8n In, n. n 3 0 ( 4 x) dx, n 0, (6) Given that ( 4 x ) dx, (b) use the result in part (a) to find the exact value of 4 0 x ( 4 x) dx. [FP/P5 January 006 Qn 7] 7. [48] The point S, which lies on the positive x-axis, is a focus of the ellipse with x equation + y =. 4 Given that S is also the focus of a parabola P, with vertex at the origin, find (a) a cartesian equation for P, (b) an equation for the directrix of P. () [FP June 006 Qn ] Extra FP3 papers -

23 8. [50] Figure y C O x The curve C, shown in Figure, has parametric equations x = t ln t, y = 4t, t 4. (a) Show that the length of C is 3 + ln 4. (7) The curve is rotated through radians about the x-axis. (b) Find the exact area of the curved surface generated. [FP June 006 Qn 6] End of paper Extra FP3 papers - 3

24 Extra FP3 past paper - F This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 69.. [34] (a) Show that, for 0 < x, ( x ) ln ( x ) = ln x. x (b) Using the definition of cosh x or sech x in terms of exponentials, show that, for 0 < x, (c) Solve the equation arsech x = ln ( x ). x (5) 3 tanh x 4 sech x + = 0, giving exact answers in terms of natural logarithms. (5) (Total 3 marks) [FP/P5 June 005 Qn 8]. [38] Evaluate 4 ( x x 7) dx, giving your answer as an exact logarithm. (5) [FP/P5 January 006 Qn ] Extra FP3 papers - 4

25 3. [4] (a) Using the definition of cosh x in terms of exponentials, prove that 4 cosh 3 x 3 cosh x = cosh 3x. (b) Hence, or otherwise, solve the equation cosh 3x = 5 cosh x, giving your answer as natural logarithms. [FP/P5 January 006 Qn 4] 4. [44] A transformation T : R R is represented by the matrix A =, where k is a constant. Find (a) the two eigenvalues of A, (b) a cartesian equation for each of the two lines passing through the origin which are invariant under T. [*FP3/P6 January 006 Qn 3] 5. [46] The plane passes through the points P(, 3, ), Q(4,, ) and R(3, 0, c), where c is a constant. (a) Find, in terms of c, RP RQ. Given that RP RQ = 3i + dj + k, where d is a constant, (b) find the value of c and show that d = 4, (c) find an equation of in the form r.n = p, where p is a constant. () The point S has position vector i + 5j + 0k. The point S is the image of S under reflection in. (d) Find the position vector of S. (5) [FP3/P6 January 006 Qn 7] Extra FP3 papers - 5

26 6. [5] Figure y R O Figure shows a sketch of part of the curve with equation y = x arsinh x. 3 x The region R, shown shaded in Figure, is bounded by the curve, the x-axis and the line x = 3. Show that the area of R is 9 ln (3 + 0) 9 ( + 70). (0) [FP June 006 Qn 7] x 7. [53] The ellipse E has equation a c, where m > 0 and c > 0. y b + = and the line L has equation y = mx + (a) Show that, if L and E have any points of intersection, the x-coordinates of these points are the roots of the equation (b + a m )x + a mcx + a (c b ) = 0. () Hence, given that L is a tangent to E, (b) show that c = b + a m. () The tangent L meets the negative x-axis at the point A and the positive y-axis at the point B, and O is the origin. Extra FP3 papers - 6

27 (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 x-coordinate of the point of contact of L and E when the area of triangle OAB is a minimum. End of paper [FP June 006 Qn 9] Extra FP3 papers - 7

28 Extra FP3 past paper - G This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 68.. [43] (a) Show that artanh sin = ln ( + ). 4 d y (b) Given that y = artanh (sin x), show that = sec x. dx () (c) Find the exact value of 4 0 sin x artanh (sinx) dx. (5) [FP/P5 January 006 Qn 8]. [45] A = k 0 9 k, where k is a real constant. 0 (a) Find values of k for which A is singular. Given that A is non-singular, (b) find, in terms of k, A. (5) [FP3/P6 January 006 Qn 4] Extra FP3 papers - 8

29 3. [47] Find the values of x for which 5 cosh x sinh x =, giving your answers as natural logarithms. (6) [FP June 006 Qn ] 4. [49] The curve with equation has a maximum turning point A. y = x + tanh 4x, x 0, (a) Find, in exact logarithmic form, the x-coordinate of A. (b) Show that the y-coordinate of A is 4 {3 ln( + 3)}. [FP June 006 Qn 5] 5. [5] I n = x n cosh x dx, n 0. (a) Show that, for n, (b) Hence show that n n In x sinh x nx cosh x n n In. I 4 = f(x) sinh x + g(x) cosh x + C, where f(x) and g(x) are functions of x to be found, and C is an arbitrary constant. (5) (c) Find the exact value of 0 x 4 cosh x dx, giving your answer in terms of e. [FP June 006 Qn 8] Extra FP3 papers - 9

30 6. [54] A = Prove by induction, that for all positive integers n, A n = 0 0 n 0 ( n 3n) n. (5) [FP3 June 006 Qn ] x 7. [58] The ellipse D has equation 5 x 4 + y 9 =. + y 9 = 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 axes. The point S is a focus of D and the point T is a focus of E. (b) Find the length of ST. (5) [FP June 007 Qn ] 8. [59] The curve C has equation y = 4 (x ln x), x > 0. Find the length of C from x = 0.5 to x =, giving your answer in the form a + b ln, where a and b are rational numbers. (7) [FP June 007 Qn 3] 9. [65] Show that d [ln(tanh x)] = cosech x, x > 0. [FP June 008 Qn ] dx End of paper Extra FP3 papers - 30

31 Extra FP3 past paper - H This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 78.. [55] The eigenvalues of the matrix M, where M = 4, are and, where <. (a) Find the value of and the value of. (b) Find M. (c) Verify that the eigenvalues of M are and. () A transformation T : R R is represented by the matrix M. There are two lines, passing through the origin, each of which is mapped onto itself under the transformation T. (d) Find cartesian equations for each of these lines. [FP3 June 006 Qn 5] Extra FP3 papers - 3

32 . [56] The points A, B and C lie on the plane Π and, relative to a fixed origin O, they have position vectors respectively. a = i + 3j k, b = 3i + 3j 4k and c = 5i j k (a) Find (b a) (c a). (b) Find an equation for Π, giving your answer in the form r.n = p. () The plane Π has cartesian equation x + z = 3 and Π and Π intersect in the line l. (c) Find an equation for l, giving your answer in the form (r p) q = 0. The point P is the point on l that is the nearest to the origin O. (d) Find the coordinates of P. [FP3 June 006 Qn 7] 3. [57] Evaluate 3 ( x 4x 5) dx, giving your answer as an exact logarithm. (5) [FP June 007 Qn ] 4. [60] (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that cosh(a B) = cosh A cosh B sinh A sinh B. (b) Hence, or otherwise, given that cosh (x ) = sinh x, show that tanh x = e. e e [FP June 007 Qn 4] Extra FP3 papers - 3

33 5. [6] Given that I n = x n ( 8 x) dx, n 0, (a) show that I n = 4n In, n. 3n 4 (6) (b) Hence find the exact value of x ( x 5)(8 x) dx. (6) [FP June 007 Qn 6] 6. [6] Figure Figure shows part of the curve C with equation y = arsinh (x), x 0. (a) Find the gradient of C at the point where x = 4. The region R, shown shaded in Figure, is bounded by C, the x-axis and the line x = 4. (b) Using the substitution x = sinh, or otherwise, show that the area of R is k ln ( + 5) 5, where k is a constant to be found. (0) [FP June 007 Qn 7] Extra FP3 papers - 33

34 7. [73] Figure Figure shows a pyramid PQRST with base PQRS. The coordinates of P, Q and R are P (, 0, ), Q (,, ) and R (3, 3, ). Find (a) PQ PR (b) a vector equation for the plane containing the face PQRS, giving your answer in the form r. n = d. () The plane Π contains the face PST. The vector equation of Π is r. (i j 5k) = 6. (c) Find cartesian equations of the line through P and S. (d) Hence show that PS is parallel to QR. (5) () Given that PQRS is a parallelogram and that T has coordinates (5,, ), (e) find the volume of the pyramid PQRST. [FP3 June 008 Qn 7] End of paper Extra FP3 papers - 34

35 Extra FP3 past paper - I This paper is compiled from the list of sample questions which Edexcel circulated in 009, when they changed the syllabus. It may not have exactly the same balance as an actual past paper, but it is a good approximation. Each numbered question carries an extra number in square brackets [...] That is the number of the question in Edexcel's 009 list, and the number by which the mark scheme for the question can be located in the mark scheme document. Maximum marks for this paper are 80.. [64] The points A, B and C have position vectors, relative to a fixed origin O, a = i j, b = i + j + 3k, c = i + 3j + k, respectively. The plane Π passes through A, B and C. (a) Find AB AC. (b) Show that a cartesian equation of Π is 3x y + z () The line l has equation (r 5i 5j 3k) (i j k) = 0. The line l and the plane Π intersect at the point T. (c) Find the coordinates of T. (d) Show that A, B and T lie on the same straight line. (5) [FP3 June 007 Qn 7]. [66] Find the values of x for which 8 cosh x 4 sinh x = 3, giving your answers as natural logarithms. (6) [FP June 008 Qn ] Extra FP3 papers - 35

36 3. [67] Show that x ( x 9) dx = 3 ln (7) [FP June 008 Qn 3] 4. [68] The curve C has equation y = arsinh (x 3 ), x 0. The point P on C has x-coordinate. (a) Show that an equation of the tangent to C at P is y = x + ln (3 + ). (5) The tangent to C at the point Q is parallel to the tangent to C at P. (b) Find the x-coordinate of Q, giving your answer to decimal places. (5) [FP June 008 Qn 4] 5. [69] Given that I n = 0 e x sin n x dx, n 0, (a) show that, for n, I n = n( n ) I n n. (8) (b) Find the exact value of I 4. [FP June 008 Qn 5] Extra FP3 papers - 36

37 6. [70] Figure Figure shows the curve C with equation y = cosh x arctan (sinh x), x 0. 0 The shaded region R is bounded by C, the x-axis and the line x =. (a) Find cosh x arctan(sinh x) dx. (b) Hence show that, to significant figures, the area of R is (5) () [FP June 008 Qn 6] Extra FP3 papers - 37

38 7. [7] The hyperbola H has equation x 6 y 9 =. (a) Show that an equation for the normal to H at a point P (4 sec t, 3 tan t) is 4x sin t + 3y = 5 tan t. (6) The point S, which lies on the positive x-axis, is a focus of H. Given that PS is parallel to the y-axis and that the y-coordinate of P is positive, (b) find the values of the coordinates of P. (5) Given that the normal to H at this point P intersects the x-axis at the point R, (c) find the area of triangle PRS. [FP June 008 Qn 7] 0 8. [63] Given that is an eigenvector of the matrix A, where 3 4 p A q (a) find the eigenvalue of A corresponding to, (b) find the value of p and the value of q. () The image of the vector l m n when transformed by A is (c) Using the values of p and q from part (b), find the values of the constants l, m and n. [FP3 June 007 Qn 3] End of paper Extra FP3 papers - 38

39 FP3-type paper compiled from Solomon P6 papers A and B. Total: 73 marks A. With respect to a fixed origin O, the lines l and l are given by the equations l : [r ( 3i + j k)] (i + k) = 0, l : [r (i + j + 4k)] (i j k) = 0. (a) Find (i + k) (i j k). (3 marks) (b) Find the shortest distance between l and l. (3 marks). Prove by induction that, for all n +, n (r + )r! = n(n + )! r= (6 marks) 3. (a) Solve the equation z = 0, giving your answers in the form re iθ where r > 0, π < θ π. (5 marks) (b) Show the points representing your solutions on an Argand diagram. ( marks) A 4. a A =. b The matrix A has eigenvalues λ = and λ = 3. (a) Find the value of a and the value of b. (4 marks) Using your values of a and b, (b) for each eigenvalue, find a corresponding eigenvector, (3 marks) (c) find a matrix P such that P T 0 AP =. 0 3 ( marks) P6 A page Solomon Press

40 5. ( + x d y ) dx d y + 4x d x d y + y = 0 and y =, = at x =. dx Find a series solution of the differential equation in ascending powers of (x + ) up to and including the term in (x + ) 4. ( marks) 6. The variable y satisfies the differential equation d y dx = x d y dx + y with y =. at x = 0. and y = 0.9 at x = 0. dy y y d y y y Use the approximations 0 + y and dx 0 h d with a step length x h 0 of 0. to estimate the values of y at x = 0.3 and x = 0.4 giving your answers to 3 significant figures. ( marks) A 7. M = k 4 3. k (a) Find the determinant of M in terms of k. ( marks) (b) Prove that M is non-singular for all real values of k. ( marks) (c) Given that k = 3, find M, showing each step of your working. (4 marks) When k = 3 the image of the vector a b c when transformed by M is the vector (d) Find the values of a, b and c. (3 marks) Turn over Solomon Press P6 A page 3

41 . Given that x is so small that terms in x 3 and higher powers of x may be neglected, find the values of the constants a and b for which ln(+ ax) + bx = 3x + 3 x. (5 marks). Given that z + 4i =, (a) sketch, in an Argand diagram, the locus of z, ( marks) (b) find the maximum value of arg z in degrees to one decimal place. (3 marks) B 3. (a) Show that cosh ix = cos x where x. ( marks) (b) Hence, or otherwise, solve the equation cosh ix = e ix for 0 x < π. (3 marks) 4. Given that u n+ = 5u n+ 6u n for n, u = and u = 4, prove by induction that u n = n for all integers n, n. (6 marks) P6 B page Solomon Press

42 B 5. M = 0 x 3 4. (a) Given that λ = is an eigenvalue of M, find the value of x. (3 marks) (b) Show that λ = is the only real eigenvalue of M. (6 marks) (c) Find an eigenvector corresponding to the eigenvalue λ =. ( marks) 6. A student is looking at different methods of solving the differential equation d y = xy with y = at x = 0. dx The first method the student tries is to use the approximation step length of 0. to obtain an estimate for y at x = 0.4 dy dx 0 y y0 h twice with a (a) Find the value of the student s estimate for y at x = 0.4 (6 marks) The student then realises that the exact value of y at x = 0.4 can be found using integration. (b) Use integration to find the exact value of y at x = 0.4 (4 marks) (c) Find, correct to decimal place, the percentage error in the estimated value in part (a). ( marks) Turn over Solomon Press P6 B page 3

43 B 7. (a) Given that z = cosθ + i sinθ, show that z z z n n + = cos nθ and z n n = i sin nθ, where n is a positive integer. (3 marks) (b) Given that cos 4 θ + sin 4 θ = A cos 4θ + B, find the values of the constants A and B. (8 marks) (c) Hence find the exact value of π 8 cos 4 θ + sin 4 θ dθ. (3 marks) 0 B 8. The points A, B, C and D have coordinates (3,, ), (, 0, ), (,, 6) and (, 5, 8) respectively, relative to the origin O. (a) Find AB AC. (5 marks) (b) Find the volume of the tetrahedron ABCD. (3 marks) The plane Π contains the points A, B and C. (c) Find a vector equation of Π in the form r.n = p. (3 marks) The perpendicular from D to Π meets the plane at the point E. (d) Find the coordinates of E. (6 marks) END P6 B page 4 Solomon Press

44 P6 Paper A Marking Guide A i j k. (a) 0 = i(0 + ) j( ) + k( 0) = i + 4j k M A (b) d = ( 4 i + j 5 k.i )( + 4 j k ) + 6+ = = 5 8 or 5 6 M A A (6). assume true for n = k (r + )r! = k(k + )! k + k r = (r + )r! = k(k + )! + [(k + ) + ](k + )! r = = (k + )!(k + k + k + ) = (k + )!(k + 3k + ) M = (k + )!(k + )(k + ) = (k + )[(k + ) + ]! A true for n = k + if true for n = k n if n = (r + )r! =! = ; n(n + )! =! = true for n = r = by induction true for n + M A B A (6) 3. (a) z 3 = 7 (re iθ ) 3 = 7e iπ M r 3 = 7 so r = 3 A 3θ = nπ + π M n =, 0, gives θ =, π 3, π A i π i π π 3 3 z = 3e 3, z = 3e, z 3 = 3e iπ A (b) Im(z) α z z 3 α O α 3 Re(z) z α = π 3 B (7) A 4. (a) λ a = 0 ( λ)(b λ) a = 0 M A b λ λ = gives 4b a + 8 = 0 λ = 3 gives b a + 3 = 0 M solve simul. giving a =, b = A 4 x 0 (b) λ =, = y 0, 4 x+ y = 0 eigenvector k y = x x 0 λ = 3, = 4 y 0, x+ y = 0 eigenvector k x = y M A A (c) P = 5 M A (9) P6 A MARKS page Solomon Press

45 A 7. (a) det M = (8 3k) (k + 3) + (k + 4) = k 8k + 7 M A (b) det M = (k 4) = (k 4) + M (k 4) 0 det M > 0 M non-singular for all real k A (c) k = 3, M = 3 4 3, det M = 3 B 9 3 matrix of cofactors: M A M = A (d) a 0 b = M 3 = c 5 3 = = 0 4 a =, b = 0, c = M A () 8. (a) w(iz ) = z + ; iwz w = z + M w + z(iw ) = w + z = iw A z = w + = iw M w + = i(w + i) = i w + i = w + i perp. bisector of and i u = v M A x + (b) Im z = 0 y = 0 so u + iv = ix M (u + iv)( + ix) = x + u vx + i(ux v) = x + M u u vx = x + x = ; ux v = 0 x = v + v u A u = v ; + v u u u = v + v M giving (u + ) + (v + ) = circle, centre i, radius A A (c) Im(z) L C O Re(z) B3 Total (75) P6 A MARKS page 4 Solomon Press

46 P6 Paper B Marking Guide. ln ( + ax) ( + bx) = (ax a x + )( bx + ) B = ax abx a x +... M A ax + ( ab a )x = 3x + 3 x a = 3 and ab a = 3 M giving 3b 9 = 3 so a = 3, b = A (5). (a) Im(z) 4 circle centre + 4i, radius B - O Re(z) (b) Im(z) 4 α - O Re(z) tan α = 4, α = M max. arg z = 90 + α = 8. (dp) M A (5) B 3. (a) cosh ix = (eix + e ix ) = [cos x + isin x + cos ( x) + isin ( x)] M [ cos x + isin x + cos x isin x] = cos x A = (b) cosh ix = e ix cos x = cos x + isin x M sin x = 0 giving x = 0, π M A (5) 4. assume true for n = k and n = k + u k = k, u k+ = k+ M u k+ = 5( k+ ) 6( k ) M = 0( k ) 6( k ) = 4( k ) = k+ M A true for n = k + if true for n = k and n = k + if n =, u = = ; if n =, u = = 4 true for n = and n = B by induction true for integer n, n A (6) P6 B MARKS page Solomon Press

47 B 5. (a) λ =, 0 4 = 0 M x 3 0 (0 + ) (0 + 4x) (0 x) = 0 M 4 8x + x = 0 so x = 4 A (b) λ 0 λ 4 = 0 M 4 3 λ ( λ)[( λ)( λ) + ] (0 + 6) [0 4( λ)] = 0 A ( + λ)( λ) + λ λ = 0 M ( + λ)( λ) 6λ 6 = 0 ( + λ)( λ) + 6( + λ) = 0 ( + λ)[( λ) + 6] = 0 A λ = ; or ( λ) = 6, not poss. for real λ M λ = is only real eigenvalue A (c) x y = z 0 x+ y z = 0 x+ 4z z = 0 y 4z = 0 x+ 3z = 0 y = z x = 3z eigenvector 3 k 4 M A () y 6. (a) y0 x 0 y 0 y 0.x 0 y 0 + y 0 0. x 0 = 0., x = 0.3, y 0 = y.0 y 0.x y + y x = 0.3, x = 0.4, y =.0 y.0506 M A M A M A (b) y y dy = 0.4 x dx M 0. y [ ln y ] = [ x ] A ln y ln = giving y = e 0.06 M A (c) % error = 0.06 e e 00% =.% (dp) M A () Solomon Press P6 B MARKS page 3

48 B 7. (a) z n + n z z n n z = cos nθ + isin nθ + cos ( nθ) + isin ( nθ) M = cos nθ + isin nθ + cos nθ isin nθ = cos nθ A = cos nθ + isin nθ cos nθ + isin nθ = isin nθ A (b) (z + z )4 = z 4 + 4z 3 + z 6z z + 4z z z (cos θ) 4 = cos 4θ + 4(cos θ) + 6 6cos 4 θ = cos 4θ + 8cos θ + 6 (z z )4 = z 4 4z z z (isin θ) 4 = cos 4θ 8cos θ + 6 6sin 4 θ = cos 4θ 8cos θ + 6 6(cos 4 θ + sin 4 θ) = 4cos 4θ + cos 4 θ + sin 4 θ = 4 cos 4θ so A = 4, B = 3 4 M M A A M A M A (c) I = π cos 4θ dθ = [ 6 sin 4θ θ ] π 8 0 M A = π A B 8. (a) uuur uuur AB = 5i + j 3k, AC = i + 3j + 4k M A i j k uuur uuur AB AC = = i(4 + 9) j( 0 6) + k( 5 + ) = 3i + 6j 3k M A (b) uuur AD = 4i 4j + 6k B volume = 6 (3i + 6j 3k).( 4i 4j + 6k) M = = 39 units3 A (c) n = i + j k B r.n = (3i j + k).(i + j k) = 3 = M eqn. is r.(i + j k) = A (d) line through DE has eqn. r = i 5j + 8k + λ(i + j k) M A at intersection [( + λ)i + ( 5 + λ)j + (8 λ)k].(i + j k) = M + λ 0 + 4λ 8 + λ = giving λ = 3 M A E is (,, 5) A (7) Total (75) P6 B MARKS page 4 Solomon Press

49 FP3-type paper compiled from Solomon papers C, D, E for P6. 7 marks. C D

50 D E

51 E E

52 MARK SCHEME C D

53 D E

54 E E

55 FP3-type paper from AQA papers. 80 marks AQA FP January 006 AQA FP June 006

56 AQA FP February 007 AQA FP January 03

57 AQA FP June 008 AQA FP January 009

58 MARK SCHEME Jan 006

59 June 006

60 February 007

61 January 03

62 June 008

63 June 006

64 January 009