, B = (b) Use induction to prove that, for all positive integers n, f(n) is divisible by 4. (a) Use differentiation to find f (x).

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1 Edexcel FP1 FP1 Practice Practice Papers A and B Papers A and B PRACTICE PAPER A 1. A = 2 1, B = , I = (a) Show that AB = ci, stating the value of the constant c. (b) Hence, or otherwise, find A 1. (Total 4 marks) 2. f(x) = 5 2x + 3 x The equation f(x) = 0 has a root, α, between 2 and 3. Starting with the interval (2, 3), use interval bisection twice to find an interval of width 0.25 which contains α. 3. f(n) = (2n + 1)7 n 1. (Total 4 marks) (a) Show that f(k + 1) f(k) = (ak + b)7 k, stating the values of the constants a and b. (b) Use induction to prove that, for all positive integers n, f(n) is divisible by 4. (4) (Total 7 marks) 4. f(x) = x 3 + x 3. (a) Use differentiation to find f (x). The equation f(x) = 0 has a root, α, between 1 and 2. (b) Taking 1.2 as your first approximation to α, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α. Give your answer to 3 significant figures. (4) (Total 6 marks) N33681A 2

2 5. Given that 3 + i is a root of the equation f(x) = 0, where f(x) = 2x 3 + ax 2 + bx 10, a, b R, (a) find the other two roots of the equation f(x) = 0, (b) find the value of a and the value of b. (Total 8 marks) 6. (a) Write down the 2 2 matrix which represents an enlargement with centre (0, 0) and scale factor k. (1) (b) Write down the 2 2 matrix which represents a rotation about (0, 0) through 90. (c) Find the 2 2 matrix which represents a rotation about (0, 0) through 90 followed by an enlargement with centre (0, 0) and scale factor 3. The point A has coordinates (a + 2, b) and the point B has coordinates (5a + 2, 2 b). A is transformed onto B by a rotation about (0, 0) through 90 followed by an enlargement with centre (0, 0) and scale factor 3. (d) Find the values of a and b. (Total 10 marks) N33681A 3 Turn over

3 7. Given that z = 1 + 3i and that z w = 2 + 2i, find (a) w in the form a + ib, where a, b R, (b) the argument of w, (c) the exact value for the modulus of w. On an Argand diagram, the point A represents z and the point B represents w. (d) Draw the Argand diagram, showing the points A and B. (e) Find the distance AB, giving your answer as a simplified surd. (Total 11 marks) 8. The parabola C has equation y 2 = 4ax, where a is a constant. The point (3t 2, 6t) is a general point on C. (a) Find the value of a. (1) (b) Show that an equation for the tangent to C at the point (3t 2, 6t) is ty = x + 3t 2. (4) The point Q has coordinates (3q 2, 6q). The tangent to C at the point Q crosses the x-axis at the point R. (c) Find, in terms of q, the coordinates of R. The directrix of C crosses the x-axis at the point D. Given that the distance RD = 12 and q > 1, (d) find the exact value of q. (4) (Total 12 marks) N33681A 4

4 9. (a) Prove by induction that, for all positive integers n, 1 ( 1)(2 1) n 2 r = n n+ n+. r= 1 6 (6) (b) Show that = 6 1 n(n+ 7)(2n + 7). (c) Hence calculate the value of 40 ( r+ 1)( r+ 5). r= 10 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS N33681A 5

5 1. q 3 A = 2 q 1, PRACTICE PAPER B where q is a real constant. (a) Find det A in terms of q. (b) Show that A is non-singular for all values of q. (Total 5 marks) z 2. Given that z = i and = 6 8i, find w z (a) w, (b) w in the form a + bi, where a and b are real, (c) the argument of z, in radians to 2 decimal places. (Total 7 marks) n 1 3. (a) Show that ( r 1)( r+ 2) = ( n 1) n( n+ 4). 3 r = 1 (b) Hence calculate the value of 20 ( r 1)( r+ 2). r = 5 (Total 7 marks) f( x) = x + 5 x The root α of the equation f(x) = 0 lies in the interval [0.5, 0.6]. (a) Using the end points of this interval find, by linear interpolation, an approximation to α, giving your answer to 3 significant figures. (4) (b) Taking 0.55 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 significant figures. (Total 9 marks)

6 5. (a) Given that 2 + i is a root of the equation z 2 + bz + c = 0, where b and c are real constants, (i) write down the other root of the equation, (ii) find the value of b and the value of c. (b) Given that 2 + i is a root of the equation z 3 + mz 2 + nz 5 = 0, where m and n are real constants, find the value of m and the value of n. (Total 10 marks) 6. A, B and C are non-singular 2 2 matrices such that AB = C. (a) Show that B = A 1 C. The triangle T 1 has vertices at the points with coordinates (0, 0), (5, 0) and (0, 3). 1 2 A = C = 1 1, Triangle T 1 is mapped onto triangle T 2 by the transformation given by C. (b) Find det C. (c) Hence, or otherwise, find the area of triangle T 2. (1) Triangle T 1 is mapped onto triangle T 2 by the transformation given by B followed by the transformation given by A. (d) Using part (a) or otherwise, find B. (e) Describe fully the geometrical transformation represented by B. (4) (Total 12 marks)

7 7. (a) Show that the normal to the rectangular hyperbola xy = 4, at the general point t 0 has equation P 2t, 2, t 2 y = t 2 x+ 2 t 3. t The normal to the hyperbola at the point A ( 4, 1) meets the hyperbola again at the point B. (b) Find the coordinates of B. (7) (Total 12 marks) 8. (a) f(n) = n 3 10n Given that f(k + 1) f(k) = ak 2 + bk + c, (i) find the values of a, b and c. (ii) Use induction to prove that, for all positive integers n, f(n) is divisible by 3. (b) Prove by induction that, for n Z +, n r n r2 = 2{ 1+ ( n 1) 2 }. r = 1 (4) (6) (Total 13 marks) TOTAL FOR PAPER: 75 MARKS END

8 Further Pure Mathematics FP1 (6667) Practice paper A mark scheme Question 1. (a) AB = = 10I, c = 10 A1 (b) A 1 = A1 (4 marks) 2. f = f = B1 f(2.5) = < α < 3 A1 f(2.75) = < α < 2.75 A1 (4 marks) 3. (a) f(k + 1) = (2k + 3)7 k+1 1 B1 f(k + 1) f(k) = (2k + 3)7 k+1 1 [(2k + 1)7 k 1] = (12k + 20)7 k a = 12, b = 20 A1 (b) f(1) = = 20 ; divisible by 4 B1 f(k + 1) f(k) = 4 (3k + 5)7 k true for n = k + 1 if true for n = k A1 Conclusion, with no wrong working seen. A1 (4) (7 marks) 4. (a) f (x) = 3x A1 (b) f(1.2) = f (x) = 5.32 B1 α = 1.2 = = 1.21 (3 sf) A1 α = 1.21 (3 sig figs) A1 cso (4) (6 marks) GCE Further Pure Mathematics FP1 practice paper A mark scheme 1

9 Question 5. (a) Second root = 3 i B1 Product of roots = (3 + i)(3 i) = 10 or quadratic factor is x 2 6x + 10 Complete method for third root or linear factor A1 Third root = A1 (b) Use candidate s 3 roots to find cubic with real coefficients (x 2 6x + 10)(2x 1) = 2x 3 13x + 26x 10 Equating coefficients a = 13, b = 26 A1 (8 marks) 6. (a) B1 (1) (b) A1 (c) A1 (d) 3b = 5a + 2, 3a 6 = 2 b Eliminate a or b a = 5.5, b = 8.5 A1 A1 (10 marks) 7. (a) w = (2 + 2i)(1 + i) i A1, A1 (b) arg w = or adds two args e.g. = or 105 or 1.83 radians A1 2 GCE Further Pure Mathematics FP1 practice paper A mark scheme

10 Question (c) ( = 4 ) A1 (d) B Im z O A Re z B1 B1 ft in quadrant other than first (e) (=20), then square root AB = A1 (e) Alternative: w z = = A1 (11 marks) 8. (a) a = 3 B1 (1) (b) y = and attempt sub x = Tangent is ( ) A1 cso (4) (c) Equation of tangent at Q is B1 At R, y = 0 0 = x + 3q 2 x = 3q 2 A1 GCE Further Pure Mathematics FP1 practice paper A mark scheme 3

11 Question (d) Equation of directrix is x = 3 RD = 3q 2 3 3q 2 3 = 12 q 2 = 5 B1 q = A1 (4) (12 marks) 9. (a) If n = 1, = 1, = 1 B1 true for n = 1 A1 = = = = A1 true for n = k + 1 if true for n = k, true for n Z + by induction A1 cso (6) (b) Expand brackets and attempt to use appropriate formulae. A1 = A1 = A1 = ( ) A1 cso (c) Use S(40) S(9) =, = , A1 (13 marks) 4 GCE Further Pure Mathematics FP1 practice paper A mark scheme

12 Further Pure Mathematics FP1 (6667) Practice paper B mark scheme Question 1. (a) det A = q(q 1) + 6 A1 (b) A1 > 0 for all real q A is non-singular ( ) A1 cso (b) Alternative: If A is singular, det A = 0 q 2 q + 6 = 0 b 2 4ac = 1 24 < 0 no real roots A is non-singular A1 A1 (5 marks) 2. (a) = 10 A1 (b) = = 1 + 2i A1 A1 (c) = 0.18 A1 (7 marks) 3. (a) [A1 for 2n, A1 for rest] A1 A1 Use factor n and use common denominator. (e.g.3, 6, 12) ( ) A1 cso Attempt complete factorisation GCE Further Pure Mathematics FP1 practice paper B mark scheme 1

13 Question (b) Use S(20) S(4) = = 3008 A1 (7 marks) 4. (a) f(0.5) = 0.75, f(0.6) = 0.36 B1 B1 = (3sf) A1 (4) (b) f (x) = A1 f(0.55) = f (0.55) = B1 α = 0.55 = (3sf) A1 (9 marks) 5. (a) (i) 2 i B1 (ii) (2 i) 2 + b(2 + i) + c = 0 o.e. Imaginary parts b = 4 B1 Real parts c b = 0 c = 5 A1 (b) (2 + i) 3 = i B1 α = + m(3 + 4i) + n(2 + i) 5 = 0 Real parts 3m + 2n = 3, Imaginary parts 8m + 2n = 22 m = 5, n = 9 A1 A1 (10 marks) 6. (a) A 1 AB = A 1 C B = A 1 C ( ) A1 cso (b) det C = 2 B1 (1) (c) Area of T 2 = det C 5 3 = = 15 A1 2 GCE Further Pure Mathematics FP1 practice paper B mark scheme

14 Question (c) Alternative Area = = 15 A1 (d) det A = 1 A 1 = A1 B = A1 (4) (e) Enlargement centre (0, 0), scale factor B1 B1 (12 marks) 7. (a) y = and attempt substitute x = 2t Gradient of normal = t 2 Equation of normal is y = t 2 (x 2t) y = t 2 x + 2t 3 ( ) A1 cso (b) At A 2t = 4, t = 2 Equation of normal is y = 4x + 15 B1 At B 4x + 15 = A1 4x x 4 = 0 (4x 1)(x + 4) = 0 At B x = A1 y = 16 A1 (7) (12 marks) GCE Further Pure Mathematics FP1 practice paper B mark scheme 3

15 Question 8. (a) (i) f(k + 1) f(k) = k 3 + 3k 2 + 3k k (k 3 10k + 15) = 3k 2 + 3k 9 A2, 1, 0 (b) f(1) = 6 = 3 2 true for n = 1 B1 f(k + 1) f(k) = 3k 2 + 3k 9 = 3(k 2 + k 3) true for n = k + 1 if true for n = k, true for n Z + by induction A1 A1 cso (4) (ii) When n = 1, LHS = 1 1 = 2; RHS = 2{1 + 0} = 2 true for n = 1 B1 A1 = 2 + k 2 k k k 2 k k + 1 = 2(1 + k 2 k + 1 ) = A1 true for n = k + 1 if true for n = k, true for n Z + by induction A1 cso (6) (12 marks) 4 GCE Further Pure Mathematics FP1 practice paper B mark scheme

FP1 practice papers A to G

FP1 practice papers A to G Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced Subsidiary Practice Paper A Time: 1 hour 30 minutes Materials required for examination Mathematical