Edexcel FP1 FP1 Practice Practice Papers A and B Papers A and B PRACTICE PAPER A 1. A = 2 1, B = 4 3 3 1, I = 4 2 1 0. 0 1 (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
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
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
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
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 = 22 + 4i 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) 4. 2 3 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)
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 = 1 2 1 2 C = 1 1, 1. 1 1 2 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. (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 + 15. 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
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) = 0.06415 2.5 < α < 3 A1 f(2.75) = 0.45125 2.5 < 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) = 3 7 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 2 + 1 A1 (b) f(1.2) = 0.072 f (x) = 5.32 B1 α = 1.2 = 1.21353 = 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
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
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
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) =, = 26 660, A1 (13 marks) 4 GCE Further Pure Mathematics FP1 practice paper A mark scheme
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
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 = 0.568 (3sf) A1 (4) (b) f (x) = A1 f(0.55) = 0.1520 f (0.55) = 11.0173 B1 α = 0.55 = 0.564 (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 + 3 + 2b = 0 c = 5 A1 (b) (2 + i) 3 = 2 + 11i 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 = 2 5 3 = 15 A1 2 GCE Further Pure Mathematics FP1 practice paper B mark scheme
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 2 + 15x 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
Question 8. (a) (i) f(k + 1) f(k) = k 3 + 3k 2 + 3k + 1 10k 10 + 15 (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 + 1 2 k + 1 + k 2 k + 1 + 2 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