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 Formulae Items included with question papers Answer Booklet Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Instructions to Candidates In the boxes on the answer book, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g.. There are 9 questions in this question paper. The total mark for this paper is 75. There are 8 pages in this question paper. Any blank pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. Printer s Log. No. N33681A W850/R6667/57570 2/2/ *N33681A* Turn over This publication may be reproduced only in accordance with Edexcel Limited copyright policy. 2007 Edexcel Limited.
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. (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. (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. (5) (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. (5) (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. 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. (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). (5) (c) Hence calculate the value of 40 ( r+ 1)( r+ 5). r= 10 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS N33681A 5
Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced Subsidiary Practice Paper B Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae Items included with question papers Answer Booklet Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Instructions to Candidates In the boxes on the answer book, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g.. There are 8 questions in this question paper. The total mark for this paper is 75. There are 4 pages in this question paper. Any blank pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. Printer s Log. No. N33682A W850/XXXX/57570 2/ *N33682A* Turn over This publication may be reproduced only in accordance with Edexcel Limited copyright policy. 2007 Edexcel Limited.
1. q 3 A = 2 q 1, 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 (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. (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. (5) (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. (5) (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. (5) (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. (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 (5) 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 (6) (Total 13 marks) TOTAL FOR PAPER: 75 MARKS END
FP1 practice paper C - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 77 marks. 1.[#1] Given that z = 22 + 4i and w z = 6 8i, find (a) w in the form a + bi, where a and b are real, (b) the argument of z, in radians to 2 decimal places. [P4 January 2002 Qn 1] 2.[#2] (a) Prove by induction that ( r 1)( r 1) = 1 6 n (n 1)(2n + 5). (b) Deduce that n(n 1)(2n + 5) is divisible by 6 for all n > 1. n r 1 (5) [P4 January 2002 Qn 3] 3.[#3] f(x) = x 3 + x 3. The equation f(x) = 0 has a root,, between 1 and 2. (a) By considering f (x), show that is the only real root of the equation f(x) = 0. (b) Taking 1.2 as your first approximation to, apply the Newton- Raphson procedure once to f(x) to obtain a second approximation to. Give your answer to 3 significant figures. (c) Prove that your answer to part (b) gives the value of correct to 3 significant figures. 4.[#4] 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, [P4 January 2002 Qn 4] (ii) find the value of b and the value of c. (5) FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers L Version March 2009 [*P4 January 2002 Qn 5]
5.[#5] Prove by using standard results that n r 1 2 6 ( r 1) (n 1)n(2n + 5). [P4 June 2002 Qn 1] 6.[#6] Given that z = 3 + 4i and w = 1 + 7i, (a) find w. The complex numbers z and w are represented by the points A and B on an Argand diagram. (1) (b) Show points A and B on an Argand diagram. (c) Prove that OAB is an isosceles right-angled triangle. (d) Find the exact value of arg z w. (1) (5) [P4 June 2002 Qn 5] 7.[#7] The point P (2p, 2 p ) and the point Q (2q, 2 ), where p q, lie on q the rectangular hyperbola with equation xy = 4. The tangents to the curve at the points P and Q meet at the point R. Show that at the point R, x = 4pq p q and y = 4 p q. (8) [*P5 June 2002 Qn 7] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 2
8.[#8] For n Z + prove that (a) 2 3n + 2 + 5 n + 1 is divisible by 3, (9) (b) ( 2 1 9 4 ) n 1 3n n = ( 9n 3n+1). (7) [P6 June 2002 Qn 6] 9.[#9] f(x) = 2 sin 2x + x 2. The root of the equation f(x) = 0 lies in the interval [2, ]. Using the end points of this interval find, by linear interpolation, an approximation to. [You won't have sine in linear interpolation questions in the real exam, but really having since in this question doesn't make it harder. P4 January 2003 Qn 4] 10.[#10] Given that z = 3 3i express, in the form a + ib, where a and b are real numbers, (a) z 2, (b) z 1. (c) Find the exact value of each of z, z 2 and 1. z The complex numbers z, z 2 and z 1 are represented by the points A, B and C respectively on an Argand diagram. The real number 1 is represented by the point D, and O is the origin. (d) Show the points A, B, C and D on an Argand diagram. (e) Prove that triangle OAB is similar to triangle OCD. END OF PAPER [P4 January 2003 Qn 6] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 3
FP1 practice paper D - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 66 marks. 1.[#11] (a) Using the fact that 3 is the real root of the cubic equation x 3 27 = 0, show that the complex roots of the cubic satisfy the quadratic equation x 2 + 3x + 9 = 0. (b) Hence, or otherwise, find the three cube roots of 27, giving your answers in the form a + ib, where a, b R. (c) Show these roots on an Argand diagram. [#P4 June 2003 Qn 3] 2.[#12.] f(x) = 3 x x 6. (a) Show that f(x) = 0 has a root between x = 1 and x = 2. (b) Starting with the interval (1, 2), use interval bisection three times to find an interval of width 0.125 which contains. [*P4 June 2003 Qn 4] 3.[#13.] z = a 3i, a R. 2 ai (a) Given that a = 4, find z. (b) Show that there is only one value of a for which arg z = 4, and find this value. (6) [P4 June 2003 Qn 5] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 4
4. [#14] f(n) = (2n + 1)7 n 1. Prove by induction that, for all positive integers n, f(n) is divisible by 4. (6) [P6 June 2003 Qn 2] 5. [#15.] Given that z = 2 2i and w = 3 + i, (a) find the modulus and argument of wz 2. (b) Show on an Argand diagram the points A, B and C which represent z, w and wz 2 respectively, and determine the size of angle BOC. [It's unusual these days to be asked to find the size of an angle, but angle BOC is just 2π plus arg wz 2 minus arg w. P4 January 2004 Qn 3] (6) 6. [#16] (a) Show by induction that ( r 1)( r 5) 1 = n(n+ 7)(2n + 7). 6 (b) Hence calculate the value of n r 1 40 r 10 ( r 1)( r 5). [P4 June 2004 Qn 1] 7. [17] f(x) = 2 x + x 4. The equation f(x) = 0 has a root in the interval [1, 2]. Use linear interpolation on the values at the end points of this interval to find an approximation to. [*P4 June 2004 Qn 2] 8.[#18] The complex number z = a + ib, where a and b are real numbers, satisfies the equation z 2 + 16 30i = 0. (a) Show that ab = 15. FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 5
(b) Write down a second equation in a and b and hence find the roots of z 2 + 16 30i = 0. [P4 June 2004 Qn 3] 9.[#19] 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. [P4 June 2004 Qn 5] 10.[#20] Show that the normal to the rectangular hyperbola xy = c 2, at the point P ( ct, c t ), t 0 has equation y = t 2 x + t c ct3. (5) [*P5 June 2004 Qn 8] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 6
FP1 practice paper E - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 70 marks. 1.[#21] Given that z 2 2 2 2i and w = 1 i 3, find (a) z, w (b) arg ( z w ). (c) On an Argand diagram, plot points A, B, C and D representing the complex numbers z, w, ( z w ) and 4, respectively. (d) Show that AOC = DOB. (e) Find the area of triangle AOC. [#FP1/P4 January 2005 Qn 8] 2.[#22] Given that 2 is a root of the equation z 3 + 6z + 20 = 0, (a) find the other two roots of the equation, (b) show, on a single Argand diagram, the three points representing the roots of the equation, (1) (c) prove that these three points are the vertices of a right-angled triangle. [#FP1/P4 June 2005 Qn 2] 3.[#23] f(x) = 1 e x + 3 sin 2x The equation f(x) = 0 has a root in the interval 1.0 < x < 1.4. Starting with the interval (1.0, 1.4), use interval bisection three times to find the value of to one decimal place. [You won't have e x or sine in interval bisection questions this year, but... FP1/P4 June 2005 Qn 4] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 7
4.[#24] z = 4 + 6i. (a) Calculate arg z, giving your answer in radians to 3 decimal places. The complex number w is given by w = constant. Given that w = 20, A, where A is a positive 2 i (b) find w in the form a + ib, where a and b are constants, (c) calculate arg z w. [FP1/P4 June 2005 Qn 5] 5.[#25] The point P(ap 2, 2ap) lies on the parabola M with equation y 2 = 4ax, where a is a positive constant. (a) Show that an equation of the tangent to M at P is py = x + ap 2. The point Q(16ap 2, 8ap) also lies on M. (b) Write down an equation of the tangent to M at Q. [*FP2/P5 June 2005 Qn 5] 6.[#26] The sequence of real numbers u 1, u 2, u 3,... is such that u 1 = 5.2 and u n + 1 = 6 8 u n +3. (b) Prove by induction that u n > 5, for n Z +. [It's an unusual induction question, but there's an outside chance you could see one like it this year. FP3/P6 June 2005 Qn 1] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 8
7.[#27] 1 Prove by using standard results that ( r 1)( r 2) = (n 1)n(n + 4). 3 n r 1 (6) [FP1/P4 January 2006 Qn 1] 8.[#28] Given that z 2i z i = i, where is a positive, real constant, (a) show that z = ( λ 2 +1 ) + i ( λ 2 1 ). (5) Given also that arg z = arctan 2 1, calculate (b) the value of, (c) the value of z 2. [FP1/P4 January 2006 Qn 3] 9.[#29] The temperature C of a room t hours after a heating system has been turned on is given by = t + 26 20e 0.5t, t 0. The heating system switches off when = 20. The time t =, when the heating system switches off, is the solution of the equation 20 = 0, where lies in the interval [1.8, 2]. (a) Using the end points of the interval [1.8, 2], find, by linear interpolation, an approximation to. Give your answer to 2 decimal places. (b) Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. (1) [You won't have e t in questions this year, but... FP1/P4 January 2006 Qn 5] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 9
10.[#30] The parabola C has equation y 2 = 4ax, where a is a constant. (a) Show that an equation for the normal to C at the point P(ap 2, 2ap) is y + px = 2ap + ap 3. The normals to C at the points P(ap 2, 2ap) and Q(aq 2, 2aq), p q, meet at the point R. (b) Find, in terms of a, p and q, the coordinates of R. (5) [*FP2/P5 January 2006 Qn 9] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 10
FP1 practice paper F - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 68 marks. 1.[#31] A transformation T : R 2 R 2 is represented by the matrix A = 4 2 2 1, where k is a constant. Find the image under T of the line with equation y = 2x + 1. [You won't have a question this year quite like this. But you should be able to do it with just the added hint that you're asked to find what happens to ( x 2x+1) when multiplied by A. FP3/P6 January 2006 Qn 3] 2.[#32] Prove by induction that, for n Z + r, r2 = 2{1 + (n 1)2 n }. n r 1 (5) [*FP3/P6 January 2006 Qn 5] 3.[#33] The complex numbers z and w satisfy the simultaneous equations 2z + iw = 1, z w = 3 + 3i. (a) Use algebra to find z, giving your answers in the form a + ib, where a and b are real. (b) Calculate arg z, giving your answer in radians to 2 decimal places. [FP1 June 2006 Qn 1] 4.[#34] f(x) = 0.25x 2 + 4 sin x. (a) Show that the equation f(x) = 0 has a root between x = 0.24 and x = 0.28. FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 11
(b) Starting with the interval [0.24, 0.28], use interval bisection three times to find an interval of width 0.005 which contains. [You won't have sine in questions like this, but... FP1 June 2006 Qn 6] 5.[#35] (a) Find the roots of the equation 2 z 2z 17 0, giving your answers in the form a + ib, where a and b are integers. (b) Show these roots on an Argand diagram. (1) [FP1 January 2007 Qn 1] 6.[#36] The complex numbers z 1 and z 2 are given by where p is an integer. z 5 3i, 1 z 1 pi, 1 (a) Find p. z2, in the form a + ib, where a and b are expressed in terms of z 1 Given that z z 2 arg, 1 4 (b) find the value of p. [FP1 January 2007 Qn 3] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 12
7.[#37] f (x) = x 3 + 8x 19. (a) Show that the equation f(x) = 0 has only one real root. (b) Show that the real root of f(x) = 0 lies between 1 and 2. (c) Obtain an approximation to the real root of f(x) = 0 by performing two applications of the Newton-Raphson procedure to f(x), using x = 2 as the first approximation. Give your answer to 3 decimal places. (d) By considering the change of sign of f(x) over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places. [FP1 June 2007 Qn 4] 8.[#38] z = 3 i. z* is the complex conjugate of z. (a) Show that (b) Find the value of z 1 = i. z 2 23 z z (c) Verify, for z = 3 i, that arg (d) Display z, z* and z z. z z = arg z arg z*. on a single Argand diagram. (e) Find a quadratic equation with roots z and z* in the form ax 2 + bx + c = 0, where a, b and c are real constants to be found. [FP1 June 2007 Qn 6] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 13
9.[#39] The points P(ap 2, 2ap) and Q(aq 2, 2aq), p q, lie on the parabola C with equation y 2 = 4ax, where a is a constant. (a) Show that an equation for the chord PQ is (p + q) y = 2(x + apq). The normals to C at P and Q meet at the point R. (b) Show that the coordinates of R are ( a(p 2 + q 2 + pq + 2), apq(p + q) ). (7) [*FP2 June 2007 Qn 8] 10.[#40] Prove by induction that n for n Z + 2, ( 2r 1) = 1 3 n(2n 1)(2n + 1). r 1 (5) [FP3 June 2007 Qn 5] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 14
FP1 practice paper G - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted, and with two questions addedfrom AQA papers. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 74 marks. 1.[#41] Given that f(n) = 3 4n + 2 4n + 2, (a) show that, for k Z +, f(k + 1) f(k) is divisible by 15, (b) prove that, for n Z +, f (n) is divisible by 5, [*FP3 June 2007 Qn 6] 2.[#42.] Given that x = 2 1 is the real solution of the equation 2x 3 11x 2 + 14x + 10 = 0, find the two complex solutions of this equation. (6) [Watch out! It's 2x 3, not x 3... FP1 January 2008 Qn 2] 3.[#43] f(x) = 3x 2 + x tan ( x 2) 2, < x <. The equation f(x) = 0 has a root in the interval [0.7, 0.8]. Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to. Give your answer to 3 decimal places. [You won't have tan in your questions this year. Still, having tan in it doesn't make it really any harder. FP1 January 2008 Qn 4] 4.[#44] z = 2 + i. (a) Express in the form a + ib (i) 1 z FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 15
(ii) z 2. (b) Show that z 2 z = 5 2. (c) Find arg (z 2 z). (d) Display z and z 2 z on a single Argand diagram. [FP1 January 2008 Qn 6] 5.[#45] (a) Write down the value of the real root of the equation x 3 64 = 0. (1) (b) Find the complex roots of x 3 64 = 0, giving your answers in the form a + ib, where a and b are real. (c) Show the three roots of x 3 64 = 0 on an Argand diagram. [FP1 June 2008 Qn 1] 6.[#46] The complex number z is defined by z = a a 2i i, a R, a > 0. Given that the real part of z is 2 1, find (a) the value of a, (b) the argument of z, giving your answer in radians to 2 decimal places. [FP1 June 2008 Qn 3] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 16
7.[#47] A = ( k 1 k 2 k ), where k is constant. A transformation T : R 2 R 2 is represented by the matrix A. (a) Find the value of k for which the line y = 2x is mapped onto itself under T. (You won't have a question quite like this. But all you have to do is find a k where multiplying an answer where the bottom number is twice the top number) ( x 2x) produces (b) Show that A is non-singular for all values of k. (c) Find A 1 in terms of k. A point P is mapped onto a point Q under T. The point Q has position vector ( 3) 4 relative to an origin O. (This just means that its coordinates relative to O are x=4, y=-3. That's all!) Given that k = 3, (d) find the position vector of P. (i.e. what its x-coordinate and y-coordinate are) [FP3 June 2008 Qn 5] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 17
8.[#48] 9.[#49] (from AQA rather than Edexcel, but will test your matrix skills) FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 18
10.[#50] FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers Version 2 March 2009 19