Jan 23, 2006 - i is a root of the equation. Ð2 Ð i. ÐÐÐ. 5 p. Ðz Ð 3zâ where zâ is the conjugate of z. (2 marks). (b) The quadratic equation x2 Ð px Ð q Ð 0.
General Certificate of Education January 2006 Advanced Subsidiary Examination
MATHEMATICS Unit Further Pure 1 Monday 23 January 2006
MFP1
1.30 pm to 3.00 pm
For this paper you must have: * an 8-page answer book * the blue AQA booklet of formulae and statistical tables * an insert for use in Question 6 (enclosed) You may use a graphics calculator.
Time allowed: 1 hour 30 minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MFP1. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. Fill in the boxes at the top of the insert. * *
* * *
Information The maximum mark for this paper is 75. The marks for questions are shown in brackets. * *
Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. *
P80785/Jan06/MFP1 6/6/6/
MFP1
2
Answer all questions.
1
(a) Show that the equation x3 2x
20
has a root between 0.5 and 1.
(2 marks)
(b) Use linear interpolation once to find an estimate of this root. Give your answer to two decimal places. (3 marks)
2
(a) For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value: 9 1 p dx ; (i) (3 marks) 0 x (ii)
9
1 p dx . 0x x
(3 marks)
(b) Explain briefly why the integrals in part (a) are improper integrals.
(1 mark)
3 Find the general solution, in degrees, for the equation sin 4x 10 sin 50
(5 marks)
4 A curve has equation y
6x x
1
(a) Write down the equations of the two asymptotes to the curve.
Jan 25, 2008 - Unless stated otherwise, you may quote formulae, without proof, from the booklet .... (ii) State, giving a reason, whether the roots of the equation.
Jun 16, 2008 - (d) Find a quadratic equation, with integer coefficients, which has roots a b and b a . (2 marks). 2 It is given that z ¼ x Ñ iy, where x and y are ...
Jan 15, 2009 - 2 The complex number 2 Ñ 3i is a root of the quadratic equation ... (a) (i) Write down the equations of the three asymptotes of this curve.
Jun 20, 2007 - (c) Find a quadratic equation with integer coefficients such that the roots of ... (a) Write down the equations of the two asymptotes to the curve.
Jun 1, 2009 - (a) Write down the values of a Ñ b and ab . (2 marks). (b) Find the value of a2 Ñ b2 . (2 marks). (c) Find a quadratic equation which has roots 4a.
Jun 20, 2007 - 2. P93925/Jun07/MFP1. Answer all questions. 1 The matrices A and B are given by ... (a) Find, in terms of x and y, the real and imaginary parts of ... 5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this ...
Jan 26, 2007 - 3 The quadratic equation. 2x. 2 Ñ 4x Ñ 3 ¼ 0 has roots a and b . (a) Write down the values of a Ñ b and ab . (2 marks). (b) Show that a2 Ñ b2 ¼ ...
Jun 2, 2009 - an 8-page answer book ... Write the information required on the front of your answer book. ..... Responses to this question were good, and the.
Jan 2, 2008 - Advice. â¢. Unless stated otherwise, you may quote formulae, without proof, from the booklet. ... (ii) Find a similar expression for sin3y . (1 mark).
Jun 4, 2009 - General Certificate of Education. June 2009. Advanced Level ..... matrices of 3-d transformations, and was mostly handled very well. ..... referred to. Grade boundaries. Grade. A. B. C. D. E. Mark. Max 75. 60. 52. 44. 36. 28.
Jan 3, 2009 - Use black ink or black ball-point pen. Pencil should only be used for drawing. â¢. Write the information required on the front of your answer book.
Jan 27, 2006 - (iii) Calculate the value of y when x Ð 1.2, giving your answer to three ... (c) The circle C1 intersects the curve C2 at the points P and Q. Find, ...
Jun 2, 2007 - The points A and B represent the complex numbers z1 ... (c) Find, in terms of z1 , the complex number representing the point of intersection of ...
Jan 2, 2007 - They expressed sinh x in exponential form and solved the ensuing .... expression for γ which needed to be simplified. .... Sinht Cosh tdt. Cosh t.
Jan 4, 2009 - 1 The line l has equation r ¼ Ñ1 Ñ 4tЮi Ñ ÑÐ2 Ñ 12tЮj Ñ Ñ1 Ð 3tЮk. ..... The discriminant of the quadratic equation is ( 6) 4 1 1 32. 6. 32.
Jan 4, 2009 - 1 The line l has equation r ¼ Ñ1 Ñ 4tЮi Ñ ÑÐ2 Ñ 12tЮj Ñ Ñ1 Ð 3tЮk. ..... The discriminant of the quadratic equation is ( 6) 4 1 1 32. 6. 32.
Jun 3, 2009 - Write the information required on the front of your answer book. The Examining ..... Although it was the best answered question on the paper,.
Jun 4, 2008 - (a) State the equations of the invariant lines of the transformation and .... Solving quadratic to find evals. M1 .... M0 if no inverse matrix is given. 1.
Jan 4, 2007 - 2 (a) Show that рa А bЮ is a factor of the determinant. A ¼ a b c b ю c c ю a a ... 3 The points P, Q and R have position vectors p, q and r respectively relative to an origin O, where p ¼ ... (a) Find the equation of the line of invari
Jun 3, 2008 - 1 The function yÑxЮ satisfies the differential equation dy dx. ¼ fÑx, yЮ where ..... square root (or the second solution of the quadratic equation).
Jun 2, 2006 - (a) Find the six roots of the equation z6 Р1, giving your answers in the form eif, ..... was to substitute β for z in the quadratic equation in z and.
Jun 4, 2008 - General Certificate of Education ...... that 3-d shapes become 2-d ones, or ... Grade boundaries. Grade. A. B. C. D. E. Mark. Max 75. 66. 58. 51.
