January 2009 QP – C2 OCR.pdf

ADVANCED SUBSIDIARY GCE
MATHEMATICS
4722
Core Mathematics 2
Candidates answer on the Answer Booklet
OCR Supplied Materials:
• 8 page Answer Booklet
• List of Formulae (MF1)
Other Materials Required:
None
Tuesday 13 January 2009
Morning
Duration: 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
• Use black ink. Pencil may be used for graphs and diagrams only.
• Read each question carefully and make sure that you know what you have to do before starting your answer.
• Answer all the questions.
• Do not write in the bar codes.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• You are reminded of the need for clear presentation in your answers.
• The total number of marks for this paper is 72.
• This document consists of 4 pages. Any blank pages are indicated.
1. Find (i) ∫(x3 + 8x −5) dx, [3] (ii) ∫12√x dx. [3]
2. The diagram shows a sector OAB of a circle, centre O and radius 7 cm. The angle AOB is 140°. (i) Express 140° in radians, giving your answer in an exact form as simply as possible. [2] (ii) Find the perimeter of the segment shaded in the diagram, giving your answer correct to 3 significant figures. [4]
3. A sequence of terms u1, u2, u3, … is defined by un = 24 − 2/3n. (i) Write down the exact values of u1, u2 and u3. [2] (ii) Find the value of k such that uk = 0. [2] (iii) Find ∑20n=1 un. [3]
4. The diagram shows the curve y = x4 + 3 and the line y = 19 which intersect at (−2, 19) and (2, 19). Use integration to find the exact area of the shaded region enclosed by the curve and the line. [7]
5. Some walkers see a tower, T, in the distance and want to know how far away it is. They take a bearing from a point A and then walk for 50 m in a straight line before taking another bearing from a point B. They find that angle TAB is 70° and angle TBA is 107° (see diagram). (i) Find the distance of the tower from A. [2] (ii) They continue walking in the same direction for another 100 m to a point C, so that AC is 150 m. What is the distance of the tower from C? [3] (iii) Find the shortest distance of the walkers from the tower as they walk from A to C. [2]
6. A geometric progression has first term 20 and common ratio 0.9. (i) Find the sum to infinity. [2] (ii) Find the sum of the first 30 terms. [2] (iii) Use logarithms to find the smallest value of p such that the pth term is less than 0.4. [4]
7. In the binomial expansion of (k + ax)4 the coefficient of x2 is 24. (i) Given that a and k are both positive, show that ak = 2. [3] (ii) Given also that the coefficient of x in the expansion is 128, find the values of a and k. [4] (iii) Hence find the coefficient of x3 in the expansion. [2]
8. (a) Given that loga x = p and loga y = q, express the following in terms of p and q. (i) loga(xy) [1] (ii) loga(a2x3/y) [3] (b) (i) Express log10(x2 −10) −log10 x as a single logarithm. [1] (ii) Hence solve the equation log10(x2 −10) −log10 x = 2 log10 3. [5]
9. (i) The polynomial f(x) is defined by f(x) = x3 −x2 −3x + 3. Show that x = 1 is a root of the equation f(x) = 0, and hence find the other two roots. [6] (ii) Hence solve the equation tan3x −tan2x −3 tan x + 3 = 0 for 0 ≤x ≤2π. Give each solution for x in an exact form. [6]