Document Sample

ADDITIONAL MATHEMATICS SPM 2012 PAPER 1 ( PART ONE ) LAST KOPEK REVISION TRY TO ANSWER THESE QUESTIONS AND CHECK YOUR ANSWERS FROM THE FOLLOWING SCHEME OF ANSWER AND MARKING PAPER 1 1. Diagram shows part of the relation f(x) State f(x) f(x) = (x – 2)2 − 3 (a) the type of the relation, x (b) the range. . (2, -3) Answer: (a) Many-to-one √1 (b) f(x) ≥ -3 √ 1 2. Given the function f(x) = 3 – x, g(x) = px2 – q and gf(x) = 3x2 – 18x + 5. Find (a) gf(- 2), (b) the value of p and q. Answer: (a) f (-2) = 3 – (−2) = 5 √ M1 gf(- 2) = g(5) = 3(5)2 – 18(5) + 5 = −10 √ 2 (b) gf(x) = p(3 – x)2 – q = px2 – 6px + 9p - q √ M1 Compare to 3x2 – 18x + 5 p = 3, 9p – q = 5 9(3) – q = 5 q = 22 √ 2 Both correct 3. A function g is defined as g(x) = 7x – 4. Find (a) g -1, (b) the value of h if g -1(h) = 2 Answer: Using the principle: x+4 √1 cx – b (a) g -1 = ax + b = 7 c a h+4 (b) g -1(h) = =2 √ M1 7 h = 10 √2 4. A quadratic equation with roots p and q is x2 + mx + m = 0 where p, q and m are constants. Express p in terms of q. Answer: (x – p) (x – q) = 0 OR: Use SOR/POR x2 – (p + q) x + pq = 0 √ M1 method: Compare to x2 + mx + m = 0 SOR = p + q − (p + q ) = m and pq = m POR = pq pq = − (p + q) √ M2 pq + p = −q −q p= √3 1+q 5. Find the range of values of x for 4x2 ≥ 3 – 4x Answer: a = 4 > 0 minimum graph 4x2 + 4x – 3 ≥ 0 √ M1 Let 4x2 + 4x – 3 = 0 x (2x – 1) (2x + 3) = 0 -³ 2 ½ √ M2 x=½; x=−³2 ? √ M2 x≤−³2; x≥½ √3 6. Diagram shows the graph of a quadratic function y = f(x). The straight line y = −15 is a tangent of the curve y = f(x). (a) State the equation of the axis of symmetry of the curve. (b) Express f(x) in the form of (x + p) 2 + q, where p and q are constants. Answer: y −2 + 6 √ M1 the mid-point of the two y = f(x) (a) x = points on the curve that 2 intersect at x-axis x −2 0 6 y = −15 x=2 √2 (x + p)² + q x + p = 0 x = −p (axis of symmetry) (b) (x – 2)² − 15 √ 2 q = optimum value (either minimum or q = −15 √ M1 maximum based on the graph) 2 7. Solve the equation: 16x = 8 2−x Answer: √ M1 2 24(x) = √ M1 23(2 − x) 24x = 2 [2 −3(2 − x)] 24x = 21 – 6 + 3x Use the concept of the rules of 4x = – 5 + 3x √ M2 equation of indices x = − 5 √3 8. Solve the equation: 2 logx4 + logx8 = −7 Answer: √ M1 logx 42 + logx 8 = −7 Converting log to indices logx 16(8) = −7 √ M1 OR logx 16(8) = logx x −7 Use the concept of the rules of equation of logarithms. 128 = x −7 √M2 27 = x −7 OR 2 7 = ( 1 /x ) 7 (½ )−7 = x −7 x=½ √3 9. The nth term of an arithmetic progression is given by Tn = 12 – 3n. Find the common difference of the progression. Answer: a = T1 = 12 – 3(1) = 9 √ M1 T2 = 12 – 3(2) = 6 d = T2 − a =6–9 √ M2 = −3 √3 10. Given that 12, 6, 3 … is a geometric progression, find the sum of the first 7 terms after the 3 th. term of the progression Answer: OR: T4 = 12 ( ½ )3 = 12/8 a = 12 S7* = (12/8) [1 – ( ½ )7] r = ½ √ M1 = 2.977 S7* = S10 – S3 12 [1 – ( ½ )10] – 12 [1 – ( ½ )3] = √ M2 1–½ 1–½ = 2.977 √3 S7* = Sum from T4 to T10 or first 7 terms after the 3rd. term k 11. Given 0.471 + 0.000471 + 0.000000472 + … = ------- Find the value of k. 333 Answer: a = 0.471 r = 0.001 √ M1 0.471 k Use the concept of S∞ = = √ M2 the sum of infinity. 1 – 0.001 333 = 157 333 k = 157 √3 12. Diagram shows a circle with center O. The length of minor arc AB is 4.8 cm and the angle of minor sector BOC is 1 rad. Using π = 3.142, find (a) the value of θ, (b) the length, in cm, of the major arc AC. Answer: 4.8 (a) θ = = 0.75 rad. √1 A 6.4 O θ B (b) θAC (major) = 2π – (1 + 0.75) √ M1 = 4.534 rad. C SAC (major) = 6.4 ( 4.534 ) = 29.02 cm. √ 2 dy 13. Given that y = x3 (3x + 1)4, find the value of dx when x = −1. Answer: dy = (3x + 1)4 3x2 + x3 (4)(3x + 1)3 (3) √ M1 dx dy x = −1; = [3(−1) + 1]4 3(−1)2 + (−1)3 (4)[3(−1) + 1]3(3) dx √3 √ M2 = 144

DOCUMENT INFO

Shared By:

Categories:

Tags:
Additional, Mathematics, Paper, 2012, Last, Kopek, Revision

Stats:

views: | 870 |

posted: | 11/19/2012 |

language: | |

pages: | 15 |

OTHER DOCS BY nklye

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.