Geometric Sequences Ans

Transcription

1 IB Questionbank Mathematical Studies 3rd edition Geometric Sequences Ans 0 min 0 marks 1. (a) a 1 8 = 2 a = = a 2 a = 4 (C1) (b) = = (ft) (ft) (C2) (c) = 16.0(3 s. f ) ( = 4095/256) Note: Award for using correct formula and correct substitution, for correct answer ( ). for correct answer to 3 s.f. (ft) (ft) (C3)

2 2. (a) = (C1) (b) Note: Award for correct substitution into correct formula. = (C2) (c) = n n = 33.0 (Accept 33) Note: Award for n or equivalent, for = (C3) 3. (a) For obtaining an equation in r 2, can be implied 28 = 7r 2 r = 2 (C3) (b) For using their value of r in the GP sum formula For obtaining (accept fewer s.f. up to ) (C3) (a) r = (C1) Note: Accept (b) u 1 = 36 3 u 1 = Note: Award for correct substitution in formula for n th term of a GP. Accept equivalent forms. Notes: Accept Follow through from their common ratio found in part (a). If used from part (a) award (ft) for an answer of or irrespective of whether working is shown. (ft) (C2)

3 k (c) = 1 3 Notes: Award for correct substitution in the sum of a GP formula, for equating their sum to Follow through from parts (a) and (b). k 1 3 Sketch of the function y = Indication of point where y = = k = 10 Note: Award for a list of at least 8 correct terms, for the sum of the terms equated to Notes: Follow through from parts (a) and (b). If k is not an integer, do not award final. Accept alternative methods. If and used award (ft) for k = 5. If and used award (A0). (ft) (C3) 5. (a) 0.5 (C1) 2 (b) (i) 4 (ii) 1 (C2)

4 n 16(1 0.5 ) (c) = (1 0.5) Note: Award for correct substitution in the GP formula, for equating their sum to sketch of the function y = 16(1 0.5 n (1 0.5) ) indication of point where y = = n = 9 Note: Award for a list of at least 7 correct terms, for the sum of the terms equated to Note: Follow through from their answer to part (a) but answer mark is lost if n is not a whole number. (ft) (C3) 6. (a) u 1 r 4 = 324 u 1 r = 12 r 3 = 27 r = 3 Note: Award at most (G3) for trial and error. (G3) (b) = or = (ft)(g3) Note: Award for u 1 = 4 if n = 9, or u 1 = 12 if n = 8, for correctly substituted formula. (ft) from their (a). (c) 4 3 k 1 > 2000 k > 6 k = 7 Note: Award for correct substitution in correct formula. Accept an equation. (ft)(g2) Notes: If second line not seen award (A2) for correct answer. (ft) from their (a). Accept a list, must see at least 3 terms including the 6 th and 7 th. Note: If arithmetic sequence formula is used consistently in parts (a), (b) and (c), award (A0)(A0)(M0)(A0) for (a) and (ft) for parts (b) and (c). [10]

5 7. (a) = (using table function of the GDC) (G2) (C2) (b) 100(1.08) 10 = (using table function of the GDC) (G2) (C2) (c) x = 100(1.08) x After 16 years Note: Candidate can use trial and error so not necessary to see the first line to award (A2). 16 years (using table function of the GDC). (G2) (C2) 8. (a) Let the population at the end of 1999 be x = x x x = (b) r = r = 1.05 u n = u 1 r n = u 1 (1.05) 6 u 1 = (or to 3 s.f.) [4] 9. (a) (i) a = $250 d = $200 T 10 = = 2050 Note: Award the marks for the values of a and d if they are correctly substituted into the formula without being explicitly stated.

6 (ii) a = $10 r = 2 T 0 = = Note: Award the marks for the values of a and r if they are correctly substituted into the formula without being explicitly stated. (b) 10 S 10 = ( ) 2 = S 10 = { (10 1) 200} 2 = (c) Option One: $10000 Option Two: $ (2 10 1) Option Three: S 10 = 2 1 = Therefore, Option Two would be best. (R1) 4 [12] 10. (a) u n = 2(0.9) 7 = m 2 Note: Award for substitution into formula, list or suitable diagram. 5 2(1 (0.9) ) (b) S n = = 8.19 m 1 (0.9) Note: Award for substitution into formula, list or suitable diagram. Total distance travelled = = 16.4 m. 4